{h0 Amit's Thoughts on Path-Finding and A-Star}
{# Old archive at http://zach.in.tu-clausthal.de/teaching/info_literatur/A_Star/Amit/index.html }
{set title "Amit's A* Pages"}
{set meta-description "Thoughts on pathfinding, including discussions of performance, memory consumption, and path quality."}
{set meta-keywords "pathfinding, A*, a-star, unit movement"}
{# pathfinding = strategy }
{# movement = path following = tactics }
The problem we're trying to solve is to get a game object from the
starting point to a goal. {emph Pathfinding} addresses the problem of
finding a good path from the starting point to the goal---avoiding
obstacles, avoiding enemies, and minimizing costs (fuel, time,
distance, equipment, money, etc.). {emph Movement} addresses the
problem of taking a path and moving along it. It's possible to spend
your efforts on only one of these. At one extreme, a sophisticated
pathfinder coupled with a trivial movement algorithm would find a path
when the object begins to move and the object would follow that path,
oblivious to everything else. At the other extreme, a movement-only
system would not look ahead to find a path (instead, the initial
"path" would be a straight line), but instead take one step at a time,
considering the local environment at every point. Best results are
achieved by using both pathfinding and movement algorithms.
{h1 Pathfinding}
{h2 Introduction to A*}
{filename AStarComparison.html}
Movement for a single object seems easy. Pathfinding is complex. Why
bother with pathfinding? Consider the following situation:
{figure {img 450 318 concave1.png}}
The unit is initially at the bottom of the map and wants to get to the
top. There is nothing in the area it scans (shown in pink) to
indicate that the unit should not move up, so it continues on its way.
Near the top, it detects an obstacle and changes direction. It then
finds its way around the "U"-shaped obstacle, following the red path.
In contrast, a pathfinder would have scanned a larger area (shown in
light blue), but found a shorter path (blue), never sending the unit
into the concave shaped obstacle.
You can however extend a movement algorithm to work around traps like
the one shown above. Either avoid creating concave obstacles, or mark
their convex hulls as dangerous (to be entered only if the goal is
inside):
{figure {img 450 318 concave2.png}}
Pathfinders let you plan ahead rather than waiting until the last
moment to discover there's a problem. There's a tradeoff between
planning with pathfinders and reacting with movement algorithms.
Planning generally is slower but gives better results; movement is
generally faster but can get stuck. If the game world is changing
often, planning ahead is less valuable. I recommend using both:
pathfinding for big picture, slow changing obstacles, and long paths;
and movement for local area, fast changing, and short paths.
{h3 Algorithms}
{id S1}
{emph I have written a {link
https://www.redblobgames.com/pathfinding/a-star/introduction.html newer
version of this one page}, but not the rest of the pages. It has
interactive diagrams and sample code.}
The pathfinding algorithms from computer science textbooks work on
{emph graphs} in the mathematical sense---a set of vertices with edges
connecting them. A tiled game map can be considered a graph with each
tile being a vertex and edges drawn between tiles that are adjacent to
each other:
{figure {img 246 246 ../game-programming/a-star/map-as-graph.png}}
For now, I will assume that we're using {link
http://www-cs-students.stanford.edu/~amitp/game-programming/grids/
two-dimensional grids}. If you haven't worked with graphs before,
{link https://www.redblobgames.com/pathfinding/grids/graphs.html see
this primer}. Later on, I'll discuss {link {ref Polygonal maps} how to
build other kinds of graphs out of your game world}.
Most pathfinding algorithms from AI or Algorithms research are
designed for arbitrary graphs rather than grid-based games. We'd like
to find something that can take advantage of the nature of a game map.
There are some things we consider common sense, but that algorithms
don't understand. We know something about distances: in general, as
two things get farther apart, it will take longer to move from one
to the other, assuming there are no wormholes. We know something about
directions: if your destination is to the east, the best path is more
likely to be found by walking to the east than by walking to the
west. On grids, we know something about symmetry: most of the time,
moving north then east is the same as moving east then north. This
additional information can help us make pathfinding algorithms run
faster.
{h3 Dijkstra's Algorithm and Best-First-Search}
{id S2} Dijkstra's Algorithm works by visiting vertices in the graph starting
with the object's starting point. It then repeatedly examines the
closest not-yet-examined vertex, adding its vertices to the set of
vertices to be examined. It expands outwards from the starting point
until it reaches the goal. Dijkstra's Algorithm is guaranteed to find
a shortest path from the starting point to the goal, as long as none
of the edges have a negative cost. (I write "a shortest path" because
there are often multiple equivalently-short paths.) In the following
diagram, the pink square is the starting point, the blue square is the
goal, and the teal areas show what areas Dijkstra's Algorithm scanned.
The lightest teal areas are those farthest from the starting point,
and thus form the "frontier" of exploration:
{figure {img 526 376 ../game-programming/a-star/dijkstra.png}}
The Greedy Best-First-Search algorithm works in a similar way, except
that it has some estimate (called a {emph heuristic}) of how far from
the goal any vertex is. Instead of selecting the vertex closest to
the starting point, it selects the vertex closest to the goal.
Greedy Best-First-Search is {emph not} guaranteed to find a shortest
path. However, it runs much quicker than Dijkstra's Algorithm because
it uses the heuristic function to guide its way towards the goal very
quickly. For example, if the goal is to the south of the starting
position, Greedy Best-First-Search will tend to focus on paths that lead
southwards. In the following diagram, yellow represents those nodes
with a high heuristic value (high cost to get to the goal) and black
represents nodes with a low heuristic value (low cost to get to the
goal). It shows that Greedy Best-First-Search can find paths very
quickly compared to Dijkstra's Algorithm:
{figure {img 526 376 ../game-programming/a-star/best-first-search.png}}
However, both of these examples illustrate the simplest case---when
the map has no obstacles, and the shortest path really is a straight
line. Let's consider the concave obstacle as described in the
previous section. Dijkstra's Algorithm works harder but is guaranteed
to find a shortest path:
{figure {img 526 376 ../game-programming/a-star/dijkstra-trap.png}}
Greedy Best-First-Search on the other hand does less work but its path
is clearly not as good:
{figure {img 526 376 ../game-programming/a-star/best-first-search-trap.png}}
The trouble is that Greedy Best-First-Search is “greedy” and tries to
move towards the goal even if it's not the right path. Since it only
considers the cost to get to the goal and ignores the cost of the path
so far, it keeps going even if the path it's on has become really
long.
Wouldn't it be nice to combine the best of both? A* was developed in
1968 to combine heuristic approaches like Greedy Best-First-Search and
formal approaches like Dijsktra's Algorithm. It's a little unusual in
that heuristic approaches usually give you an approximate way to solve
problems without guaranteeing that you get the best answer. However,
A* is built on top of the heuristic, and although the heuristic itself
does not give you a guarantee, A* {emph can} guarantee a shortest
path.
{h3 The A* Algorithm}
{id S3} I will be focusing on the {strong {link
http://en.wikipedia.org/wiki/A-star_search_algorithm A* Algorithm}}.
A* is the most popular choice for pathfinding, because it's fairly
flexible and can be used in a wide range of contexts.
A* is like Dijkstra's Algorithm in that it can be used to find a
shortest path. A* is like Greedy Best-First-Search in that it can use
a heuristic to guide itself. In the simple case, it is as fast as
Greedy Best-First-Search:
{figure {img 526 376 ../game-programming/a-star/a-star.png}}
In the example with a concave obstacle, A* finds a path as good as
what Dijkstra's Algorithm found:
{figure {img 526 376 ../game-programming/a-star/a-star-trap.png}}
The secret to its success is that it combines the pieces of
information that Dijkstra's Algorithm uses (favoring vertices that are
close to the starting point) {emph and} information that
Greedy Best-First-Search uses (favoring vertices that are close to the goal).
In the standard terminology used when talking about A*, {code g(n)}
represents the {emph exact cost} of the path from the starting point
to any vertex {code n}, and {code h(n)} represents the heuristic
{emph estimated cost} from vertex {code n} to the goal. In the above
diagrams, the yellow ({code h}) represents vertices far from the goal
and teal ({code g}) represents vertices far from the starting point.
A* balances the two as it moves from the starting point to the goal.
Each time through the main loop, it examines the vertex {code n} that
has the lowest {code f(n) = g(n) + h(n)}.
The rest of this article will explore {link {ref Heuristics} heuristic
design}, {link {ref Implementation notes} implementation}, {link {ref Map
representations} map representation}, and a variety of other topics
related to the use of pathfinding in games. Some sections are
well-developed and others are rather incomplete.
{h2 Heuristics}
{filename Heuristics.html}
The heuristic function {code h(n)} tells A* an {emph estimate} of the
minimum cost from any vertex {code n} to the goal. It's important to
choose a good heuristic function.
{h3 A*'s Use of the Heuristic}
The heuristic can be used to control A*'s behavior.
{list
{hbox At one extreme, if {code h(n)} is 0, then only {code g(n)}
plays a role, and A* turns into Dijkstra's Algorithm, which is
guaranteed to find a shortest path.}
{hbox If {code h(n)} is always lower than (or equal to) the cost of
moving from {code n} to the goal, then A* is guaranteed to find a
shortest path. The lower {code h(n)} is, the more node A*
expands, making it slower.}
{hbox If {code h(n)} is exactly equal to the cost of moving from
{code n} to the goal, then A* will only follow the best path and
never expand anything else, making it very fast. Although you
can't make this happen in all cases, you can make it exact in some
special cases. It's nice to know that given perfect information,
A* will behave perfectly.}
{hbox If {code h(n)} is sometimes greater than the cost of moving from
{code n} to the goal, then A* is not guaranteed to find a shortest
path, but it can run faster.}
{hbox At the other extreme, if {code h(n)} is very high relative to
{code g(n)}, then only {code h(n)} plays a role, and A* turns into
Greedy Best-First-Search.}
}
{margin-note
{vbox {strong Note:}
{hbox Technically, the {strong A*} algorithm should be called simply
{strong A} if the heuristic is an underestimate of the
actual cost. However, I will continue to call it
{strong A*} because the implementation is the same and
the game programming community does not distinguish
{strong A} from {strong A*}.}
}
}
So we have an interesting situation in that we can decide what we want
to get out of A*. With 100% accurate estimates, we'll get shortest
paths really quickly. If we're too low, then we'll continue to get
shortest paths, but it'll slow down. If we're too high, then we give
up shortest paths, but A* will run faster.
In a game, this property of A* can be very useful. For example, you
may find that in some situations, you would rather have a "good" path
than a "perfect" path. To shift the balance between {code g(n)} and
{code h(n)}, you can modify either one.
{h3 Speed or accuracy?}
A*'s ability to vary its behavior based on the heuristic and cost
functions can be very useful in a game. The tradeoff between speed and
accuracy can be exploited to make your game faster. For most games,
you don't {emph really} need the {strong best} path between two
points. You {link http://realtimecollisiondetection.net/blog/?p=56
need something that's close}. What you need may depend on what's going
on in the game, or how fast the computer is. Using a function that
{emph guarantees} it never overestimates the cost means that it will
sometimes underestimate the cost by quite a bit.
Suppose your game has two types of terrain, Flat and Mountain, and the
movement costs are 1 for flat land and 3 for mountains, A* is going to
search three times as far along flat land as it does along mountainous
land. This is because it's {emph possible} that there is a path along
flat terrain that goes around the mountains. You can speed up A*'s
search by using 1.5 as the heuristic distance between two map spaces.
A* will then compare 3 to 1.5, and it won't look as bad as comparing 3
to 1. It is not as dissatisfied with mountainous terrain, so it won't
spend as much time trying to find a way around it. Alternatively, you
can speed up up A*'s search by decreasing the amount it searches for
paths around mountains---tell A* that the movement cost on
mountains is 2 instead of 3. Now it will search only twice as far
along the flat terrain as along mountainous terrain. Either approach
gives up ideal paths to get something quicker.
The choice between speed and accuracy does not have to be static. You
can choose dynamically based on the CPU speed, the fraction of time
going into pathfinding, the number of units on the map, the importance
of the unit, the size of the group, the difficulty level, or any other
factor. One way to make the tradeoff dynamic is to build a heuristic
function that assumes the minimum cost to travel one grid space is 1
and then build a cost function that scales:
{snippet
{line g'(n) = 1 + alpha * (g(n) - 1)}
}
If {code alpha} is 0, then the modified cost function will always be
1. At this setting, terrain costs are completely ignored, and A*
works at the level of simple passable/unpassable grid spaces. If
{code alpha} is 1, then the original cost function will be used, and
you get the full benefit of A*. You can set {code alpha} anywhere in
between.
You should also consider switching from the heuristic returning the
{emph absolute} minimum cost to returning the {emph expected} minimum
cost. For example, if most of your map is grasslands with a movement
cost of 2 but some spaces on the map are roads with a movement cost of
1, then you might consider having the heuristic assume no roads, and
return {code 2 * distance}.
The choice between speed and accuracy does not have to be global. You
can choose some things dynamically based on the importance of having
accuracy in some region of the map. For example, it may be more
important to choose a good path near the current location, on the
assumption that we might end up recalculating the path or changing
direction at some point, so why bother being accurate about the
faraway part of the path? Or perhaps it's not so important to have
the shortest path in a safe area of the map, but when sneaking past an
enemy village, safety and quickness are essential.
{h3 Scale}
A* computes {code f(n) = g(n) + h(n)}. To add two values, those two
values need to be at the same scale. If {code g(n)} is measured in
hours and {code h(n)} is measured in meters, then A* is going to
consider {code g} or {code h} too much or too little, and you either
won't get as good paths or you A* will run slower than it could.
{h3 Exact heuristics}
If your heuristic is exactly equal to the distance along the optimal
path, you'll see A* expand very few nodes, as in the diagram shown in
{link {ref Manhattan distance} the next section}. What's happening
inside A* is that it is computing {code f(n) = g(n) + h(n)} at every
node. When {code h(n)} exactly matches {code g(n)}, the value of
{code f(n)} doesn't change along the path. All nodes not on the right
path will have a higher value of {code f} than nodes that are on the
right path. Since A* doesn't consider higher-valued {code f} nodes
until it has considered lower-valued {code f} nodes, it never strays
off the shortest path.
{h4 Precomputed exact heuristic}
One way to construct an exact heuristic is to precompute the length of
the shortest path between every pair of points. This is not feasible
for most game maps. However, there are ways to approximate this
heuristic:
{list
{hbox Fit a coarse grid on top of the fine grid. Precompute the
shortest path between any pair of coarse grid locations.}
{hbox Precompute the shortest path between any pair of {link {ref
Waypoints} waypoints}. This is a generalization of the coarse
grid approach.}
}
Then add in a heuristic {code h'} that estimates the cost of going
from any location to nearby waypoints. (The latter too can be
precomputed if desired.) The final heuristic will be:
{snippet
{line h(n) = h'(n, w1) + distance(w1, w2) + h'(w2, goal)}
}
or if you want a better but more expensive heuristic, evaluate the
above with all pairs {code w1}, {code w2} that are close to the node
and the goal, respectively.
{h4 Linear exact heuristic}
In a special circumstance, you can make the heuristic exact without
precomputing anything. If you have a map with no obstacles and no
slow terrain, then the shortest path from the starting point to the
goal should be a straight line.
If you're using a simple heuristic (one which does not know about the
obstacles on the map), it should match the exact heuristic. If it
doesn't, then you may have a problem with scale or the type of heuristic
you chose.
{h3 Heuristics for grid maps}
{id S7} On a grid, there are well-known heuristic functions to use.
{strong Use the distance heuristic that matches the allowed movement:}
{list
{hbox On a square grid that allows {strong 4 directions} of movement, use Manhattan distance (L{sub 1}).}
{hbox On a square grid that allows {strong 8 directions} of movement, use Diagonal distance (L{sub ∞}).}
{hbox On a square grid that allows {strong any direction} of movement, you might or might not want Euclidean distance (L{sub 2}). If A* is finding paths on the grid but you are allowing movement not on the grid, you may want to consider {link https://www.redblobgames.com/pathfinding/grids/algorithms.html other representations of the map}.}
{hbox On a hexagon grid that allows {strong 6 directions} of movement, use Manhattan distance {link https://www.redblobgames.com/grids/hexagons/#distances adapted to hexagonal grids}.}
}
Multiply the distance in steps by the minimum cost for a step. For
example, if you're measuring in meters, the distance is 3 squares, and
each square is 15 meters, then the heuristic would return 3 ⨉ 15 = 45
meters. If you're measuring in time, the distance is 3 squares, and
each square takes at least 4 minutes to cross, then the heuristic
would return 3 ⨉ 4 = 12 minutes. The units (meters, minutes, etc.)
returned by the heuristic should match the units used by the cost
function.
{h4 Manhattan distance}
The standard heuristic for a square grid is the {link
http://en.wikipedia.org/wiki/Taxicab_geometry Manhattan distance}.
Look at your cost function and find the minimum cost {code D} for
moving from one space to an adjacent space. {emph In the simple case,
you can set {code D} to be 1.} The heuristic on a square grid where
you can move in 4 directions should be {code D} times the Manhattan
distance:
{snippet
{line function heuristic(node) =}
{line {sp} dx = abs(node.x - goal.x)}
{line {sp} dy = abs(node.y - goal.y)}
{line {sp} return D * (dx + dy)}
}
How do you pick D? Use a scale that matches your cost function. For
the best paths, and an “admissible” heuristic, set D to the lowest
cost between adjacent squares. In the absence of obstacles, and on
terrain that has the minimum movement cost D, moving one step closer
to the goal should {emph increase {code g}} by D and {emph decrease
{code h}} by D. When you add the two, {code f} (which is set to {code g +
h}) will stay the same; that's a sign that the heuristic and cost
function scales match. You can also give up optimal paths to make A*
run faster by increasing D, or by decreasing the ratio between the
lowest and highest edge costs.
{figure {img 526 226 ../game-programming/a-star/manhattan.png}}
(Note: the above image has a {link {ref Breaking ties} tie-breaker}
added to the heuristic.)
{h4 Diagonal distance}
If your map allows diagonal movement you need a different heuristic.
The Manhattan distance for (4 east, 4 north) will be 8⨉D. However,
you could simply move (4 northeast) instead, so the heuristic should
be 4⨉D2, where D2 is the cost of moving diagonally.
{figure {img 526 226 ../game-programming/a-star/diagonal.png}}
{snippet
{line function heuristic(node) =}
{line {sp} dx = abs(node.x - goal.x)}
{line {sp} dy = abs(node.y - goal.y)}
{line {sp} return D * (dx + dy) + (D2 - 2 * D) * min(dx, dy)}
}
Here we compute the number of steps you take if you can't take a
diagonal, then subtract the steps you save by using the
diagonal. There are {code min(dx, dy)} diagonal steps, and each one
costs {code D2} but saves you {code 2⨉D} non-diagonal steps.
When D = 1 and D2 = 1, this is called the {link
http://en.wikipedia.org/wiki/Chebyshev_distance Chebyshev distance}.
When D = 1 and D2 = sqrt(2), this is called the {emph octile
distance}.
Another way to write this is {code D * max(dx, dy) + (D2-D) * min(dx, dy)}.
{link http://www.policyalmanac.org/games/heuristics.htm Patrick Lester
writes it yet a different way}, with {code if (dx > dy) (D * (dx-dy) + D2 * dy) else (D * (dy-dx) + D2 * dx)}. These are all equivalent.
{h4 Euclidean distance}
If your units can move at any angle (instead of grid directions), then
you should probably use a straight line distance:
{snippet
{line function heuristic(node) =}
{line {sp} dx = abs(node.x - goal.x)}
{line {sp} dy = abs(node.y - goal.y)}
{line {sp} return D * sqrt(dx * dx + dy * dy)}
}
However, if this is the case, then you may have trouble with using A*
directly because the cost function {code g} will not match the
heuristic function {code h}. Since Euclidean distance is shorter than
Manhattan or diagonal distance, you will still get shortest paths, but
A* will take longer to run:
{figure {img 526 226 ../game-programming/a-star/euclidean.png}}
{h4 Euclidean distance, squared}
I've seen several A* web pages recommend that you avoid the expensive
square root in the Euclidean distance by using distance-squared:
{snippet
{line function heuristic(node) =}
{line {sp} dx = abs(node.x - goal.x)}
{line {sp} dy = abs(node.y - goal.y)}
{line {sp} return D * (dx * dx + dy * dy)}
}
{strong Do not do this!} This definitely runs into the scale
problem. The scale of {code g} and {code h} need to match, because
you're adding them together to form {code f}. When A* computes {code
f(n) = g(n) + h(n)}, the square of distance will be much higher than
the cost {code g} and you will end up with an overestimating
heuristic. For longer distances, this will approach the extreme of
{code g(n)} not contributing to {code f(n)}, and A* will degrade into
Greedy Best-First-Search:
{figure {img 526 376 ../game-programming/a-star/best-first-search-trap.png}}
To attempt to fix this you can scale the heuristic down. However, then
you run into the opposite problem: for shorter distances, the
heuristic will be too small compared to {code g(n)} and A* will
degrade into Dijkstra's Algorithm.
If, after profiling, you find the cost of the square root is
significant, either use a fast square root approximation with
Euclidean distance or use the diagonal distance as an approximation to
Euclidean.
{h4 Multiple goals}
If you want to search for {emph any} of several goals, construct a
heuristic {code h'(x)} that is the minimum of {code h1(x), h2(x),
h3(x), ...} where {code h1, h2, h3} are heuristics to each of the
nearby spots.
One way to think about this is that we can add a new zero-cost edge
from each of the goals to a new graph node. A path to that new node
will necessarily go through one of the goal nodes.
If you want to search for paths to {emph all} of several goals, your
best option may be Dijkstra's Algorithm with early exit when you find
all the goals. There may be a variant of A* that can calculate these
paths; I don't know.
If you want to search for spot {emph near} a single goal, ask A*
search to find a path to the center of the goal area. While processing
nodes from the OPEN set, exit when you pull a node that is near
enough.
{h4 Breaking ties}
In some grid maps there are many paths with the same length. For example,
in flat areas without variation in terrain, using a grid will lead to
many equal-length paths. A* might explore all the paths with the same
{code f} value, instead of only one.
{figure
{vbox
{img 491 351 ../game-programming/a-star/tie-breaking-off.png}
{hbox Ties in {code f} values.}
}
}
The quick hack to work around this problem is to either adjust the
{code g} or {code h} values. The tie breaker needs to be
deterministic with respect to the vertex ({emph i.e.,} it shouldn't
be a random number), and it needs to make the {code f} values
differ. Since A* sorts by {code f} value, making them different
means only one of the "equivalent" {code f} values will be explored.
One way to break ties is to nudge the scale of {code h} slightly. If
we scale it downwards, then {code f} will increase as we move towards
the goal. Unfortunately, this means that A* will prefer to expand
vertices close to the starting point instead of vertices close to the
goal. We can instead scale {code h} upwards slightly (even by 0.1%).
A* will prefer to expand vertices close to the goal.
{snippet
{line heuristic *= (1.0 + p)}
}
The factor {code p} should be chosen so that {code p <} {emph (minimum
cost of taking one step)} {code /} {emph (expected maximum path
length)}. Assuming that you don't expect the paths to be more than
1000 steps long, you can choose p = 1/1000. (Note that this slightly
breaks "admissibility" of the heuristic but in games it almost never
matters.) The result of this tie-breaking nudge is that A*
explores far less of the map than previously:
{figure
{vbox
{img 491 351 ../game-programming/a-star/tie-breaking-scale-1.png}
{hbox Tie-breaking scaling added to heuristic.}
}
}
When there are obstacles of course it still has to explore to find a
way around them, but note that after the obstacle is passed, A*
explores very little:
{figure
{vbox
{img 491 351 ../game-programming/a-star/tie-breaking-scale-2.png}
{hbox Tie-breaking scaling added to heuristic, works nicely with obstacles.}
}
}
Steven van Dijk suggests that a more straightforward way to do this
would to pass {code h} to the comparison function. When the {code f}
values are equal, the comparison function would break the tie by
looking at {code h}.
Another way to break ties is to add a deterministic random number to
the heuristic or edge costs. (One way to choose a deterministic random
number is to compute a hash of the coordinates.) This breaks more
ties than adjusting {code h} as above. Thanks to Cris Fuhrman for
suggesting this.
A different way to break ties is to prefer paths that are along the
straight line from the starting point to the goal:
{snippet
{line dx1 = current.x - goal.x}
{line dy1 = current.y - goal.y}
{line dx2 = start.x - goal.x}
{line dy2 = start.y - goal.y}
{line cross = abs(dx1*dy2 - dx2*dy1)}
{line heuristic += cross*0.001}
}
This code computes the vector cross-product between the start to goal
vector and the current point to goal vector. When these vectors don't
line up, the cross product will be larger. The result is that this
code will give some slight preference to a path that lies along the
straight line path from the start to the goal. When there are no
obstacles, A* not only explores less of the map, the path looks very
nice as well:
{figure
{vbox
{img 491 351 ../game-programming/a-star/tie-breaking-cross-1.png}
{hbox Tie-breaking cross-product added to heuristic, produces pretty paths.}
}
}
However, because this tie-breaker prefers paths along the straight
line from the starting point to the goal, weird things happen when
going around obstacles (note that the path is still optimal; it will
look strange):
{figure
{vbox
{img 491 351 ../game-programming/a-star/tie-breaking-cross-2.png}
{hbox Tie-breaking cross-product added to heuristic, less pretty with obstacles.}
}
}
To interactively explore the improvement from this tie breaker, see
{link http://www.ccg.leeds.ac.uk/people/j.macgill/xaStar/ James
Macgill's A* applet} [or try {link
http://www.vision.ee.ethz.ch/~cvcourse/astar/AStar.html this mirror}
or {link http://web.archive.org/web/www.ccg.leeds.ac.uk/james/aStar/
this mirror}]. Use "Clear" to clear the map, and choose two points on
opposite corners of the map. When you use the "Classic A*" method,
you will see the effect of ties. When you use the "Fudge" method, you
will see the effect of the above cross product added to the heuristic.
Yet another way to break ties is to carefully construct your A*
priority queue so that {emph new} insertions with a specific {code f}
value are always ranked better (lower) than {emph old} insertions with
the same {code f} value.
And yet another way to break ties on grids is to minimize turns. The
change in x,y from the {emph parent} to the {emph current} node tells
you what direction you were moving in. For all edges being considered
from {emph current} to {emph neighbor}, if the change in x,y is
different than the one from parent to current, then add a small
penalty to the movement cost.
{strong The above modifications to the heuristic are a “band aid” fix
to an underlying inefficiency.} Ties occur when there are lots of
paths that are equally good, leading to a large number of nodes to
explore. Consider ways to “work smarter, not harder”:
{list
{hbox Alternate {link {ref Map representations} map representations}
can solve the problem by {strong reducing the number of nodes in
the graph}. Collapsing multiple nodes into one, or by remove all
but the important nodes. {link
http://aigamedev.com/open/tutorial/symmetry-in-pathfinding/
Rectangular Symmetry Reduction} is a way to do this on square
grids; also look at “framed quad trees”. {link {ref Hierarchical}
Hierarchical pathfinding} uses a high level graph with few nodes
to find most of the path, then a low level graph with more nodes
to refine the path.}
{hbox Some approaches leave the number of nodes alone but {strong
reduce the number of nodes visited}. {link
http://aigamedev.com/open/tutorial/symmetry-in-pathfinding/ Jump
Point Search} skips over large areas of nodes that would contain
lots of ties; it's designed for square grids. {link {ref Skip
links} Skip links} add “shortcut” edges that skip over areas of
the map. The {link https://web.archive.org/web/20060303021605/http://home1.stofanet.dk/breese/astaralpha-submitted.pdf.gz
AlphA* algorithm (PDF)} adds some depth-first searching to the usual
breadth-first behavior of A*, so that it can explore a single path
instead of processing all of them simultaneously.}
{hbox {link http://cswww.essex.ac.uk/cig/2005/papers/p1039.pdf
Fringe Search (PDF)} solves the problem instead by {strong
making node processing fast}. Instead of keeping the OPEN set
sorted and visiting nodes one at a time, it processes nodes in
batches, expanding only the nodes that have low f-values. This
is related to the {link {ref HOT queues} HOT queues} approach.}
}
{h3 Approximate heuristics}
A heuristic that has the exact distance is ideal for making A* fast
but it's usually impractical. We can often preprocess the graph to
construct an approximate distance, and use that approximation in the
A* heuristic.
{link http://research.microsoft.com/pubs/154937/soda05.pdf ALT A*}
uses “landmarks” and the triangle inequality to preprocess the
pathfinding graph in order to make pathfinding much faster. ALT also
does a few other things, but the heuristic improvement is the part
that got my attention. It's surprisingly simple to implement,
sometimes under 15 lines of code, and produces impressive speedups.
The name “landmark” is a little misleading. These points need to be
placed on the outer edges of the map. Some authors call it
“differential heuristics”.
The landmark approach stores lots of data that could be compressed.
{link
https://www.aaai.org/ocs/index.php/SOCS/SOCS11/paper/viewFile/4020/4340
The Compressed Differential Heuristic} shows the results of
compressing the landmark data. You can store a lot more landmarks in
the same space, so you get improved heuristic values.
Landmarks may be a special case of a more general approach. {link
http://webdocs.cs.ualberta.ca/~bowling/papers/11aaai-heuristicopt.pdf
This paper} explores transforming a map into a map where a regular
distance metric works.
{link https://en.wikipedia.org/wiki/Distance_oracle Distance oracles}
seem to be related but I haven't looked into them yet.
{h2 Implementation notes}
{filename ImplementationNotes.html}
{set meta-description "Notes on pathfinding implementations including performance issues."}
{set shortname Implementation}
{h3 Sketch}
The A* algorithm, stripped of all the code, is fairly simple. There
are two sets, OPEN and CLOSED. The OPEN set contains those nodes that
are candidates for examining. Initially, the OPEN set contains only
one element: the starting position. The CLOSED set contains those
nodes that have already been examined. Initially, the CLOSED set is
empty. Graphically, the OPEN set is the "frontier" and the CLOSED set
is the "interior" of the visited areas. Each node also keeps a
pointer to its parent node so that we can determine how it was found.
There is a main loop that repeatedly pulls out the best node {code n}
in OPEN (the node with the lowest {code f} value) and examines it. If
{code n} is the goal, then we're done. Otherwise, node {code n} is
removed from OPEN and added to CLOSED. Then, its neighbors {code n′}
are examined. A neighbor that is in CLOSED has already been seen, so
we don't need to look at it ⁽¹⁾. A neighbor that is in OPEN is
scheduled to be looked at, so we don't need to look at it now.
Otherwise, we add it to OPEN, with its parent set to {code n}. The
path cost to {code n′}, {code g(n′)}, will be set to {code g(n) +
movementcost(n, n′)}.
{strong I go into a lot more detail {link
https://www.redblobgames.com/pathfinding/a-star/introduction.html
here}}, with interactive diagrams.
⁽¹⁾ I'm skipping a small detail here. You do need to check to see if
the node's {code g} value can be lowered, and if so, you re-open it.
{snippet
{line "OPEN = priority queue containing START"}
{line "CLOSED = empty set"}
{line "while lowest rank in OPEN is not the GOAL:"}
{line " current = remove lowest rank item from OPEN"}
{line " add current to CLOSED"}
{line " for neighbors of current:"}
{line " cost = g(current) + movementcost(current, neighbor)"}
{line " if neighbor in OPEN and cost less than g(neighbor):"}
{line " remove neighbor from OPEN, because new path is better"}
{line " if neighbor in CLOSED and cost less than g(neighbor): ⁽²⁾"}
{line " remove neighbor from CLOSED"}
{line " if neighbor not in OPEN and neighbor not in CLOSED:"}
{line " set g(neighbor) to cost"}
{line " add neighbor to OPEN"}
{line " set priority queue rank to g(neighbor) + h(neighbor)"}
{line " set neighbor's parent to current"}
{line }
{line "reconstruct reverse path from goal to start"}
{line "by following parent pointers"}
}
⁽²⁾ This should never happen if you have an consistent admissible
heuristic. However in games we {link http://realtimecollisiondetection.net/blog/?p=56 often want inadmissible heuristics}.
{strong See Python and C++ implementations {link https://www.redblobgames.com/pathfinding/a-star/implementation.html here}.}
{h4 Connectivity}
If your game has situations in which the start and goal are not
connected at all by the graph, A* will take a long time to run, since
it has to explore every node connected from the start before it
realizes there's no path. Calculate the {link
http://en.wikipedia.org/wiki/Connected_component_(graph_theory)
Connected Components} first and only use A* if the start and goal are
in the same region.
{h3 Performance}
The main loop of A* reads from a priority queue, analyzes it, and
inserts nodes back into the priority queue. In addition, it tracks
which nodes have been visited. To improve performance, consider:
{list
{hbox Can you decrease the size of the graph? This will reduce the number of nodes that are processed, both those on the path and those that don't end up on the final path. Consider {link {ref Polygonal maps} navigation meshes} instead of grids. Consider {link {ref Hierarchical} hierarchical map representations}.}
{hbox Can you improve the accuracy of the heuristic? This will reduce the number of nodes that are not on the final path. The closer the hheuristic to the actual path length (not the distance), the fewer nodes A* will explore. Consider {link {ref Exact heuristics} these heuristics for grids}. Consider ALT (A*, Landmarks, Triangle Inequality) for graphs in general (including grids).}
{hbox Can you make the priority queue faster? Consider {link {ref Set representation} other data structures} for your priority queue. Consider processing nodes in batches, as {link http://en.wikipedia.org/wiki/Fringe_search fringe search} does. Consider approximate sorting.}
{hbox Can you make the heuristic faster? The heuristic function is called for every open node. Consider caching its result. Consider inlining the call to it.}
}
For grid maps, {link https://www.redblobgames.com/pathfinding/grids/algorithms.html see these suggestions}.
{h3 Source code and demos}
{h4 Demos}
These demos run in your browser:
{list
{hbox {strong I have written {link https://www.redblobgames.com/pathfinding/a-star/introduction.html an introduction to A*} with interactive demos.}}
{hbox I have written Flash demos for {link
http://theory.stanford.edu/~amitp/game-programming/a-star-flash/main-square.swf
square grids},
{link http://theory.stanford.edu/~amitp/game-programming/a-star-flash/main-hexagon.swf
hexagonal grids}, and {link
http://theory.stanford.edu/~amitp/game-programming/a-star-flash/main-triangle.swf
triangular grids}. The Actionscript 3 code for these demos is {link
http://www-cs-students.stanford.edu/~amitp/game-programming/a-star-flash/
available here} (see Pathfinder.as for the main algorithm, and
Graph.as for the abstract interface to graphs).}
{hbox {link https://github.com/anvaka/ngraph.path This Javascript library and demo} is very fast, works on graphs, and also includes bidirectional A*.}
{hbox {link http://kevanahlquist.com/osm_pathfinding/ This site} has demos of A*, Breadth-First Search, Dijkstra's Algorithm, and Greedy Best-First Search on road maps (not grids)}
{hbox {link http://mikolalysenko.github.io/l1-path-finder/www/ This Javascript library and demo} has lots of optimizations for grid maps.}
{hbox {link http://easystar.nodejitsu.com/demo.html This Javascript A* demo} lets you change the road weight; source code on {link https://github.com/prettymuchbryce/easystarjs github}, using the MIT open source license. The calculation is interruptible so you can run a few iterations per frame.}
{hbox {link http://www.vision.ee.ethz.ch/~cvcourse/astar/AStar.html James Macgill's Java applet}.}
{hbox {link http://www.growingwiththeweb.com/projects/pathfinding-visualiser/ This interactive demo lets you choose A* or Dijkstra's Algorithm}.}
{hbox {link http://qiao.github.com/PathFinding.js/visual/ This demo} is nice and has {link https://github.com/qiao/PathFinding.js Javascript source code}.}
{hbox {link http://will.thimbleby.net/a-shortest-path-in-javascript/ This demo} is nice and also has code available.}
{hbox {link http://www.briangrinstead.com/files/astar/ Another Javascript demo}}
{hbox {link http://zerowidth.com/2013/05/05/jump-point-search-explained.html This page} describes Jump Point Search and also has an online demo.}
{hbox {link
http://www.untoldentertainment.com/blog/2010/08/20/introduction-to-a-a-star-pathfinding-in-actionscript-3-as3-2/
This Actionscript A* tutorial} has a demo near the end.}
{hbox {link http://cs.williams.edu/~morgan/codeheartjs/examples/pathfinding/play.html This demo} is in Javascript with readable source but I don't know the license for the source code.}
{hbox {link http://singul4rity.com/misc/astar/ This A* demo} and {link http://singul4rity.com/2013/04/jump-point-search-unity3d-online-demo/ this Jump Point Search demo} use Unity.}
}
{h4 Code}
If you're using C++, be sure to look at {link
http://code.google.com/p/recastnavigation/ Recast}, from {link
http://digestingduck.blogspot.com/ Mikko Mononen}.
If you're planning to implement graph search yourself, {strong {link
https://www.redblobgames.com/pathfinding/a-star/implementation.html
here's my guide to Python and C++ implementations}}. My code is
much shorter and simpler than most other A* code I have found, and
works on any graphs, not only on grids.
I've collected some links to source code but haven't looked into these
projects and can't make specific recommendations. {strong These are
old} links. I started collecting them in 1997, before Google existed,
before Github or npm or pypy or other ways of finding code online.
{strong I no longer maintain this list.}
{list
{hbox C++:
{link https://code.google.com/p/policy-based-search/ "[1]"}
{link https://gist.github.com/codemonkey-uk/8240501 "[2]"}
{link https://github.com/justinhj/astar-algorithm-cpp "[3]"}
{link https://github.com/vandersonmr/A_Star_Algorithm/tree/master/libs/c%2B%2B "[4]"}
}
{hbox Java:
{link http://memoization.com/2008/11/30/a-star-algorithm-in-java/ "[1]"}
{link http://code.google.com/p/a-star/source/browse/trunk/java/AStar.java?r=8 "[2]"}
{link http://www.vision.ee.ethz.ch/~cvcourse/astar/AStar.html "[3]"}
}
{hbox Javascript:
{link http://will.thimbleby.net/a-shortest-path-in-javascript/ "[1]"}
{link http://www.briangrinstead.com/blog/astar-search-algorithm-in-javascript "[2]"}
{link http://ahuynh.posterous.com/javascript-a-implementation "[3]"}
{link http://jsastar.tapirpirates.net/dfdemo/ "[4]"}
{link https://github.com/qiao/PathFinding.js "[5]"}
{link http://mikolalysenko.github.io/l1-path-finder/www/ "[6]"} (extremely fast pathfinding for grid maps)
}
{hbox Python:
{link https://github.com/vandersonmr/A_Star_Algorithm/tree/master/libs/python "[1]"}
}
{hbox Objective C + Cocos2D:
parts {link http://www.raywenderlich.com/4946/introduction-to-a-pathfinding "[1]"}
and {link http://www.raywenderlich.com/4970/how-to-implement-a-pathfinding-with-cocos2d-tutorial "[2]"}
}
{hbox Lua:
{link https://github.com/Yonaba/Jumper "[1]"}
{link https://love2d.org/forums/viewtopic.php?f=5&t=7983 "[2]"}
{link https://www.love2d.org/forums/viewtopic.php?f=4&t=78021&p=168522#p168522 "more"}
}
{hbox Ruby:
{link https://github.com/shawn42/polaris/tree/master/lib/ "[1]"}
{link https://github.com/DanaDanger/a_star/tree/master/ruby "[2]"}
{link https://github.com/vandersonmr/A_Star_Algorithm/blob/master/libs/ruby/AlgorithmA.rb "[3]"}
}
{hbox C#:
{link http://www.codeproject.com/KB/recipes/csharppathfind.aspx "[1]"}
{link https://bitbucket.org/BlueRaja/high-speed-priority-queue-for-c/wiki/Home "[priority queue helper class]"}
}
{hbox Unity:
{link http://www.arongranberg.com/unity/a-pathfinding/ "[1]"}
}
{hbox Assembly:
{link http://magervalp.github.io/2014/05/07/astar-in-asm.html "[1]"}
}
{hbox Actionscript 3 (Flash):
{link http://www-cs-students.stanford.edu/~amitp/game-programming/a-star-flash/ "[1]"}
{link http://www.antimodal.com/astar/ "[2]"}
{link http://lassieadventurestudio.wordpress.com/2008/12/09/a-star-pathfinding/ "[3]"}
{link http://proto.layer51.com/d.aspx?f=998 "[4]"}
{link http://www.dauntless.be/astar/ "[5]"}
}
{hbox Flex (Flash):
{link http://web.archive.org/web/http://blog.topholt.com/2007/12/07/a-pathfinding-in-an-isometric-world-with-flex/
"[1]"}
}
{hbox Go:
{link https://github.com/beefsack/go-astar "[1]"}
}
{hbox Prolog:
{link http://www.csupomona.edu/~jrfisher/www/prolog_tutorial/5_1.html "[1]"}
}
{hbox Processing:
{link http://www.robotacid.com/PBeta/AILibrary/Pathfinder/index.html "[1]"}
}
}
{h3 Set representation}
What's the first thing you'll think of using for the OPEN and CLOSED
sets? If you're like me, you probably thought "array". You may have
thought "linked list", too. There are many different data structures
we can use, but to pick one we should look at what operations are
needed.
There are three main operations we perform on the OPEN set: the main
loop repeatedly finds the best node and removes it; the neighbor
visiting will check whether a node is in the set; and the neighbor
visiting will insert new nodes. Insertion and remove-best are
operations typical of a {link
http://en.wikipedia.org/wiki/Priority_queue priority queue}.
The choice of data structure depends not only on the operations but on
the number of times each operations runs. The membership test runs
once for each neighbor for each node visited. Insertion runs once for
each node being considered. Remove-best runs once for each node
visited. Most nodes that are considered will be visited; the ones
that are not are the {emph fringe} of the search space. When evaluating
the cost of operations on these data structures, we need to consider
the maximum size of the fringe (F).
{margin-note Do you really need the priority-adjustment operation? When I wrote this document in 1997 I believed you did. I have since come to believe that you don't always need it, and in many cases you're better off not implementing it. See the {link https://www.redblobgames.com/pathfinding/a-star/implementation.html#optimizations “what happens if you don't reprioritize?” paragraph} in my newer A* page.}
In addition, {emph there's a fourth operation}, which is relatively
rare but still needs to be implemented. If the node being examined is
already in the OPEN set (which happens frequently), and if its {code
f} value is better than the one already in the OPEN set (which is
rare), then the value in the OPEN set must be adjustment. The
adjustment operation involves removing the node (which is not the best
{code f}) and re-inserting it. These two steps may be optimized into
an increase-priority operation that moves the node (this is also
called decrease-key).
{strong My recommendation:} The best generic choice is {link {ref
Binary heaps} a binary heap}. If you have a binary heap library
available, use it. If not, start with {link {ref Sorted arrays} sorted
arrays} or {link {ref Unsorted arrays or linked lists} unsorted
arrays}, and switch to binary heaps if you want more performance. If
you have more than 10,000 elements in your OPEN set, then consider
more complicated structures such as a bucketing system.
{h4 Unsorted arrays or linked lists}
The simplest data structure is an unsorted array or list. Membership
test is slow, O(F) to scan the entire structure. Insertion is fast,
O(1) to append to the end. Finding the best element is slow, O(F) to
scan the entire structure. Removing the best element is O(1) with
linked lists, and O(1) with unsorted arrays since we can change the
order. The increase-priority operation is O(F) to find the node and
O(1) to change its value.
{h4 Sorted arrays}
To make remove-best fast, we can keep the array sorted. Membership is
then O(log F), since we can use binary search. Insertion is slow, O(F)
to move all the elements to make space for the new one. Finding the
best is fast, O(1) since it's already at the end. And removing the
best is O(1) if we make sure the best sorts to the {emph end} of the
array. The increase-priority operation is O(log F) to find the node
and O(F) to change its value/position.
Make sure the array is sorted so that the best element is at the end.
{h4 Sorted linked lists}
With sorted arrays, insertion is slow. If we use a linked list, we
can make that fast. Membership in the linked list is slow, O(F) to
scan the list. Insertion is fast, O(1) to insert a new link, but it
was O(F) to find the right position for it. Finding the best remains
fast, O(1) because the best is at the end. Removing the best also is
O(1). The increase-priority operation is O(F) to find the node and
O(1) to change its value/position.
{h4 Binary heaps}
A {link http://en.wikipedia.org/wiki/Heap_(data_structure) binary
heap} (not to be confused with a memory heap) is a tree structure that
is stored in an array. Unlike most trees, which use pointers to refer
to children, the binary heap uses indexing to find children.
In a binary heap, membership is O(F), as you have to scan the entire
structure. Insertion is O(log F) and remove-best is O(log F).
The increase-priority operation is tricky, with O(F) to find the node
and surprisingly, only O(log F) to increase-priority it. Unfortunately
most priority queue libraries don't include this operation.
Fortunately, {emph it's not strictly necessary}. So I recommend not
worrying about it unless you absolutely need to. Instead of increasing
priority, insert a new element into the priority queue. You'll
potentially end up processing the node twice but that's relatively
cheap compared to implementing increase-priority.
In C++, use the {link
http://en.cppreference.com/w/cpp/container/priority_queue
priority_queue} class, which doesn't have increase-priority, or {link
http://www.boost.org/doc/libs/1_50_0/doc/html/heap/concepts.html#heap.concepts.mutability
Boost's mutable priority queue}, which does. In Python, use the {link
https://docs.python.org/3/library/heapq.html heapq library}.
You can combine a hash table or indexed array for membership and a
priority queue for managing priorities; see {link {ref Hybrid
representations} the hybrid section below}.
As part of my {link
https://www.redblobgames.com/pathfinding/a-star/introduction.html newer
A* tutorial}, I have a {link
https://www.redblobgames.com/pathfinding/a-star/implementation.html
complete A* implementation in Python and C++} using binary heaps for
the priorities and hash tables for the for membership. I {emph do not
implement increase-priority} and explain why in the optimization
section.
A variant of the binary heap is a {link
http://en.wikipedia.org/wiki/D-ary_heap d-ary heap}, which has more
than 2 children per node. Inserts and increase-priority become a
little bit faster, but removals become a little bit slower. They may
have better cache performance.
{strong I have only used binary heaps and bucket approaches for my
own pathfinding projects.} If binary heaps aren't good enough, then
consider pairing heaps, sequence heaps, or a bucket based approach.
The paper {link
http://www3.cs.stonybrook.edu/~rezaul/papers/TR-07-54.pdf Priority
Queues and Dijkstra's Algorithm} is worth a read if you are unable to
shrink the graph and need a faster priority queue.
{h4 Sorted skip lists}
Searching an unsorted linked list is slow. We can make that faster if
we use a {link http://en.wikipedia.org/wiki/Skip_list skip list}
instead of a linked list. With a skip list, membership is fast if you
have the sort key: O(log F). Insertion is O(1) like a linked list if
you know where to insert. Finding the best node is fast if the sort
key is {code f}, O(1), and removing a node is O(1). The
increase-priority operation involves finding a node, removing it, and
reinserting it.
If we use skip lists with the map location as the key, membership is
O(log F), insertion is O(1) after we've performed the membership test,
finding the best node is O(F), and removing a node is O(1). This is
better than unsorted linked lists in that membership is faster.
If we use skip lists with the {code f} value as the key, membership is
O(F), insertion is O(1), finding the best node is O(1), and removing a
node is O(1). This is no better than sorted linked lists.
{h4 Indexed arrays}
If the set of nodes is finite and reasonably sized, we can use a
direct indexing structure, where an index function {code i(n)} maps
each node {code n} to an index into an array. Unlike the unsorted and
sorted arrays, which have a size corresponding to the largest size of
OPEN, with an indexed array the array size is always {code max(i(n))}
over all {code n}. If your function is dense ({emph i.e.,} there are
no indices unused), then {code max(i(n))} will be the number of nodes
in your graph. Whenever your map is a grid, it's easy to make the
function dense.
Assuming {code i(n)} is O(1), membership test is O(1), as we merely
have to check whether {code "Array[i(n)]"} contains any data.
Insertion is O(1), as we set {code "Array[i(n)]"}. Find and
remove best is O(numnodes), since we have to search the entire
structure. The increase-priority operation is O(1).
{h4 Hash tables}
Indexed arrays take up a lot of memory to store all the nodes that are
{emph not} in the OPEN set. An alternative is to use a hash table,
with a hash function {code h(n)} that maps each node {code n} into a
hash code. Keep the hash table twice as big as N to keep the chance of
collisions low. Assuming {code h(n)} is O(1), membership test is
expected O(1), insertion is expected O(1), and remove best is
O(numnodes), since we have to search the entire structure. The
increase-priority operation is O(1).
Hash tables are best for set membership but not for managing
priorities. In my my {link
https://www.redblobgames.com/pathfinding/a-star/introduction.html newer
A* tutorial}, I use {link
https://www.redblobgames.com/pathfinding/a-star/implementation.html
hash tables for membership and binary heaps for priorities}. I
combined the OPEN and CLOSED sets into one, which I call VISITED.
{h4 Splay trees}
Heaps are a tree-based structure with expected O(log F) time
operations. However, the problem is that with A*, the common behavior
is that you have a low cost node that is removed (causing O(log F)
behavior, since values have to move up from the very bottom of the
tree) followed by low cost nodes that are added (causing O(log F)
behavior, since these values are added at the bottom and bubble up to
the very top). The {emph expected case} behavior of heaps here is
equivalent to the {emph worst case} behavior. We may be able to do
better if we find a data structure where {emph expected case} is
better, even if the {emph worst case} is no better.
Splay trees are a self adjusting tree structure. Any access to a node
in the tree tends to bring that node up to the top. The result is a
"caching" effect, where rarely used nodes go to the bottom and don't
slow down operations. It doesn't matter how big your splay tree is,
because your operations are only as slow as your "cache size". In A*,
the low cost nodes are used a lot, and the high cost nodes aren't used
for a long time, so those high cost nodes can move to the bottom of
the tree.
With splay trees, membership, insertion, remove-best, and increase-priority
are all expected O(log F), worst case O(F). Typically however, the
caching keeps the worst case from occurring. Dijkstra's Algorithm and
A* with an underestimating heuristic however have some peculiar
characteristics that may keep splay trees from being the best. In
particular, {code f(n') >= f(n)} for nodes {code n} and neighboring
node {code n'}. When this happens, it may be that the insertions all
occur on one side of the tree and end up putting it out of balance. I
have not tested this.
{h4 Bucket queues}
A {link https://en.wikipedia.org/wiki/Bucket_queue bucket queue} is
useful when the range of movement costs is limited to small integers,
especially for Dijkstra's Algorithm. Given a restricted range, there
are often better algorithms. For example, sorting can be done on
arbitrary values in O(N log N) time, but when there is a fixed range
the bucket or radix sorts can perform sorting in O(N) time.
Consider Dijkstra's Algorithm: the frontier has priorities {code p}
through {code p+k} where {code k} is the largest movement cost. For
example if your movement costs are 1, 2, or 3, then everything in the
frontier has priority {code p}, {code p+1}, {code p+2}, {code p+3}.
Use 4 buckets, one for each bucket. There's no need to sort within a
bucket because the priorities are all the same. And there's no need to
sort the buckets. That means insertion is O(1) and remove-best is O(1).
{#
to take advantage of {code f(n') >= f(n)} where
{code n'} is a neighbor of {code n}. We are removing the node {code
n} with minimal {code f(n)}, and inserting neighbors {code n'} with
{code f(n) <= f(n') <= f(n) + delta} where {code delta <= C}. The
constant {code C} is the {emph maximum} change in cost from one point
to an adjacent point. Since {code f(n)} was the minimal {code f}
value in the OPEN set, and everything being inserted is {code <= f(n)
+ delta}, we know that all {code f} values in the OPEN set are within
a range of {code 0 .. delta}.
}
Note that Breadth First Search can be viewed as using a bucketed
priority queue with exactly two buckets.
With A* it's a little more complicated because you also have to look
at the effect of the heuristic on the priority. There will be more
buckets than with Dijkstra's Algorithm.
{h4 HOT queues}
{link http://www.star-lab.com/goldberg/pub/neci-tr-97-104.ps HOT
Queues} are a variant of bucket queues that accept a larger range of
values. Instead of each bucket having {emph exactly} one priority,
each bucket has a range of priorities. Since we're only removing
elements from the top bucket, only the top bucket has to be ordered.
{# With K buckets, we reduce any O(N) cost to an average of O(N/K). }
HOT queues make the topmost bucket use a binary heap. All other
buckets are unsorted arrays. Membership test is O(F) because we don't
know which bucket the node is in. Insertion and remove-best in the top
bucket are O(log (F/K)), for K buckets. Insertion into other buckets
is O(1), and remove-best never runs on other buckets. If the top
bucket empties, then we need to convert the next bucket, an unsorted
array, into a binary heap. It turns out this operation ("heapify") can
be run in O(F/K) time. The increase-priority operation is best treated
as a O(F/K) removal followed by an O(log (F/K)) or O(1) insertion.
In A*, many of the nodes we put into OPEN we never actually need. HOT
Queues are a big win because the elements that are not needed are
inserted in O(1) time. Only elements that are needed get heapified
(which is not too expensive). The only operation that is more than
O(1) is node deletion from the heap, which is only O(log (F/K)).
One person reported that HOT queues are as fast as heaps for at most
800 nodes in the OPEN set, and are 20% faster when there are at most
1500 nodes. I would expect that HOT queues get faster as the number of
nodes increases.
A cheap variant of a HOT queue is a two-level queue: put good nodes
into one data structure (a heap or an array) and put bad nodes into
another data structure (an array or a linked list). Since most nodes
put into OPEN are "bad", they are never examined, and there's no harm
putting them into the big array.
{h4 Pairing heaps}
Fibonacci heaps are good priority queues for A*, in theory. However,
in practice they're not used. A {link
http://en.wikipedia.org/wiki/Pairing_heap pairing heap} can be thought
of as a simplified Fibonacci heap. They are said to work well in
practice; I have never used them.
Here's {link https://www.cs.cmu.edu/~sleator/papers/pairing-heaps.pdf
the original paper describing them}.
{h4 Soft heaps}
A {link https://en.wikipedia.org/wiki/Soft_heap soft heap} is a type of heap that gives the nodes in approximately the right order. By approximating, it can provide results faster than a regular heap.
I haven't tried soft heaps. The algorithms described in {link http://epubs.siam.org/doi/pdf/10.1137/1.9781611973068.53 this paper} seem fairly short and straightforward to implement. However {link https://stackoverflow.com/a/26126781 this answer on stackoverflow} says it's useful in theory but “unlikely to be useful in practice”.
For pathfinding in games, we often do not need the {emph exact}
shortest path and usually would prefer to have a reasonably short path
computed quickly. So this may be one of the applications where it
{emph is} useful in practice. I don't know yet; if you have studied
this data structure, please {link mailto:redblobgames@gmail.com email
me}.
{h4 Sequence heaps}
I haven't looked into them. See Peter Sanders's paper {link
http://algo2.iti.kit.edu/sanders/papers/falenex.ps.gz Fast Priority
Queues for Cached Memory}.
{emph “Sequence heaps may currently be the fastest available data
structure for large comparison based priority queues both in cached
and external memory This is particularly true if the queue elements
are small and if we do not need deletion of arbitrary elements or
decreasing keys.”}
{h4 Data Structure Comparison}
It is important to keep in mind that we are not merely looking for
asymptotic ("big O") behavior. We also want to look for a low
constant. To see why, consider an algorithm that is O(log F) and
another that is O(F), where F is the number of elements in the heap.
It may be that on your machine, an implementation of the first
algorithm takes 10,000 * log(F) seconds, while an implementation of
the second one takes 2 * F seconds. For F = 256, the first would take
80,000 seconds and the second would take 512 seconds. The "faster"
algorithm takes more time in this case, and would only start to be
faster when F > 200,000.
{emph You cannot merely compare two algorithms.} You should also
compare the implementations of those algorithms. You also have to
know what size your data might be. In the above example, the first
implementation is faster for F > 200,000, but if in your game, F stays
under 30,000, then the second implementation would have been better.
None of the basic data structures is entirely satisfactory. Unsorted
arrays or lists make insertion very cheap and membership and removal
very expensive. Sorted arrays or lists make membership somewhat
cheap, removal very cheap and insertion very expensive. Binary heaps
make insertion and removal somewhat cheap, but membership is very
expensive. Splay trees make everything somewhat cheap. HOT queues
make insertions cheap, removals fairly cheap, and membership tests
somewhat cheap. Indexed arrays make membership and insertion very
cheap, but removals are incredibly expensive, and they can also take
up a lot of memory. Hash tables perform similarly to indexed arrays,
but they can take up a lot less memory in the common case, and
removals are merely expensive instead of extremely expensive.
For a good list of pointers to more advanced priority queue papers and
implementations, see {link http://www.leekillough.com/heaps/
Lee Killough's Priority Queues page}.
{h4 Hybrid representations}
To get the best performance, you will want a hybrid data structure.
For my A* code, I used an indexed array for O(1) membership test and a
binary heap for O(log F) insertion and O(log F) remove-best. For
increase-priority, I used the indexed array for an O(1) test whether I
really needed to perform the change in priority (by storing the {code
g} value in the indexed array), and then in those rare cases that I
did need to increase priority, I used the O(F) increase-priority on
the binary heap. You can also use the indexed array to store the
location in the heap of each node; this would give you O(log F) for
increase-priority.
In my {link
https://www.redblobgames.com/pathfinding/a-star/introduction.html newer
A* tutorial}, I use {link
https://www.redblobgames.com/pathfinding/a-star/implementation.html a
hash table for membership and a binary heap for priorities}. I further
simplified by combining OPEN and CLOSED into the same set, which I
call Visited.
{h3 Interaction with the game loop}
Interactive (especially real-time) games introduce requirements that
affect your ability to compute the best path. It may be more
important to get {emph any} answer than to get the {emph best} answer.
Still, all other things being equal, a shorter path is better than a
longer one.
In general, computing the part of the path close to the starting point
is more important than the path close to the goal. The principle of
{emph immediate start}: get the unit moving as soon as possible, even
along a suboptimal path, and then {link {ref Recalculating paths}
compute a better path later}. In real-time games, the {emph latency}
of A* is often more important than its {emph throughput}.
Units can be programmed to follow either their instincts (simple
movement) or their brains (a precalculated path). Units will follow
their instincts unless their brains tell them otherwise. (This
approach is used in nature and also in Rodney Brook's robot
architecture.) Instead of calculating all the paths at once, limit
the game to finding one path every one, two, or three game cycles.
Then let the units start walking according to instinct (which could
simply be moving in a straight line towards the goal), and come back
later to find paths for them. This approach allows you to even out
the cost of pathfinding so that it doesn't occur all at once.
{h4 Early exit}
It is possible to exit early from the A* main loop and get a partial
path. Normally, the loop exits when it finds the goal node. However,
at any point before that, it can return a path to the currently best
node in OPEN. That node is our best chance of getting to the goal, so
it's a reasonable place to go.
Candidates for early exit include having examined some number of
nodes, having spent some number of milliseconds in the A* algorithm,
or exploring a node some distance away from the starting position.
When using path splicing, the spliced path should be given a smaller
limit than a full path.
{h4 Interruptible algorithm}
If few objects need pathfinding services or if the data structures
used to store the OPEN and CLOSED sets are small, it can be feasible
to store the state of the algorithm, exit to the game loop, then
continue where A* left off.
{h4 Group movement}
Path requests do not arrive evenly distributed. A common situation in
a real-time strategy game is for the player to select multiple units
and order them to move to the same goal. This puts a high load on the
pathfinding system.
{margin-note
{strong See Also:}
Putting paths together is called {link {ref Path splicing} path
splicing} in another section of these notes.
}
In this situation, it is very likely that the path found for one will
be useful for other units. One idea is to find a path {code P} from
the center of the units to the center of the destinations. Then, use
most of that path for all the units, but replace the first 10 steps
and the last 10 steps by paths that are found for each individual
unit. Unit {emph i} will receive a path from its starting location to
{code "P[10]"}, followed by the shared path {code "P[10..len(P)-10]"},
followed by a path from {code "P[len(P)-10]"} to the destination.
The paths found for each unit are short (approximately 10 steps on
average), and the long path is shared. Most of the path is found only
once and shared among all the units. However, the user may not be
impressed if he sees all the units moving on the same path. To
improve the appearance of the system, make the units follow slightly
different paths. One way to do this is to alter the paths themselves,
by choosing adjacent locations.
Another approach is to make the units aware of each other (perhaps by
picking a "leader" unit randomly, or by picking the one with the best
sense of what's going on), and only find a path for the leader. Then
use a {link http://www.red3d.com/cwr/boids/ flocking algorithm} to make
them move in a group.
There are some {link
http://en.wikipedia.org/wiki/Incremental_heuristic_search variants of
A*} that handle a moving destination, or updated knowledge of the
destination. Some of them might be adapted to handle multiple units
going to the same destination, by performing A* in reverse (finding a
path from the destination to the unit).
{h4 Refinement}
If the map contains few obstacles, but instead contains terrain of
varying costs, then an initial path can be computed by treating
terrain as being cheaper than normal. For instance, if grasslands are
cost 1, hills are cost 2, and mountains are cost 3, then A* will
consider walking through 3 grasslands to avoid 1 mountain. Instead,
compute an initial path by treating grasslands as 1, hills as 1.1, and
mountains as 1.2. A* will then spend less time trying to avoid the
mountains, and it will find a path quicker. (It approximates the
benefits of an {link {ref Linear exact heuristic} exact heuristic}.)
Once a path is known, the unit can start moving, and the game loop can
continue. When spare CPU is available, compute a better path using
the real movement costs.
{h2 Variants of A*}
{filename Variations.html}
{set meta-description "Variations of A*."}
{set shortname Variations}
{h3 Beam search}
In the main A* loop, the OPEN set stores all the nodes that may need
to be searched to find a path. The {strong Beam Search} is a
variation of A* that places a limit on the size of the OPEN set. If
the set becomes too large, the node with the worst chances of giving a
good path is dropped. One drawback is that you have to keep your set
sorted to do this, which limits the kinds of data structures you'd
choose.
{h3 Iterative deepening}
Iterative Deepening is an approach used in many AI algorithms to start
with an approximate answer, then make it more accurate. The name
comes from game tree searches, where you look some number of moves
ahead (for example, in Chess). You can try to deepen the tree by
looking ahead more moves. Once your answer doesn't change or improve
much, you assume that you have a pretty good answer, and it won't
improve when you try to make it more accurate again. In IDA*, the
"depth" is a cutoff for {code f} values. When the {code f} value is
too large, the node won't even be considered ({emph i.e.,} it won't be
added to the OPEN set). The first time through you process very few
nodes. Each subsequent pass, you increase the number of nodes you
visit. If you find that the path improves, then you continue to
increase the cutoff; otherwise, you can stop. For more details, read
{link http://www.apl.jhu.edu/~hall/AI-Programming/IDA-Star.html these
lecture nodes on IDA*}.
I personally don't see much need for IDA* for finding paths on game
maps. ID algorithms tend to increase computation time while reducing
memory requirements. In map pathfinding, however, the "nodes" are
very small---they are simply coordinates. I don't see a big win from
not storing those nodes.
{h3 Dynamic weighting}
With dynamic weighting, you assume that at the beginning of your
search, it's more important to get (anywhere) quickly; at the
end of the search, it's more important to get to the goal.
{snippet
{line "f(p) = g(p) + w(p) * h(p)"}
}
There is a weight ({code w >= 1}) associated with the heuristic. As
you get closer to the goal, you decrease the weight; this decreases
the importance of the heuristic, and increases the relative importance
of the actual cost of the path.
{h3 Bandwidth search}
There are two properties about {emph Bandwidth Search} that some
people may find useful. This variation assumes that {code h} is
an {emph overestimate}, but that it doesn't overestimate by more than
some number {code e}. If this is the case in your search, then
the path you get will have a cost that doesn't exceed the best path's
cost by more than {code e}. Once again, the better you make
your heuristic, the better your solution will be.
Another property you get is that if you can drop some nodes in the
OPEN set. Whenever {code h+d} is greater then the true cost of
the path (for some {code d}), you can drop any node that has an
{code f} value that's at least {code e+d} higher than the
{code f} value of the best node in OPEN. This is a strange
property. You have a "band" of good values for {code f};
everything outside this band can be dropped, because there is a
guarantee that it will not be on the best path.
Curiously, you can use different heuristics for the two properties,
and things still work out. You can use one heuristic to guarantee
that your path isn't too bad, and another one to determine what to
drop in the OPEN set.
{emph Note:} When I wrote this in 1997, Bandwidth search looked potentially useful, but I've never used it and I don't see much written about it in the game industry, so I will probably remove this section. You can {link https://www.google.com/search?q=bandwidth+condition+heuristic search Google for more information}, especially from textbooks.
{h3 Bidirectional search}
Instead of searching from the start to the finish, you can start two
searches in parallel---one from start to finish, and one from finish
to start. When they meet, you should have a good path.
It's a good idea that will help in some situations. The idea behind
bidirectional searches is that searching results in a "tree" that fans
out over the map. A big tree is much worse than two small trees, so
it's better to have two small search trees.
The {emph front-to-front} variation links the two searches together.
Instead of choosing the best forward-search node---{code g(start,x) +
h(x,goal)}---or the best backward-search node---{code g(y,goal) +
h(start,y)}---this algorithm chooses a pair of nodes with the best {code
g(start,x) + h(x,y) + g(y,goal)}.
The {emph retargeting} approach abandons simultaneous searches in the
forward and backward directions. Instead, it performs a forward
search for a short time, chooses the best forward candidate, and then
performs a backward search---not to the starting point, but to that
candidate. After a while, it chooses a best backward candidate and
performs a forward search from the best forward candidate to the best
backward candidate. This process continues until the two candidates
are the same point.
{link https://arxiv.org/pdf/1703.03868.pdf Holte, Felner, Sharon,
Sturtevant's 2016 paper {emph Front-to-End Bidirectional Heuristic
Search with Near-Optimal Node Expansions}} is a recent result with a
near-optimal bidirectional variant of A*. {link
https://repub.eur.nl/pub/16100/ei2009-10.pdf Pijls and Post's 2009
paper {emph Yet another bidirectional algorithm for shortest paths}
proposes an unbalanced bidirectional A* that runs faster than balanced
bidirectional search.}
{h3 Dynamic A* and Lifelong Planning A*}
There are variants of A* that allow for changes to the world after the
initial path is computed. D* is intended for use when you don't have
complete information. If you don't have all the information, A* can
make mistakes; D*'s contribution is that it can correct those mistakes
without taking much time. LPA* is intended for use when the costs are
changing. With A*, the path may be invalidated by changes to the map;
LPA* can re-use previous A* computations to produce a new path.
{emph However}, both D* and LPA* require a lot of space---essentially
you run A* and keep around its internal information ({code
OPEN/CLOSED} sets, path tree, {code g} values), and then when the map
changes, D* or LPA* will tell you if you need to adjust your path to
take into account the map changes.
For a game with lots of moving units, you usually don't want to keep all
that information around, so D* and LPA* aren't applicable. They were
designed for robotics, where there is only one robot---you don't need
to reuse the memory for some other robot's path. If your game has
only one or a small number of units, you may want to investigate D* or
LPA*.
{list
{link http://www.frc.ri.cmu.edu/~axs/dynamic_plan.html Overview of D*}
{link http://www.frc.ri.cmu.edu/~axs/doc/icra94.ps D* Paper 1}
{link http://www.frc.ri.cmu.edu/~axs/doc/ijcai95.ps D* Paper 2}
{link http://idm-lab.org/project-a.html Lifelong planning overview}
{link http://csci.mrs.umn.edu/UMMCSciWiki/pub/CSci3903s03/KellysPaper/seminar.pdf Lifelong planning paper (PDF)}
{link http://idm-lab.org/applet.html Lifelong planning A* applet}
}
{h3 Jump Point Search}
Many of the techniques for speeding up A* are really about reducing
the number of nodes. In a square grid with uniform costs it's quite a
waste to look at all the individual grid spaces one at a time. One
approach is to build a graph of key points (such as corners) and use
that for pathfinding. However, you don't want to precompute a waypoint
graph, look at Jump Point Search, a variant of A* that can skip ahead
on square grids. When considering children of the current node for
possible inclusion in the OPEN set, Jump Point Search skips ahead to
faraway nodes that are visible from the current node. Each step is
more expensive but there are fewer of them, reducing the number of
nodes in the OPEN set. See {link
http://harablog.wordpress.com/2011/09/07/jump-point-search/ this blog
post} for details, {link http://zerowidth.com/2013/05/05/jump-point-search-explained.html this blog post} for a nice visual explanation, and {link
http://www.reddit.com/r/programming/comments/1es39b/speed_up_a_using_jump_point_search_explained/
this discussion on reddit} of pros and cons.
Also see {link
http://harablog.wordpress.com/2011/09/01/rectangular-symmetry-reduction/
Rectangular Symmetry Reduction}, which analyzes the map and {link {ref
Skip links} embeds jumps into the graph itself}. Both techniques were
developed for square grids. {link
http://adityasubramanian.weebly.com/jump-point-search-on-hexagonal-grids.html
Here's the algorithm extended for hexagonal grids}.
{h3 Theta*}
Sometimes grids are used for pathfinding because the map is made on a
grid, not because you actually want movement on a grid. A* would run
faster and produce better paths if given a graph of key points (such
as corners) instead of the grid. However if you don't want to
precompute the graph of corners, you can use Theta*, a variant of A*
that runs on square grids, to find paths that don't strictly follow
the grid. When building {emph parent} pointers, Theta* will point
directly to an ancestor if there's a line of sight to that node, and
skips the nodes in between. Unlike {link {ref Path smoothing} path
smoothing}, which straightens out paths after they're found by A*,
Theta* can analyze those paths as part of the A* process. This can
lead to shorter paths than postprocessing a grid path into an
any-angle path. {link
http://aigamedev.com/open/tutorials/theta-star-any-angle-paths/ This
article} is a reasonable introduction to the algorithm; also see {link
http://aigamedev.com/open/tutorial/lazy-theta-star/ Lazy Theta*}.
The ideas from Theta* likely can be applied to navigation
meshes as well.
Also see {link
http://www.aaai.org/ocs/index.php/AIIDE/AIIDE11/paper/download/4055/4436
Block A*}, which claims to be much faster than Theta* by using a
hierarchical approach.
{h2 Dealing with moving obstacles}
{filename MovingObstacles.html}
{set meta-description "Pathfinding and how to avoid moving obstacles such as enemy units."}
{set shortname Moving Obstacles}
A pathfinding algorithm will compute a path around stationary
obstacles, but what if the obstacles move? By the time a unit reaches
a particular point, an obstacle may no longer be there, or a new
obstacle may be there. If the typical obstacle can be routed around,
use a separate obstacle avoidance algorithm (steering) along with your
pathfinder. The pathfinder will find the desired path, and then while
following it, move around obstacles. If however obstacles can cause
the path to change significantly, consider using the pathfinder for
obstacle avoidance.
{h3 Recalculating paths}
As time passes we expect the game world to change. A path found some
time ago may no longer be the optimal path. It may be worth updating
old paths with new information. Listed below are some criteria that
could be used for determining when a recalculation is needed:
{list
{hbox Every {emph N} steps: this guarantees that the information
used to calculate the path is not more than {emph N} steps old.}
{hbox Whenever extra CPU time is available: this allows dynamic
adjustment of path quality; as more units are deployed, or if the
game is running on a slower computer, CPU usage per unit can be
decreased.}
{hbox Whenever the unit turns a corner or passes a waypoint.}
{hbox Whenever the world near the unit has changed.}
}
The main drawback of path recalculation is that a lot of path
information is thrown away. For example, if the path is 100 steps
long and it is recalculated every 10 steps, the total number of path
steps is 100+90+80+70+60+50+40+30+20+10 = 550. For a path of {emph M}
steps, approximately {emph M}^{super 2} path steps are computed over
time. Therefore path recalculation is not a good idea if you expect
to have many long paths. It would be better to reuse the path
information instead of throwing it away.
{h3 Path splicing}
When a path needs to be recalculated, it means the world is changing.
Given a changing world, nearby parts of the map are better known than
faraway parts of the map. We can follow a {emph local repair
strategy}: find a good path nearby, and assume the path farther away
need not be recomputed until we get closer to it. Instead of
recalculating the entire path, we can recalculate the first {emph M}
steps of the path:
{enumerate
{hbox Let {code "p[1]..p[N]"} be the remainder of the path
({emph N} steps)}
{hbox Compute a new path from {code "p[1]"} to {code "p[M]"}}
{hbox {emph Splice} this new path into the old path by removing
{code "p[1]..p[M]"} and inserting the new path in its place}
}
{figure {img 400 300 "mtn_path.png"}}
Since {code "p[1]"} and {code "p[M]"} are fewer than {emph M} steps
apart, it's unlikely that the new path will be long. Unfortunately,
situations can arise in which the new path is long and not very good.
The accompanying figure shows such a situation. The original red path
is 1-2-3-4; brown areas are obstacles. If we reach 2 and discover
that the path from 2 to 3 has been blocked, path splicing would
replace 2-3 with the green path 2-3-5 and splice it in, resulting in
the unit moving along path 1-2-5-3-4. We can see this is not a good
path; the blue path 1-2-5-4 would be better.
Bad paths can often be detected by looking at the length of the new
path. If it is significantly longer than {emph M}, it could be
bad. A simple solution is to add a limit (maximum path length) to
the path finding algorithm. If a short path isn't found, the
algorithm returns an error code; in this case, use path recalculation
instead of path splicing to get a path such as 1-2-5-4.
{margin-note
{vbox {strong Implementation Note:}
{hbox Store the path in
{emph reverse} order: it is easy to remove
the beginning of the path and splice in a new
path with a different length; because both
operations occur at the {emph end} of the array.
Essentially you treat the array as a {emph stack}
where the top element is the next move to make.}
}
}
For cases not involving these situations, for a path with {emph N}
steps, path splicing will compute {emph 2N} to {emph 3N} path steps,
depending on how often a new path is spliced in. This is a fairly low
cost for the ability to respond to changes in the world. Surprisingly,
the cost is independent of {emph M}, the number of steps for splicing.
Instead of affecting CPU time, {emph M} controls a tradeoff between
responsiveness and path quality. If {emph M} is high, the unit's
movement will not respond quickly to changes in the map. If {emph M}
is too low, the paths being spliced out may be too short to allow the
replacement path to go around the obstacle cleanly; more suboptimal
paths (such as 1-2-5-3-4) will be found. Try different values of
{emph M} and different criteria for splicing (such as every 3/4 {emph
M} steps) to see what's right for your map.
Path splicing is significantly faster than path recalculation, but it
does not respond well to major changes in the path. It is possible to
detect many of these situations and use path recalculation instead.
It also has a few variables that can be adjusted, such as {emph M} and
the choice of when to find a new path, so it can be adjusted (even at
run-time) for different conditions. Path splicing also does not handle
situations where the units must coordinate in order to pass each
other.
{h3 Watching for map changes}
An alternative to recalculating all or part of the path at certain
intervals is to have changes to the map trigger a recalculation. The
map can be divided into regions, and every unit can express an
interest in certain regions. (All the regions that contain part of
the path could be of interest, or only nearby regions that contain
part of the path.) Whenever an obstacle enters or leaves a region,
that region is marked {emph changed}, and all units that have an
interest in that region are notified, so that paths can be
recalculated to take into account the change in obstacles.
Many variations of this technique are possible. For example, instead
of immediately notifying the units, only notify them at regular
intervals. Many changes can be grouped into one notification, so that
excessive path recalculations are not needed. Another example is for
the unit to poll the regions instead of the regions notifying the
unit.
Watching for map changes allows units to avoid recalculation whenever
the obstacles on the map do not change, so consider it if you have
many regions that do not change often.
{h3 Predicting obstacle movement}
If obstacle movement can be predicted, it is possible to take into
account future position of obstacles for pathfinding. An algorithm
such as A* has a cost function that determines how difficult it is to
pass a point on the map. A* can be modified to keep track of the time
required to reach a point (determined by current path length), and
this time can be passed in to the cost function. The cost function
can then take time into account, and use the predicted obstacle
position at that time to determine whether the map space is
impassable. This modification is not perfect, however, as it will not
take into account the possibility of waiting at a point for the
obstacle to move out of the way, and A* is not designed to
differentiate between paths along the same route, but at different
points in time.
{link http://www0.cs.ucl.ac.uk/staff/d.silver/web/Publications_files/coop-path-AIWisdom.pdf Cooperative Pathfinding A*} formalizes this, building a table of the paths of all other units and treating them as obstacles. Also see {link
http://www0.cs.ucl.ac.uk/staff/D.Silver/web/Applications_files/coop-path-AIIDE.pdf
the Windowed Hierarchical Cooperative A* algorithm}.
{h2 Space used by precalculated paths}
{filename SpaceUsage.html}
{set shortname Space}
Sometimes, it's not the time needed to calculate a path, but the space
used by paths for hundreds of units that is the limiting factor.
Pathfinders require space for the algorithm to run, plus space to
store a path. The temporary space required for the algorithm to run
(with A*, the OPEN and CLOSED sets) typicaly is larger than the space
required to store the resulting path. By restricting your game to
compute only one path at a time, you can minimize the amount of
temporary space needed. In addition, the {link {ref Set
representation} choice of data structure} for your OPEN and CLOSED
sets can make a big difference for minimizing temporary space. This
section will instead focus on minimizing the space used by the
resulting paths.
{h3 Locations vs. directions}
A path can be either locations or directions. Locations take more
space, but have the advantage that it is easy to determine an
arbitrary location or direction in the path without traversing the
path. When storing directions, only the direction can be determined
easily; the location can only be determined by going through the
entire path, following the directions. In a typical grid map,
locations may be stored with two 16-bit integers, making each step
take 32 bits. There are far fewer directions, however, so they can
take less space. If the unit can move in only four directions, each
step takes only 2 bits; if the unit can move in six or eight
directions, each step takes 3 bits. Either of these is a significant
savings over storing locations in the path. Hannu Kankaanpaa suggests
that you can further reduce the space requirement by storing the {emph
relative} direction ("turn right 60 degrees") instead of the absolute
direction ("go north"). Some relative directions may not make sense
for some types of units. For example, if your unit is moving north,
it's unlikely that the next step is to go south. In a six directional
game, you have only five meaningful directions. On some maps, perhaps
only three of those directions (straight, left 60 degrees, right 60
degrees) make sense, but on other maps, turning right 120 degrees may
be a valid move (for example, when going up a steep mountain path with
switchbacks).
{h3 Path compression}
Once a path can be found, it can be compressed in some way. A general
purpose compression algorithm could be used, but will not be discussed
here. A compression algorithm specific to paths could be used to
shorten either location-based paths or direction-based paths. Before
deciding, look at typical paths in your game to decide which kind of
compression will work best. In addition, consider ease of
implementation (and debugging), the size of the code, and whether it
really matters. If you have a limit of 300 units and only 50 are
walking at any one time, and paths are short (100 steps), the total
memory requirement might only be <50k anyway, and not worth worrying
about compression.
{h4 Location storage}
In maps where obstacles rather than terrain are the main influence in
determining paths, there may be many straight-line segments in the
path. If this is the case, then a path need contain only the
endpoints (sometimes called {emph waypoints}) of those line segments.
Movement consists of examining the next point on the path and moving
in a straight line towards it.
{h4 Direction storage}
When directions are stored, it may be the case that a particular
direction is followed many times in a row. You can take advantage of
that common pattern to store the path in less space.
One way to store the path is to store both a direction and a number
which indicates how many times the unit should move in that direction.
Unlike the optimization for location storage, this optimization can
make things worse if a direction is not taken many times in a row.
Also, for many straight lines where location compression is useful,
direction compression is not, since the line may not be aligned with
one of the walking directions. With relative directions, you can
eliminate "keep going forward" as a possible direction. Hannu
Kankaanpaa points out that with an eight direction map, you can
eliminate forwards, backwards, and the 135 degree left and right turns
(assuming your map allows it), and you can then store each direction
with only two bits.
Another way to store the path is to use variable length encoding.
The idea is to use a single bit (0) for the most
common step: go straight. Use a 1 to mark a turn, and follow the 1 by
some number of bits to represent the turn. In a four directional map,
you can turn only left or right, so you might use 10 for left and 11
for right.
Variable length encoding is more general and may compress better than
run length encoding for mixed paths, but not as well for long straight
paths. The sequence (north, straight six steps, turn left, straight
three steps, turn right, straight five steps, turn left, straight two
steps) is represented as [(NORTH, 6), (WEST, 3), (NORTH, 5), (WEST,
2)] with run length encoding. If each direction is two bits and each
distance is eight bits, this path requires 40 bits to store. With
variable length encoding, you use one bit for each step and two bits
for each turn---[NORTH 0 0 0 0 0 0 10 0 0 0 11 0 0 0 0 0 10 0 0]---a
total of 24 bits. If the initial direction and each turn imply one
step, you can save one bit per turn, resulting in 20 bits to store the
path. However, longer paths can take more space with variable length
encoding. The sequence (north, straight two hundred steps) is
[(NORTH, 200)] with run length encoding, a total of 10 bits. The same
sequence with variable length encoding is [NORTH 0 0 ...], a total of
202 bits.
{h3 Computed waypoints}
A {emph waypoint} is a point along a path. Instead of storing every
step along the way, after pathfinding a post-processing step can
collapse multiple steps into a single waypoint, usually at places
where the path changes direction or at major locations like cities.
The movement algorithm will then follow a path between waypoints.
{h3 Limited path length}
Given that map conditions or orders may change, it may not make sense
to store a long path, since at some point the remainder of the path
may not be of any use. Each unit can store a fixed number of steps at
the beginning of the path, and then use path recalculation when the
path has almost been traversed. This approach allows for control of
the amount of data used per unit.
{h3 Summary}
Paths can potentially take up a lot of space in a game, especially
when the paths are long and there are many units present. Path
compression, waypoints, and beacons reduce the space requirements by
storing many steps in a small amount of data. Waypoints rely on
straight-line segments being common so that we have to store only the
endpoints, while beacons rely on there being well-known paths
calculated beforehand between specially marked places on the map. If
paths still take up too much space, the path length can be limited,
resulting in the classic time-space tradeoff: to save space,
information can be forgotten and recalculated later.
{h1 Other topics}
There are many other topics related to pathfinding.
{h2 Map representations}
{filename MapRepresentations.html}
Through most of this document I've assumed that A* was being used on a
grid of some sort, where the "nodes" given to A* were grid locations
and the "edges" were directions you could travel from a grid location.
However, A* was designed to work with arbitrary graphs, not only
grids. There are a variety of map representations that can be used
with A*.
{strong The map representation can make a huge difference in the
performance and path quality.}
Pathfinding algorithms tend to be worse than {emph linear}: if you
double the distance needed to travel, it takes {emph more} than twice
as long to find the path. You can think of pathfinding as searching
some area like a circle---when the circle's diameter doubles, it has
{emph four} times the area. In general, the fewer nodes in your map
representation, the faster A* will be. Also, the more closely your
nodes match the positions that units will move to, the better your
path quality will be.
The map representation used for pathfinding does not have to be the
same as the representation used for other things in the game. However,
using the same representation is a good starting point, until you find
that you need better paths or more performance.
{h3 Grids}
{id S1} A grid map uses a uniform subdivision of the world into small regular
shapes sometimes called "tiles". Common grids in use are {link
http://www-cs-students.stanford.edu/~amitp/game-programming/grids/
square, triangular, and hexagonal}. Grids are simple and easy to
understand, and many games use them for world representation; thus, I
have focused on them in this document.
{figure {img 181 181 http://www-cs-students.stanford.edu/~amitp/game-programming/grids/square-grid.png}}
I used grids for {link
http://www-cs-students.stanford.edu/~amitp/games.html BlobCity}
because the movement costs were different in each grid location. If
your movement costs are uniform across large areas of space (as in the
examples I've used in this document), then using grids can be quite
wasteful. There's no need to have A* move one step at a time when it
can skip across the large area to the other side. Pathfinding on
a grid also yields a path on grids, which can be {link {ref Unit
movement} postprocessed to remove the jagged movement}. However, if your
units aren't constrained to move on a grid, or if your world doesn't
even use grids, then pathfinding on a grid may not be the best choice.
{h4 Tile movement}
{margin-note {img 151 151 http://www-cs-students.stanford.edu/~amitp/game-programming/grids/square-rel-face-face.png}}
Even within grids, you have a choice of {link
http://www-cs-students.stanford.edu/~amitp/game-programming/grids/
tiles, edges, and vertices} for movement. Tiles are the default
choice, especially for games in which units only move to the center of
a tile. In this diagram, the unit at A can move to any of the spots
marked B. You may also wish to allow diagonal movement, with the same
or higher movement cost.
If you're using grids for pathfinding, your units are not constrained
to grids, {emph and} movement costs are uniform, you may want to {link
{ref Path smoothing} straighten the paths} by moving in a straight
line from one node to a node far ahead when there are no obstacles
between the two.
{h4 Edge movement}
{margin-note {img 151 151 http://www-cs-students.stanford.edu/~amitp/game-programming/grids/square-rel-face-edge.png}}
If your units can move anywhere within a grid space, or if the tiles
are large, think about whether edges or vertices would be a better
choice for your application.
A unit usually enters a tile at one of the edges (often in the middle)
and exits the tile at another edge. With pathfinding on tiles, the
unit moves to the center of the tile, but with pathfinding on edges,
the unit will move directly from one edge to the other. I wrote a
{link
http://theory.stanford.edu/~amitp/game-programming/road-applet/roads.html
java applet demo of road drawing between edges}; that might help
illustrate how edges can be used.
{h4 Vertex movement}
{margin-note
{img 117 116 http://www-cs-students.stanford.edu/~amitp/game-programming/grids/square-rel-vertex-vertex.png}}
Obstacles in a grid system typically have their corners at
vertices. The shortest path around an obstacle will be to go around
the corners. With pathfinding on vertices, the unit moves from corner
to corner. This produces the least wasted movement, but paths need to
be adjusted to account for the size of the unit.
{h3 Polygonal maps}
The most common alternative to grids is to use a polygonal
representation. If the movement cost across large areas is uniform,
and if your units can move in straight lines instead of following a
grid, you may want to use a non-grid representation. You can use a
non-grid graph for pathfinding even if your game uses a grid for other
things.
Here's a simple example of one kind of polygonal map
representation. In this example, the unit needs to move around two
obstacles:
{figure {img 450 256 polygon-graph.png}}
Imagine how your unit will move in this map. The shortest path will be
between corners of the obstacles. So we choose those corners (red
circles) as the key "navigation points" points to tell A* about; these
can be computed once per map change. If your obstacles are aligned on
a grid, the navigation points will be aligned with the vertices of the
grid. In addition, the start and end points for pathfinding need to be
in the graph; these are added once per call to A*.
In addition to the navigation points, A* needs to know which points
are connected. The simple algorithm is to build a {link
http://en.wikipedia.org/wiki/Visibility_graph {strong visibility
graph}}: pairs of points that can be seen from each other. The simple
algorithm may be fine for your needs, especially if the map doesn't
change during gameplay, but you may need a more sophisticated
algorithm if the simple one is too slow. In addition, since we have
added the start and end navigation points to the graph, we check line
of sight from those to existing vertices and each other, and add edges
where needed.
The third piece of information A* needs is travel times between the
points. That will be {link {ref Manhattan distance} manhattan
distance} or {link {ref Diagonal distance} diagonal grid distance} if
your units move on a grid, or {link {ref Euclidean distance} straight
line distance} if they can move directly between the navigation
points.
A* will then consider paths from navigation point to navigation point.
The pink line is one such path. This is {emph much} faster than
looking for paths from grid point to grid point, when you have only a
few navigation points, instead of lots of grid locations. When there
are no obstacles in the way, A* will do very well---the start point
and end point will be connected by an edge, and A* will find that path
immediately, without expanding any other navigation points. Even when
there are obstacles to consider, A* will jump from corner to corner
until it finds the best path, which will still be much faster than
looking for a path from a grid location to another.
Wikipedia has more about {link
http://en.wikipedia.org/wiki/Visibility_graph visibility graphs} from
the robotics literature. {link http://www.cs.kent.edu/~dragan/ST-Spring2016/visibility%20graphs.pdf This slide deck} is a nice introduction as well.
{h4 Managing complexity}
The above example was rather simple and the graph is reasonable. In
some maps with lots of open areas or long corridors, a problem with
visibility graphs becomes apparent. A major disadvantage of
connecting every pair of obstacle corners is that if there are N
corners (vertices), you have up to N{super 2} edges. This example
demonstrates the problem:
{figure {img 401 294 polygon-unmanageable.png}}
These extra edges primarily affect memory usage. Compared to grids,
these edges provide "shortcuts" that greatly speed up pathfinding.
There are algorithms for simplifying the graph by removing redundant
edges. However, even after removing redundancies, there will still be
a large number of edges.
Another disadvantage of the visibility graphs is that we have to add
start/end nodes along with their new edges to the graph for every
invocation of A*, and then remove them after we find a path. The nodes
are easy to add but adding edges requires line of sight from the new
nodes to all existing nodes, and that can be slow in large maps. One
optimization is to only look at nearby nodes. Another option is to use
a {emph reduced visibility graph} that removes the edges that aren't
tangent to both vertices (these will never be in the shortest path).
{h3 Navigation Meshes}
Instead of representing the {emph obstacles} with polygons, we can
represent the {emph walkable} areas with non-overlapping polygons,
also called a {emph navigation mesh}. The walkable areas can have
additional information attached to them (such as "requires swimming"
or "movement cost 2"). Obstacles don't need to be stored in this
representation.
The previous example becomes this:
{figure {img 481 413 polygon-navmesh-plain.png}}
We can then treat this much like we treat a grid. As with a grid, we
have a choice of using polygon centers, edges, or vertices as
navigation points.
{h4 Polygon movement}
As with grids, the center of each polygon provides a reasonable set of
nodes for the pathfinding graph. In addition, we have to add the start
and end nodes, along with an edge to the center of the polygon we're
in. In this example, the yellow path is the what we'd find using a
pathfinder through the polygon centers, and the pink path is the
ideal path.
{figure {img 481 413 polygon-navmesh-faces.png}}
The visibility graph representation would produce the pink
path, which is ideal. Using a navigation mesh makes the map manageable
but the path quality suffers. We can make the path look better
by smoothing it.
{h4 Polygon edge movement}
Moving to the center of the polygon is usually unnecessary. Instead,
we can move through the edges between adjacent polygons. In this
example, I picked the center of each edge. The yellow path is what
we'd find with a pathfinder through the edge centers, and it compares
pretty well to the ideal pink path.
{figure {img 481 413 polygon-navmesh-edges.png}}
You can pick more points along the edge to produce a better path, at
increased cost.
{h4 Polygon vertex movement}
The shortest way around an obstacle is to go around the corner. This
is why we used corners for the visibility graph representation. We can
use vertices with navigation meshes:
{figure {img 481 413 polygon-navmesh-vertices.png}}
There's only one obstacle in the way in this example. When we need to
go around the obstacle, the yellow path goes through a vertex, as
the pink (ideal) path does. However, whereas the visibility graph
approach would have a straight line from the start point to the corner
of the obstacle, the navigation mesh adds some more steps. These steps
typically should not go through vertices, so the path looks unnatural,
with "wall hugging" behavior.
{h4 Hybrid movement}
There aren't any restrictions on what parts of each polygon can be
made into navigation points for pathfinding. You can add multiple
points along an edge, and the vertices are good points too. Polygon
centers are rarely useful. Here's a hybrid scheme that uses both the
edge centers and vertices:
{figure {img 481 413 polygon-navmesh-edges-and-vertices.png}}
Note that to get around the obstacle, the path goes through a vertex,
but elsewhere, it can go through edge centers.
{h4 Path smoothing}
Path smoothing is fairly easy with the resulting paths, as long as the
movement costs are constant. The algorithm is simple: if there's line of
sight from the navigation point {emph i} to point {emph i+2}, remove
point {emph i+1}. Repeat this until there is no line of sight between
adjacent points in the path.
What will be left is only the navigation points that go around the
corners of obstacles. These are vertices of the navigation mesh. If
you use path smoothing, there's no need to use edge or polygon centers
as navigation points; use only the vertices.
In the above examples, path smoothing would turn the yellow path into
the pink one. However, the pathfinder has no knowledge of these
shorter paths, so its decisions won't be optimal. Shortening the path
found in an approximate map representation (navigation meshes) will
not always produce paths that are as good as those found in a more
exact representation (visibility graphs).
{h3 Hierarchical}
{id S4} A flat map has but one level in its representation. It's rare for
games to have only one level---often there is a "tile" level and then
a "sub-tile" level in which objects can move within a tile. However,
it's common for {emph pathfinding} to occur on only the larger
level. You can also add higher levels such as "rooms".
Having fewer nodes in the map representation is better for pathfinding
speed. One way to reduce the problem is to have multiple levels of
searching. For example, to get from your home to a location in
another city, you would find a path from your chair to your car, from
the car to the street, from the street to a freeway, from the freeway
to the edge of the city, from there to the other city, then to a
street, to a parking lot, and finally to the door of the destination
building. There are several levels of searching here:
{list
{hbox At the {emph street} level, you are concerned with walking
from one location to a nearby location, but you do not go out
on the street.}
{hbox At the {emph city} level, you go from one street to another
until you find the freeway. You do not worry about going into
buildings or parking lots, nor do you worry about going on freeways.}
{hbox At the {emph state} level, you go from one city to another
on the freeway. You do not worry about streets within cities
until you get to your destination city.}
}
Dividing the problem into levels allows you to ignore most of your
choices. When moving from city to city, it is quite tedious to
consider every street in every city along the way. Instead, you
ignore them all, and only consider freeways. The problem becomes
small and manageable, and solving it becomes fast.
A hierarchical map has many levels in its representation. A
heterogenous hierarchy typically has a fixed number of levels, each
with different characteristics. Ultima V, for example, has a "world"
map, on which are cities and dungeons. You can enter a city or
dungeon and be in a second map level. In addition, there are "layers"
of worlds on top of one another, making for a three-level
hierarchy. The levels can be of different types (tile grids,
visibility, navigation mesh, waypoints). A homogeneous hierarchy has an
arbitrary number of levels, each with the same characteristics. Quad
trees and oct trees can be considered to be homogeneous hierarchies.
In a hierarchical map, pathfinding may occur on several levels. For
example, if a 1024x1024 world was divided into 64x64 "zones", it may
be reasonable to find a path from the player's location to the edge of
the zone, then from zone to zone until reaching the desired zone, then
from the edge of that zone to the desired location. At the coarser
levels, long paths can be found more easily because the pathfinder
does not consider all of the details. When the player actually walks
across each zone, the pathfinder can be invoked again to find a short
path through that zone. By keeping the problem size small, the
pathfinder can run quicker.
You can use multiple levels with graph-searching algorithms such as
A*, but you do not need to use the same algorithm at each level. For
small levels, you may be able to precompute the shortest path between
all pairs of nodes (using Floyd-Warshall or other algorithms). In
general, pathfinding in a hierarchical map will not produce optimal
paths, but they are usually close.
A similar approach is to use varying resolution. First, plot a path
with low resolution. As you get closer to a point, refine the path
with a higher resolution. This approach can be used with {link {ref
Path splicing} path splicing} to avoid moving obstacles.
Some papers to read: {link
http://aigamedev.com/open/review/near-optimal-hierarchical-pathfinding/
Near-Optimal Hierarchical Pathfinding (HPA*)} builds a two-level
hierarchy on a grid, {link
http://aaai.org/ocs/index.php/AIIDE/AIIDE10/paper/view/2139/2547
Pathfinding for Dragon Age:Origins} explains several hierarchical
approaches used in a commercial game, {link
http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.138.7223
Ultrafast shortest-path queries with linear-time preprocessing} by
using "transit nodes" for a road graph [PDF], {link
http://www.aaai.org/ocs/index.php/AIIDE/AIIDE12/paper/view/5459/5688
Transit nodes for grid maps in games}, {link
http://scholar.google.com/scholar?q=Hierarchical+A*%3A+Searching+Abstraction+Hierarchies+Efficiently&hl=en&btnG=Search&as_sdt=1%2C5&as_sdtp=on
Hierarchical A*: Searching Abstraction Hierarchies Efficiently},
{link http://algo2.iti.kit.edu/schultes/hwy/schultes_diss.pdf Route
Planning in Road Networks} (Dominic Schultes's PhD thesis),
Hierarchical Annotated A* ({link
http://harablog.wordpress.com/2009/01/29/clearance-based-pathfinding/
part 1} and {link
http://harablog.wordpress.com/2009/02/05/hierarchical-clearance-based-pathfinding/
part 2} and {link http://code.google.com/p/ahastar/ source code}).
{h3 Wraparound maps}
If your world is spherical or toroidal, then objects can "wrap" around
from one end of the map to the other. The shortest path could lie in
any direction, so all directions must be explored. If using a grid,
you can adapt the {link {ref Euclidean distance}} heuristic to consider
wrapping around. Instead of {code abs(x1 - x2)} you can use {code
min(abs(x1 - x2), (x1+mapsize) - x2, (x2+mapsize) - x1)}. This will
take the {code min} of three options: either staying on the map
without wrapping, wrapping when {code x1} is on the left side, or
wrapping when {code x2} is on the left side. You'd do the same for
each axis that wraps. Essentially you calculate the heuristic assuming
that the map is adjacent to copies of itself.
{h3 Connected Components}
In some game maps, there's no path between the source and
destination. If you ask A* to find a path, it will end up exploring a
large subset of the graph before it determines that there's no
path. If the map can be analyzed beforehand, mark each of the {link
http://en.wikipedia.org/wiki/Connected_Component_Labeling connected
subgraphs} with a different marker. Then, before looking for a path,
check if the source and destination are both in the same subgraph. If
not, then you know there's no path between them. Hierarchical
pathfinding can also be useful here, especially if there are one way
edges between subgraphs.
{h3 Road maps}
If your units can only move on roads, you may want to consider giving
A* the road and intersection information. Each intersection will be a
node in the graph, and each road will be an edge. A* will find paths
from intersection to intersection, which is much faster than using a
grid representation.
For some applications, your units may not start and end on
intersections. To handle this case, each time you run A*, you will
need to modify the node/edge graph (this is the same technique we use
with visibility graph and navigation mesh map representations). Add
the starting and ending points as new nodes, and add edges between
these points and their nearest intersections. After pathfinding,
remove these extra nodes and edges from the graph so that the graph is
ready to be used for the next invocation of A*.
{figure {img 400 408 road-graph.png}}
In this diagram, the intersections become nodes in the pathfinding
graph for A*. The edges are the roads between the nodes, and these
edges should be given the driving distance along each road. In the
"roads as edges" framework, you can incorporate one-way roads as
unidirectional edges in the graph.
If you want to assign costs to turning, you can extend the framework a
bit: instead of nodes being locations, consider nodes to be a
pair (a point in {emph state space}),
where the direction indicates what direction you were facing when you
{emph arrived} at that location. Replace edges from X to Y with edges
from to to represent a straight drive, and from to to represent a "turn". Each edge represents {emph
either} a straight drive or a turn, but not both. You can then assign
costs to the edges representing turns.
If you also need to take into account turn limitations, such as "only
right turns", you can use a variation of this framework in which the
two types of edges are always combined. Each edge represents an
optional turn followed by a straight drive. In this framework, you
can represent restrictions like "you can only turn right": include an
edge from to for driving straight,
and an edge from to for the right
turn followed by a drive, but {emph don't} include to
anything west, because that would mean a left turn, and don't include
anything south, because that would mean a U-turn.
In this framework, you can model a large city downtown, in which you
have one-way streets, turn restrictions at certain intersections
(often prohibiting U-turns and sometimes prohibiting left turns), and
turn costs (to model slowing down and waiting for pedestrians before
you turn right). Compared to grid maps, A* can find paths in road
graphs environment fairly quickly, because there are few choices to
make at each graph node, and there are relatively few nodes in the
map.
For large scale road maps, be sure to read {link
http://research.microsoft.com/apps/pubs/default.aspx?id=154937
Goldberg and Harrelson's paper} on ALT (A*, Landmarks, Triangle
inequality) (PDF is {link
http://research.microsoft.com/pubs/64511/tr-2004-24.pdf here}, or
{link http://research.microsoft.com/apps/pubs/default.aspx?id=60764
this paper}.
{h3 Skip links}
A pathfinding graph constructed from a grid typically assigns a vertex
to each location and an edge to each possible movement from a location
to an adjacent location. The edges are not constrained to be between
adjacent vertices. A "skip link" or "shortcut link" is an edge between
non-adjacent vertices. It serves to shortcut the pathfinding process.
What should the movement cost be for a skip link? There are two approaches:
{list
{hbox Make the cost match the movement cost of the best path. This
preserves nice properties of A*, like finding optimal paths. To
give A* a nudge in the right direction, {link {ref Breaking ties}
break the tie} between the skip link and the regular links by
reducing the skip link's cost by 1% or so.}
{hbox Make the cost match the heuristic cost. This makes a much stronger
impact on performance but you give up optimal paths.}
}
Adding skip links is an approximation of a hierarchical map. It takes
less effort but can often give you many of the same performance
benefits.
For dungeon room-and-corridor grid maps, {link
http://aigamedev.com/open/tutorial/symmetry-in-pathfinding/
Rectangular Symmetry Reduction and Jump Point Search} offer two
different ways to build skip links. Rectangular Symmetry Reduction
statically builds additional edges (which they call {emph macro
edges}) and then uses standard graph search; Jump Point Search
dynamically builds longer edges as part of the graph search
algorithm. For road maps and other types of graphs {link
https://en.wikipedia.org/wiki/Contraction_hierarchies Contraction
Hierarchies} are worth looking at; see {link
http://algo2.iti.kit.edu/schultes/hwy/contract.pdf this paper}. When
I wrote this page in 1997 I didn't know about contraction
hierarchies, or I would've used that terminology instead
of making up the term “skip links”.
{h3 Waypoints}
A {emph waypoint} is a point along a path. Waypoints can specific to
each path or be part of the game map. Waypoints can be entered
manually or computed automatically. In many real-time strategy games,
players can manually add path-specific waypoints by shift-clicking.
When automatically computed along a path, waypoints can be used to
{link {ref Computed waypoints} compress the path representation}. Map
designers can {link {ref Marked beacons} manually add waypoints} (or
"beacons") to a map to mark locations that are along good paths or an
algorithm can be used to automatically mark waypoints on a map.
Since the goal of skip links is to make pathfinding faster when those
links are used, skip links should be placed between designer-placed
waypoints. This will maximize their benefit.
If there are not too many waypoints, the shortest paths between each
pair of waypoints can be precomputed beforehand (using an all-pairs
shortest path algorithm, not A*). The common case will then be a unit
following its own path until it reaches a waypoint, and then it will
follow the precomputed shortest path between waypoints, and finally it
will get off the waypoint highway and follow its own path to the goal.
Using waypoints or skip links with false costs can lead to suboptimal
paths. It is sometimes possible to smooth out a bad path in a
post-processing step or in the movement algorithm.
{h3 Graph Format Recommendations}
Start by pathfinding on the game world representation you already
use. If that's not satisfactory, consider transforming the game world
into a different representation for pathfinding.
In many grid games, there are large areas of maps that have uniform
movement costs. A* doesn't "know" this, and wastes effort exploring
them. Creating a simpler graph (navigation mesh, visibility graph, or
hierarchical representation of the grid map) can help A*.
The {link {ref Polygonal maps} visibility graph representation}
produces the best paths when movement costs are constant, and allows
A* to run rather quickly, but can use {link {ref Managing complexity}
lots of memory for edges}. Grids allow for fine variation in movement
costs ({link {ref Terrain} terrain}, {link {ref Altitude} slope},
{link {ref Enemies and friendly units} penalties for dangerous areas},
etc.), use very little memory for edges, but use lots of memory for
nodes, and pathfinding can be slow. Navigation meshes are in
between. They work well when movement costs are constant in a larger
area, allow for some variation in movement costs, and produce
reasonable paths. The paths are not always as short as with visibility
graph representation, but they are usually reasonable. {link {ref
Hierarchical} Hierarchical maps} use multiple levels of representation
to handle both coarse paths over long distances and detailed paths
over short distances.
You can read more about navigation meshes {link
https://web.archive.org/web/20110723003329/http://www.ai-blog.net/archives/000152.html in this well-illustrated
article}. Note that the article is comparing (a) keeping walkable
polygons to keeping only navigation points, and (b) moving along
vertices to moving along polygon centers. These are mostly
orthogonal. Keeping the walkable polygons is nice for dynamically
adjusting the path later, but not needed in all games. Using vertices
is better for obstacle avoidance, and if you're using path smoothing
it won't negatively affect path quality. Without path smoothing, edges
might perform better, so consider either edges or edges+vertices.
An alternative to building a separate non-grid representation of a
grid map is to use a variant of A* that better understands grid maps
with uniform costs. See {link {ref Jump Point Search} Jump Point
Search} to speed up A* on square grids and {link {ref Theta*} Theta*} to
generate non-grid movement on a grid.
{h2 Long and short term goals}
{filename Goals.html}
{set shortname Goals}
I've concentrated on the task of finding paths from one place to
another. However, an equally important question is: once I have a
path, how do I move along it? The most obvious answer is moving in a
straight line from one location to the next. However, you might also
want to move in a curve, or have multiple levels of movement. You may
want to treat the locations as a low-priority goal from which you
deviate. A higher level question is: where do you want to go? Unless
you first answer the higher level question, pathfinding is not very
useful. Certainly, one form of goal setting is asking the user to
click on the destination. However, you may have automated tasks as
well---exploring, spying, attacking, and building are common ones.
{h3 Unit movement}
I've concentrated on pathfinding, which reduces the problem of moving
from one location to another with the many smaller problems of moving
from one space to an adjacent space.
You could move in a straight line from one location to the next but
there are alternatives. Consider the four movement paths on this
diagram:
{figure {img 400 300 "splines.png"}}
The red path is the standard approach: move from the center of one
square to the center of the next. The green path is somewhat better:
move in straight lines between the {emph edges} between the tiles,
instead of the center of the tiles. You might also try moving in
straight lines between the {emph corners} of tiles. The blue paths
use splines, with dark blue being low order splines and light blue
being a higher order spline.
Lines between corners and edges of tiles will be the shortest
solution. However, the splines can make your units seem less
mechanical and more "alive". It's a cheap trick but not a great one.
There are better ways to handle movement.
If your units cannot turn easily, you may want to take that into
account when plotting a movement path. Craig Reynolds has a great
page about {link http://www.red3d.com/cwr/steer/ steering} that has a paper
about steering and Java applets demonstrating various behaviors. If
you have more than one unit moving along a path, you may also want to
investigate {link http://www.red3d.com/cwr/boids/ flocking}. Craig
recommends that instead of treating paths as a list of places your
unit must visit, you treat paths as ``a guideline, from which you
deviate reactively as conditions require.''
If you're using grids for pathfinding, your units are not constrained
to grids, {emph and} movement costs are uniform, you may want to {link
{ref Path smoothing} straighten the paths} by moving in a straight
line from one node to a node far ahead when there are no obstacles
between the two. If you're using navigation meshes, look at {link
http://digestingduck.blogspot.com/2010/03/simple-stupid-funnel-algorithm.html
the funnel algorithm}.
{h3 Behavior flags or stacks}
Your units may have more than one goal. For example, you may
have a general goal like "spying" but also a more immediate goal like
"go to the enemy headquarters". In addition, there may be temporary
goals like "avoid that patrol guard". Here are some ideas for goals:
{list
{hbox {strong Stop}: Stay in the current location}
{hbox {strong Stay}: Stay in one area}
{hbox {strong Flee}: Move to a safe area}
{hbox {strong Retreat}: Move to a safe area, while fighting off enemy units}
{hbox {strong Explore}: Find and learn about areas for which little
information is known}
{hbox {strong Wander}: Move around aimlessly}
{hbox {strong Search}: Look for a particular object}
{hbox {strong Spy}: Go near an object or unit to learn more about
it, without being seen}
{hbox {strong Patrol}: Repeatedly walk through an area to make sure
no enemy units go through it}
{hbox {strong Defend}: Stay near some object or unit to keep enemy
units away}
{hbox {strong Guard}: Stay near the entrance to some area to keep
enemy units out}
{hbox {strong Attack}: Move to some object or unit to capture or
destroy it}
{hbox {strong Surround}: With other units, try to surround an
enemy unit or object}
{hbox {strong Shun}: Move away from some object or unit}
{hbox {strong Avoid}: Stay away from any other units}
{hbox {strong Follow}: Stay near some unit as it moves around}
{hbox {strong Group}: Seek and form groups of units}
{hbox {strong Work}: Perform some task like mining, farming, or collecting}
}
For each unit you can have a flag indicating which behavior it is to
perform. To have multiple levels, keep a {emph behavior stack}. The
top of the stack will be the most immediate goal and the bottom of the
stack will be the overall goal. When you need to do something new but
later want to go back to what you were doing, push a new behavior on
the stack. If you instead need to do something new but {emph don't}
want to go back to the old behavior, clear the stack. Once you are
done with some goal, pop it from the stack and start performing the
next behavior on the stack.
{h3 Waiting for movement}
It is inevitable that the movement algorithm will run into obstacles
that were not there during the pathfinding process. An easily
implemented technique is based on the assumption that the {emph other}
obstacle will move first. This is particularly useful when the
obstacle is a friendly unit. When an obstacle is in the way,
wait some amount of time for it to move. If it still hasn't moved
after that period of time, recalculate a path around it or to the
destination. If the obstacle is detected ahead of time, your unit can
simply walk slower to give the other unit more time to get out of the
way.
It is possible that two units will bump into each other, and each will
wait for the other to proceed. In this case, a priority scheme can be
used: assign each unit a unique number, and then make the lower
numbered unit wait for the higher numbered unit. This will force one
of the units to proceed if both are waiting. When obstacles are
detected ahead of time, the lower numbered unit should slow down
before reaching the expected point of collision.
{h3 Coordinated movement}
The technique described above does not work when units are trying to
move through a narrow corridor. If one unit stands still while the
other tries to go around, the corridor can't be used by both units.
One unit will block it while the other one will take a long path
around.
It should be possible for the second unit to communicate with the
first one, and ask it to back up. Once the corridor is clear, the
second unit can pass through, and then the first unit can go through.
This may be complicated to implement unless you can identify the
corridors beforehand. For randomly generated maps, it could be very
difficult to determine where the corridor is and how far the first
unit needs to back up.
{h2 Movement costs for pathfinders}
{filename MovementCosts.html}
{set shortname Movement Costs}
When using a pathfinding algorithm, you may want to treat map spaces
as something other than {emph clear} and {emph blocked}. Often
there is more information available, such as the difficulty of moving
through that area. For example, swamps and mountains may be more
difficult to pass than grasslands and desert. With some algorithms,
like A*, you can put this encode this information into the cost
function. Listed below are some ideas for movement costs that might
be useful.
{h3 Altitude}
High altitudes (such as mountains) can have a higher movement cost
than low altitudes. With this cost function, your units will try to
stay in the lowlands whenever possible. For example, if the source
and destination are both at high altitudes, the unit might move
downhill, travel for a while, and then move back uphill.
{h4 Moving uphill}
Instead of high altitudes having a high cost, {emph moving} uphill
can have a high cost. This avoids the odd situation described above.
With this cost function, units try to avoid moving uphill. Faced with
the same situation, the unit will try to avoid moving back uphill at
the end; it can do this by staying at a high altitude throughout its
travels. A cost function such as this one may be good for units such
as soldiers, which can move downhill easily but have a hard time going
uphill.
{h4 Moving up- or downhill}
Some units, such as tanks, have a hard time moving uphill {emph or}
downhill. You can assign a high cost to moving downhill, and an even
higher cost to moving uphill. The units will try to avoid changing
altitudes.
{h3 Terrain}
You may want different types of terrain to have different movement costs.
{h4 Forests, mountains, and hills}
Instead of using altitudes, you may want to use terrain types, as in
Civilization. Each terrain type can have a movement cost associated
with it. This movement table might apply to all units, or different
movement tables could be associated with each unit type. For example,
soldiers might have no trouble moving through forests, but tanks might
have a very hard time. A fancier method is to assign movement costs
to {emph changing} terrain. Going from grassland into mountains could
be more expensive than going from hills to mountains, which could be
more expensive than going from mountains to mountains.
{h4 Roads}
In many games, the primary purpose of roads is to make movement
possible or easier. After choosing a movement cost function, you can
add a road modifier to it. One possibility is to divide the cost by
some constant (such as two); another is to assign a constant cost to
movement along a road.
I strongly advise that you do {emph not} make road movement free
(zero-cost). This confuses pathfinding algorithms such as A*,
because it introduces the possibility that the shortest path from one
point to another is along a winding road that seems to lead nowhere.
The algorithm has to search a very wide area to make sure that no such
roads exist. Note that in the game Civilization, railroads had
zero-cost movement, but when using the "Auto Goto" function, railroads
had a non-zero cost. This is evidence that a pathfinding algorithm
was being used.
{h4 Walls or other barriers}
Instead of checking both movement costs and for obstacles in your
pathfinding algorithm, you can use movement costs. Just assign
a very high movement cost to any obstacle. When expanding nodes (in
the A* algorithm), check if the cost is too high; if it is, then throw
the node out.
{h4 Sloped Land}
Instead of using {emph movement} up and down hills, you might want to
make movement on any hill expensive. To do this, compute the overall
slope of the terrain (by looking at the maximum difference between the
current tail and its neighbors), and use that as part of the movement
cost. Land that is very steep will have a high cost and land that is
shallow will have a low cost. This approach differs from the movement
uphill/downhill cost in that it looks for {emph land} that is steep,
while the previous approach looked for units that {emph move} in a
steep direction. In particular, if you're on a hill and can move left
or right without going up or down, the uphill/downhill approach will
consider it a low cost, while this approach will consider it a high
cost (because the land is steep even if you aren't going up or down).
Sloped land costs may not make sense for unit movement, but you can
use pathfinding for more than finding paths for units. I use it for
finding paths for roads, canals, bridges, and so on. If you want to
build these items on flat land, you can take land slope into account
when finding a path for a road or canal. See {link {ref Applications}
the section on applications} for more ideas.
{h3 Enemies and friendly units}
Another modifier can help you avoid enemy units. Using {link
http://www.slideshare.net/mobius.cn/influence-map influence maps}, you can keep
track of areas that are near enemy or friendly units, have recently
killed soldiers, have been recently explored, are close to an escape
route, or have been traversed recently. ({link
http://www.ensemblestudios.com/news/devnews/terrain2.shtml Age of
Empires 2 uses influence maps to influence pathfinding.}) An
influence map might have a positive value for friendly units and a
negative value for enemy units. By increasing the movement cost
whenever you are in negative territory, you can influence your units
to stay away from the enemy.
Even more complicated (and perhaps not possible with influence maps)
is to look at {emph visibility}: is your unit visible by an enemy
unit? Is your unit detectable in some other way? Is it possible for
that enemy unit to fire on you?
{h3 Marked beacons}
If your map is designed and not automatically generated, you can add
extra information to it. For example, ancient trade routes often
would pass particular points, which often became trading towns. These
places are {strong beacons}, places that are known to be along good
paths. The distance to a beacon would be added to the movement cost
as a way to influence paths to favor beacons.
Good choices for beacons include lighthouses, cities, mountain
passes, and bridges.
{h3 Fuel consumption}
{margin-note
{vbox {strong Note:}
{hbox To keep the state space small, you need to round off
the fuel value to a coarse measurement unit.
Unfortunately, this makes the search less effective.}
}
}
In addition to looking at the {emph time} it takes to go somewhere,
you may want to consider the {emph fuel} it takes. The fuel
consumption may be given more weight when the unit's fuel level is
lower.
To track the fuel usage through the map, you need to use state space,
as described in {link {ref Road maps}}. The state would be the pair
. However, state space can become very large, so it
may be worth looking at alternatives to using ordinary A*.
One alternative is {link
ftp://ftp.cs.bham.ac.uk/pub/authors/B.S.Logan/aaai-98.ps.gz A* with
Bounded Costs} (ABC). With ABC, you can assign a bound ("20 gallons")
to a cost ("fuel").
{h2 User experience with shortest paths}
{filename UserExperience.html}
What's most important in the game is the user. You want the user to
have fun! You don't want him (or her) to feel like the computer is
cheating, or that the game units aren't behaving properly.
{h3 Dumb movement}
If the pathfinding doesn't work well, the user will end up moving the
units manually. {emph Avoid this!} In Civilization, the rules for
the game allowed for zero-cost movement along railroads. However, the
pathfinder had a non-zero movement cost. The result was that users
avoided using the pathfinder, and instead moved units manually on the
railroads. In Command and Conquer, units could get stuck in "U"
shaped traps so users would have to guide the units manually. A
dumb pathfinder will annoy your users and make them move units
themselves, so make your pathfinder decent!
{h3 Smart movement}
Making units too smart is almost as bad as making units too dumb. If
the player has to deal with fog of war but the pathfinder has access
to the entire map, the units will mysteriously know where to go even
though the user does not. That's a clear sign to the user that
something odd is going on. On the other hand, it gives better paths.
A compromise is to scale up the movement costs on unexplored areas.
For example, if your normal movement costs are 1 for grass, 3 for
forest, and 7 for mountains, set them differently on unexplored areas:
5 for grass, 6 for forest, 7 for mountains. The unit will take
mountains vs. grass into account, but not too much; it'll be a
subtle hint. Raising costs for moving through unexplored areas will
also tend to make the unit stay in explored territory as much as
possible. You might want to do the opposite for "scout" units: they
should {emph prefer} unexplored areas.
Try to keep your units balanced between too dumb and too smart. The
goal should be to make it match what the user might have done to move
the unit around.
{h3 Multithreading}
You can use multithreading to improve the user experience. When a
unit needs a path, allow it to start moving in a straight line towards
the goal, and add a request to a pathfinding queue. In another (low
priority) thread, pull requests off the queue and find paths. Your
units will start moving immediately, so the user won't be left
wondering if something is wrong, and you won't have a high CPU load
(which will slow down the rest of the game) while the path is being
calculated.
{h3 Multiple units}
If your game allows multiple units in a group to move together, try to
make the movement look interesting. You can find a single path for
them all to follow, and then have them all follow the path
individually, but this will lead to either a line of units or units
trying to pass each other. Instead, vary the paths a little so that
they can walk in parallel. Alternatively, pick one "leader" unit to
move along the path and have the other units use a separately
programmed "follow" behavior. This following could be as simple as
moving towards the leader but stay some distance away, or it could be
as involved as {link http://www.red3d.com/cwr/boids/ flocking}.
{h3 Multiple waypoints}
Even given the optimal path, the player may prefer a different path.
You may allow the player to mark waypoints on the path: instead of
simply clicking on a destination, the player would click on two or
three points along the way to the destination. (Many real-time
strategy games use shift-click for this operation.) You now have
three or four smaller paths to compute, and you save some time. The
player also has some control over the overall path---for example, your
pathfinder may have found a path to the west of some mountains, but
for safety's sake, the player wants to stay to the east of the
mountains (near friendly guard towers).
The main change in unit movement code will be that instead of a single
destination, you will have a list of destinations. Find a path to the
first destination. Once you get there, remove it from the list and
find a path to the next destination. This reduces latency and
improves throughput as well.
{h2 Applications}
{filename Applications.html}
In addition to finding a path for a unit to move along, pathfinding
can be used for several other purposes.
{h3 Exploration}
If part of your cost function penalizes paths that are on known
territory, paths are more likely to go through unexplored territory.
These paths are good for scout units.
{h3 Spying}
If part of the cost function penalizes paths near the enemy's
watchtowers and other units, your unit will tend to stay in hiding.
Note however that to work well, you may have to update the path
periodically to take into account enemy unit movements.
{h3 Road building}
Historically, roads have been built along paths that are often used.
As the paths are used more and more often, vegetation is removed and
replaced with dirt, and later with stone or other material. One
application of pathfinding is to find roads. Given places that
people want to go (cities, lakes, springs, sources of minerals, and so
on), find paths randomly between these important locations. After
finding hundreds or perhaps thousands of paths, determine which spaces
on the map most often occur on paths. Turn those spaces into roads.
Repeat the experiment, with the pathfinder preferring roads, and you
will find more roads to build. This technique can work for multiple
types of roads as well (highways, roads, dirt paths): the most
commonly used spaces would become highways and less commonly used
spaces would become roads or dirt paths.
Pathfinding is also used for building roads over mountains that avoid
extreme slopes. {link
http://blog.runevision.com/2016/03/note-on-creating-natural-paths-in.html
This article} shows how setting the movement cost to the square of the
slope makes A* find a "natural" looking path up and over a mountain pass.
{h3 Terrain analysis}
Combining influence maps, pathfinding, and line of sight can give you
interesting ways to analyze terrain.
Using the same approach as road building, we can use pathfinding to
determine what areas are the most likely to be traversed given some
set of source and destination points. These points, and areas near
them, tend to be strategically important. Clash of Civilizations uses
this for their {link
http://web.archive.org/web/20091104094002/http://clash.apolyton.net/models/Model-MapAI.shtml Map AI}.
By further analysing the common paths we find, we can find ambush
sites---locations along a path that do not have line-of-sight access
to the location N steps further along the path. Placing an ambush at
one of these points means the enemy will not see you until they are
within distance N, so you can ambush with a large force.
{h3 City building}
Cities often form around natural resources such as farmland or sources
of mineral wealth. As people from these cities trade with each other,
they need trading routes. Use pathfinding to find their trading
routes, and then mark a day's worth of travel on these routes. After
a caravan travels for a day, it will need a place to stop: a perfect
place for a city! Cities that lie along more than one travel route
are great places for trading villages, which eventually grow into
cities.
A combination of the road building and city building may be useful for
producing realistic maps, either for scenarios or for randomized maps.
{h2 AI techniques}
{filename AITechniques.html}
Pathfinding is often associated with AI, because the A* algorithm and
many other pathfinding algorithms were developed by AI researchers.
Several biology-inspired AI techniques are currently popular, and I
receive questions about why I don't use them. Neural Networks model a
brain learning by example---given a set of right answers, it learns
the general patterns. Reinforcement Learning models a brain learning
by experience---given some set of actions and an eventual reward or
punishment, it learns which actions are good or bad. Genetic
Algorithms model evolution by natural selection---given some set of
agents, let the better ones live and the worse ones die. Typically,
genetic algorithms do not allow agents to learn during their
lifetimes, while neural networks allow agents to learn only during
their lifetimes. Reinforcement learning allows agents to learn during
their lifetimes and share knowledge with other agents.
{h3 Neural Networks}
Neural networks are structures that can be "trained" to recognize
patterns in inputs. They are a way to implement {link
http://webdocs.cs.ualberta.ca/~sutton/book/ebook/node85.html function
approximation}: given y{sub 1} = f(x{sub 1}), y{sub 2} = f(x{sub 2}),
..., y{sub n} = f(x{sub n}), construct a function f' that approximates
f. The approximate function f' is typically {emph smooth}: for x'
close to x, we will expect that f'(x') is close to f'(x). Function
approximation serves two purposes:
{list
{hbox {strong Size:} the representation of the approximate function
can be significantly smaller than the true function.}
{hbox {strong Generalization:} the approximate function can be used
on inputs for which we do not know the value of the function.}
}
Neural networks typically take a vector of input values and produce a
vector of output values. Inside, they train weights of "neurons".
Neural networks use {emph supervised learning}, in which inputs and
outputs are known and the goal is to build a representation of a
function that will approximate the input to output mapping.
In pathfinding, the function is f(start, goal) = path. We do not
already know the output paths. We could compute them in some way,
perhaps by using A*. But if we are able to compute a path given
(start, goal), then we already know the function f, so why bother
approximating it? There is no use in generalizing f because we know
it completely. The only potential benefit would be in reducing the
size of the representation of f. The representation of f is a fairly
simple algorithm, which takes little space, so I don't think that's
useful either. In addition, neural networks produce a fixed-size
output, whereas paths are variable sized.
Instead, function approximation may be useful to construct components
of pathfinding. It may be that the movement cost function is unknown.
For example, the cost of moving across an orc-filled forest may not be
known without actually performing the movement and fighting the
battles. Using function approximation, each time the forest is
crossed, the movement cost f(number of orcs, size of forest) could be
measured and fed into the neural network. For future pathfinding
sessions, the new movement costs could be used to find better paths.
Even when the function is unknown, function approximation is useful
primarily when the function varies from game to game. If a single
movement cost applies every time someone plays the game, the game
developer can precompute it beforehand.
Another function that is could benefit from approximation is the
heuristic. The heuristic function in A* should estimate the minimum
cost of reaching the destination. If a unit is moving along path P =
p{sub 1}, p{sub 2}, ..., p{sub n}, then after the path is traversed,
we can feed n updates, g(p{sub i}, p{sub n}) = (actual cost of moving
from i to n), to the approximation function h. As the heuristic gets
better, A* will be able to run quicker.
Neural networks, although not useful for pathfinding itself, can be
used for the functions used by A*. Both movement and the heuristic
are functions that can be measured and therefore fed back into the
function approximation.
{h3 Genetic Algorithms}
{margin-note
{vbox {strong Note:}
{hbox Function approximation can be transformed into a
function optimization problem. To find f'(x) that
approximates f(x), set g(f') = Sum of (f'(x)-f(x)){super 2} over
all input x.}
}
}
Genetic Algorithms allow you to explore a space of parameters to find
solutions that score well according to a "fitness function". They are
a way to implement {emph {link
http://mathworld.wolfram.com/StochasticOptimization.html function
optimization}}: given a function g(x) (where x is typically a vector
of parameter values), find the value of x that maximizes (or
minimizes) g(x). This is an {emph unsupervised learning}
problem---the right answer is not known beforehand. For pathfinding,
given a starting position and a goal, x is the path between the two
and g(x) is the cost of that path. Simple optimization approaches
like hill-climbing will change x in ways that increase g(x).
Unfortunately in some problems, you reach "local maxima", values of x
for which no nearby x has a greater value of g, but some faraway value
of x is better. Genetic algorithms improve upon hill-climbing by
maintaining multiple x, and using evolution-inspired approaches like
mutation and cross-over to alter x. Both hill-climbing and genetic
algorithms can be used to learn the best value of x. For pathfinding,
however, we already have an algorithm (A*) to find the best x, so
function optimization approaches are not needed.
Genetic Programming takes genetic algorithms a step further, and
treats {emph programs} as the parameters. For example, you would
breeding pathfinding {emph algorithms} instead of {emph paths}, and
your fitness function would rate each algorithm based on how well it
does. For pathfinding, we already have a good algorithm and we do not
need to evolve a new one.
It may be that as with neural networks, genetic algorithms can be
applied to some portion of the pathfinding problem. However, I do not
know of any uses in this context. Instead, a more promising approach
seems to be to use pathfinding, for which solutions are known, as one
of many tools available to evolving agents.
{h3 Reinforcement Learning}
Like genetic algorithms, Reinforcement Learning is an unsupervised
learning problem. However, unlike genetic algorithms, agents can
learn during their lifetimes; it's not necessary to wait to see if
they "live" or "die". Also, it's possible for multiple agents
experiencing different things to share what they've learned.
Reinforcement learning has some similarities to the core of A*. In
A*, reaching the end goal is propagated back to mark all the choices
that were made along the path; other choices are discarded. In
reinforcement learning, every state can be evaluated and its reward
(or punishment) is propagated back to mark all the choices that were
made leading up to that state. The propagation is made using a value
function, which is somewhat like the heuristic function in A*, except
that it's updated as the agents try new things and learn what works.
One of the key advantages of reinforcement learning and genetic
algorithms over simpler approaches is that there is a choice made
between {emph exploring} new things and {emph exploiting} the
information learned so far. In genetic algorithms, the exploration
via mutation; in reinforcement learning, the exploration is via
exlicitly allowing the probability of choosing new actions.
As with genetic algorithms, I don't believe reinforcement learning
should be used for the pathfinding problem itself, but instead as a
guide for teaching agents how to behave in the game world.
{h2 References}
{filename References.html}
This section is quite incomplete.
General graph searching algorithms can be used for pathfinding. Many
algorithms textbooks describe graph searching algorithms that do not
use heuristics (breadth-first search, depth-first search, Dijkstra's).
Reading about them may help in understanding A*, which is a variant of
Dijkstra's. Many AI textbooks will address graph searching algorithms
that do use heuristics (best-first search, A*).
For non-graph-search algorithms, see {link
http://web.archive.org/web/20020810205430/http://home.sol.no/~johncl/shorpath.htm
John Lonningdal's web page} (via Wayback Machine, since his site is no
longer up).
To see the proof that A* is optimal, see {link https://web.archive.org/web/20160322055823/http://ai.stanford.edu/%7Enilsson/OnlinePubs-Nils/PublishedPapers/astar.pdf A Formal Basis for the Heuristic Determination of Minimum Cost Paths}.
To learn more about unit movement after a path has been found, see
Pottinger's articles on {link
http://www.gamasutra.com/features/game_design/19990122/movement_01.htm
unit movement} and {link
http://www.gamasutra.com/features/game_design/19990129/implementing_01.htm
group movement}. Also highly recommended are Craig Reynold's pages on
{link http://www.red3d.com/cwr/steer/ steering} and {link
http://www.red3d.com/cwr/boids/ flocking}. Geoff Howland has a {link
http://archive.gamedev.net/archive/reference/design/features/prac_ai_2/ great
article} about unit movement in general.
{link http://www.cs.du.edu/~sturtevant/papers/GPPC-2014.pdf The
grid-based pathfinding competition results paper} {link https://www.semanticscholar.org/paper/The-Grid-Based-Path-Planning-Competition%3A-2014-and-Sturtevant-Traish/4222c3bf800c9fc2b653f9318f0727c668d2d43a (alternate link)} describes
optimizations for A* running on unweighted grids: contraction
hierarchies, {link http://idm-lab.org/bib/abstracts/papers/icaps13.pdf subgoal graphs}, jump point search, visibility graphs, compressed path
databases, and more.
Patrick Lester has a page {link
http://www.policyalmanac.org/games/twoTiered.htm describing a
two-level pathfinder}. {link
http://www.cs.ualberta.ca/~mburo/ps/hpaenh.pdf Hierarchical Planning
A*} (HPA*) can turn a grid representation into a simplified graph.
{link http://aigamedev.com/open/tutorials/clearance-based-pathfinding/
Clearance-based pathfinding} annotates the graph with the sizes of the
objects that can pass through there. This is useful if you want small
units to be able to pass through an area but large units to be
blocked.
There's an interesting paper describing how to find {link
http://citeseer.com/580850.html Simplest Paths} instead of
Shortest Paths.
{link
http://cstheory.stackexchange.com/questions/11855/how-do-the-state-of-the-art-pathfinding-algorithms-for-changing-graphs-d-d-l
This StackOverflow question} includes a summary of lots of variants of
A*.
{link http://www.cs.ualberta.ca/~mburo/ps/tra.pdf Triangulation A*}
converts a polygonal obstacle representation into a navigation mesh
using triangles, and Triangulation Reduction A* simplifies the
resulting pathfinding graph by removing nodes.
Here are some other papers I haven't classified:
{link http://ircl.cs.ualberta.ca/files/webfm/lrts/pubs/bulitko08a.pdf
Real-time heuristic search} augments A* and other algorithms with
additional data to speed up pathfinding.
{link http://webdocs.cs.ualberta.ca/~yngvi/pdf/BjornssonEHS05.pdf
Fringe Search} looks at large sets of nodes (the fringe or frontier)
at a time. They greatly reduce the cost of processing each node. A*
sorts one node at a time (either on insertion and deletion from the
set, or both), and batch sorting is faster. But even faster is not
sorting at all. In the batches of nodes that Fringe search is
processing, there's no need to sort them. The downside is that more
nodes have to be processed, sometimes more than once. But if you can
make processing them really cheap, then it's okay to process lots of
nodes.
{link http://algo2.iti.kit.edu/download/diploma_thesis_geisberger.pdf
Contraction Hierarchies} (long paper) can find paths faster by adding
shortcut edges. Also great reading is the related work section, which
summarizes A* improvements from landmarks, arc flags, transit nodes,
highway hierarchies, and other approaches.
{link http://www.rebennack.net/SEA2011/files/talks/SEA2011_Angelo.pdf
Arc flags} (note to self: need a better link here) restrict the set of
edges expanded in the main pathfinding loop. First divide the world
into regions, then precalculate which edges are part of a shortest
path to each region. Instead of looking at {emph all} edges to
neighbors, look at only the edges that are part of a shortest path to
the region the goal is in. Steve Rabin calls this approach "goal
bounding".
{link
http://ircl.cs.ualberta.ca/files/webfm/games/pubs/geramifard06-aiide.pdf
Biased Cost Pathfinding} alters the movement costs of areas where
other units are going to move, so that subsequent paths avoid
colliding with those units.
{link http://graphics.tudelft.nl/~rafa/myPapers/bidarra.CAVW2012.pdf
Parallel Ripple Search is designed for multi-core pathfinding, without
the sort bottleneck that A* has.}
{link http://people.cs.uu.nl/roland/motion_planning/cmm2.html Corridor
Maps} are a way to construct a pathfinding graph that greatly reduces
the number of nodes, especially in maps with lots of corridors.
{link http://ircl.cs.ualberta.ca/files/webfm/lrts/pubs/bulitko06a.pdf
Learning Real-Time A*} updates the heuristic as it explores the
map. {link
http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.75.6235&rep=rep1&type=pdf
Prioritized Learning Real-Time A*} prioritizes the search to favor
areas where it is learning more.
{link
http://webdocs.cs.ualberta.ca/~games/pathfind/publications/yap2002.ps IDA*
runs faster with a hex grid than a square grid}(!).
{link
https://www.aaai.org/ocs/index.php/AIIDE/AIIDE11/paper/viewFile/4050/4423
Compressed Path Databases} tests how well all-pairs shortest path
(Floyd-Warshall or Johnson's Algorithm) on grids can be compressed. It
applies if you're using a grid and the map isn't changing; I suspect
you'd be better off reducing the graph size first.
{link
http://harablog.wordpress.com/2009/02/12/clearance-based-pathfinding-with-probabilistic-roadmaps/
Probabilistic Roadmaps (PRMs)} build pathfinding graphs from polygonal
obstacle maps.