Paper list
Pick your paper below. Papers that have been claimed will be crossed out.

R. Yuster and U. Zwick. Finding even cycles even faster, SIAM Journal on Discrete Math, 1997. (Vincent)[pdf]
Description: The authors show that for any constant k>1, there is an O(n^2) time algorithm that either finds a cycle of length exactly 2k in an nnode graph, or determines no such cycle exixts. This shows that even cycles can be found faster than odd cycles, as even for just finding a triangle, the current best algorithm runs in Omega(n^{2.37}) time.

F. Kloks, D. Kratsch, H. Muller.
Finding and counting small induced subgraphs efficiently, WG'95. (Kevin) [pdf]
Description:
The authors give algorithms based on matrix multiplication, for finding and counting subgraphs of small size in a sparse host graph. For instance, they show that a 4Clique can be found in O(m^{(w+1)/2}) time in a graph with m edges, where w<2.38 is the matrix multiplication exponent.
 V. V. Williams, J. Wang, R. Williams, H. Yu. Finding FourNode Subgraphs in Triangle Time. SODA'15. [pdf]
Description:
The authors provide the fastest to date algorithms for finding any induced 4node subgraph except the independent set or clique. In particular they give algorithms that run in O(n^w) time, w<2.38, which is the currently best running time for finding triangles.
N. Alon, R. Yuster, U. Zwick. Finding and counting given length cycles, Algorithmica'97. (William) [pdf]
Description: The authors give algorithms for finding and counting fixed length cycles in directed and undirected graphs. The running time of the algorithms depends on the length of the cycle and on the number of edges in the host graph. For instance, they show that triangles can be found in O(m^{1.41}) time. This is one of the prime papers using the high degreelow degree technique.
L. Roditty, V. V. Williams. Subquadratic time approximation algorithms for the girth, SODA'12. (Koki) [pdf]
Description:
The authors give several algorithms for approximating the girth of an undirected graph. For instance, they show that one can get a (2eps)multiplicative approximation in O(n^{2delta}) time for eps,delta>0, and an +3additive approximation in ~n^3/m time in nnode, medge graphs.
D. Dor, S. Halperin, U. Zwick,
All Pairs Almost Shortest Paths,
SIAM Journal on Computing'2000. (Victor KaiserPendergrast)[pdf]
Description: The authors present fast algorithms for approximating all pairs shortest paths (APSP) in unweighted undirected graphs. In class we showed their O~(n^{7/3}) time algorithms for a +2approximation of APSP. The paper gives for instance an O(n^{2+2/(3k2)}) time algorithm for +k approximation for any even constant k. They also give some bounds for spanners and emulators.

L. Roditty, M. Thorup, U. Zwick. Roundtrip Spanners and Roundtrip Routing
in Directed Graphs, TALG'08. (Jose) [pdf]
Description:
The authors consider spanners in directed graphs. In class we showed that no nontrivial spanners can exist in directed graphs. This paper shows that if instead of approximating the distanve d(u,v) between each pair of vertices u,v, one wants to approximate d(u,v)+d(v,u), then very interesting results become possible.

M. Thorup, U. Zwick. Spanners and emulators with sublinear distance errors, SODA'06. (James) [pdf]
Description: We mentioned in class that it is an open problem to obtain constant error emulators or spanners with O(n^{4/3eps}) edges for eps>0. This paper shows that one can obtain spanners and emulators with O(n^{4/3eps}) edges, with error that depends on the distance. For instance, for any k and any undirected graph, there is a spanner with O(kn^{1+1/k}) edges and for every u,v, the estimate d'(u,v) satisfies d(u,v)<=d'(u,v)<=d(u,v) + O((d(u,v)^{11/(k1)}).

M. Cygan, F. Grandoni, T. Kavitha. On pairwise spanners. STACS'13. (Yuetong) [pdf]
Description:
This paper studies spanner constructions that are only requires to approximate distances for a small subset of points (or pairs of points).

Greg Bodwin and V. Vassilevska Williams. Better Distance Preservers and Additive Spanners, SODA'2016. (Aaron) [pdf]
Description:
This paper focuses on pairwise distance preservers (pairwise spanners with no error) and on additive spanner in the very sparse regime, where one desires potentially a linear number of edges, but might have to pay a polynomial additive error.
 M. Patrascu, L. Roditty. Distance Oracles Beyond the ThorupZwick Bound,
FOCS 2010. [pdf]
Description: This paper constitutes the first improvement over the ThorupZwick distance oracle presented in class.
The paper shows that if one is willing to have a +1 additive error, on top of the 2multiplicative error, then much less space can be used. The authors also extend their results for the case when the given graph is sparse.
 M. Patrascu, L. Roditty, M. Thorup. A New Infinity of Distance Oracles for Sparse Graphs, FOCS'12. [pdf]
Description: The distance oracles of Thorup and Zwick that we presented in class only gave approximate distances for constant approximation factors. It also had the number of edges in terms of the number of vertices of the original graph. This paper gives new bounds for nonconstant approximation ratios and where the space used by the distance oracle depends on the number of edges of the original graph instead of the number of vertices.

A. Bernstein, D. Karger. A Nearly Optimal Oracle for Avoiding Failed Vertices and
Edges, STOC'09. (Declan) [pdf]
Description: This paper considers the problem of constructing distance sensitivity oracles. These are data structures that preprocess a graph and store some data, so that the following queries can be answered efficiently: given vertices u,v and a failed vertex x or edge e, return the distance between u and v in the graph excluding x or e.
This paper shows how to construct such an oracle in ~mn time. The space used by the oracle is ~n^2 and the query time is constant. This is conjectured to be essentially optimal.

S. Chechik.
Compact Routing Schemes with Improved Stretch, PODC'13. (Brad) [pdf]
Description: This is the first improvement over the ThorupZwick compact routing scheme. The author shows how to get 3.68kapproximate paths without handshaking. In class we showed that the ThorupZwick scheme could get 4k3approximate paths without handshaking and 2k1approximate with handshaking.

C. Gavoille and C. Sommer. Sparse Spanners vs. Compact Routing. SPAA'11. (Jake) [pdf]
Description:
Among other nice results, this paper shows limitations on how good any compact routing scheme with additive stretch can be.

T. Chan, M. Patrascu, L. Roditty. Dynamic Connectivity: Connecting to Networks and Geometry, SICOMP'11. (Nishith) [pdf]
Description: In class we showed that dynamic connectivity has extremely efficient algorithms. Those algorithms, however, only supported edge updates. Suppose however, that one wants to remove or add vertices instead of edges. The dynamic connectivity algorithms that we presented, would have update time proportional to the degree of the updated vertex. This paper shows how to do better: the update time is O(n^{2/3}), while the query time is O(n^{1/3}).

L. Roditty, U. Zwick. On Dynamic Shortest Paths Problems, ESA'04. (Barak) [pdf]
Description: This paper shows that dynamic single source shortest paths (SSSP) is a difficult problem, even when only insertions or deletions are allowed. The authors give reductions from the all pairs shortest paths problem to incremental/decremental SSSP.
They also show how to use dynamic SSSP to construct sparse O(log n)spanners.
 M. Henzinger, S. Krinninger, D. Nanongkai. Decremental SingleSource Shortest Paths on Undirected Graphs in NearLinear Total Update Time, FOCS'14. [pdf]
Description:
This paper presents the first nearlinear total update time for maintaining approximate single source shortest paths under edge deletions. This is the current b
est result on the topic. (For prior results, see the papers below.)
 M. Henzinger, S. Krinninger, D. Nanongkai. A subquadratic time algorithm for decremental single source shortest paths, STOC'13. [pdf]
Description: This paper considers the problem of dynamically maintaining (1+eps)approximate SSSP when only edge deletions are to be supported. The authors show that the total update time over a sequence of m deletions is no more than ~m+n^{1.8}.
The previous best was ~n^2 total update time (see the paper below).

A. Bernstein, L. Roditty. Improved Dynamic Algorithms for Maintaining Approximate Shortest
Paths Under Deletions, SODA'11. (Hubert) [pdf]
Description: This is the first improvement over an old algorithm of Even and Shiloach that obtained O(mn) total update time for decremental SSSP. The authors give ~n^2 total update time, but the catch is that the distances returned are (1+eps)approximate rather than exact. (Better than O(mn) for exact decremental SSSP would be a big breakthrough because by RZ'04, paper 12 above, it would imply a fast algorithm for APSP.)
 M. Gupta, R. Peng. Fully dynamic (1+eps)approximate matchings, FOCS'13. [pdf]
Description: This paper gives a fully dynamic algorithm that maintains a (1+eps)approximate matching with deterministic update time ~sqrt m / eps^2. This improves upon the Neiman and Solomon 3/2approximation that ran in sqrt m time.

S. Baswana, M. Gupta, S. Sen. Fully Dynamic Maximal Matching in O(log n) Update Time, FOCS 2011. (Alexandros) [pdf]
Description: This paper gives a randomized algorithm for dynamic maximal matching, showing that with randomization the update time can be made to be O(log n). This is in contrast with the deterministic algorithm above.
 B. Haeupler, T. Kavitha, R. Mathew, S. Sen, R. Tarjan. Incremental Cycle Detection, Topological Ordering, and Strong
Component Maintenance, TALG'12. [pdf]
Description: This paper considers the problem of maintaining the topological order of a DAG under edge insertions, where a query is of the form, given u and v, is u before v in the topological order?
The paper shows that edge insertions can be performed in total update time ~min{n^{2.5}, m^{1.5}}. It also shows how to use this algorithm for maintaining the strongly connected components of a directed graph under edge insertions with the same update time.

M. Bender, J. Fineman, S. Gilbert. A New Approach to Incremental Topological Ordering, SODA'09. (Abhijit) [pdf]
Description:
This paper also considers the incremental topological order problem (just as the paper above). It gives a completely different algorithm that achieves total update time O(n^2 log n).
J. Lacki. Improved Deterministic Algorithms for Decremental Reachability
and Strongly Connected Components, TALG'13. (Jimmy) [pdf]
Description: This paper considers the problems of maintaining the transitive closure of a directed graph under edge deletions, and maintaining the strongly connected components under edge deletions. It achieves a deterministic total update time of O(mn) for the decremental strongly connected components problem, and then uses it to achieve the same time for transitive closure.
(Note that the decremental case of strongly connected components seems much harder than the incremental version from the above two papers.)
 M. Henzinger, P. Klein, S. Rao, S. Subramanian. Faster ShortestPath Algorithms for Planar Graphs, JCSS'97. [pdf]
Description: This paper shows that in nvertex planer graphs, single source shortest paths can be computed in O(n) time.
This running time is optimal.
 P. Klein, S. Mozes, C. Sommer. Structured Recursive Separator Decompositions
for Planar Graphs in Linear Time. STOC'13. [pdf]
Desription:
This paper presents a way to decompose any planar graph in linear time. The decompositions computed can be used as a building block of many fast planar graph algorithms.