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Popular conjectures imply strong lower bounds for dynamic problems
 CoRR
"... Abstract—We consider several wellstudied problems in dynamic algorithms and prove that sufficient progress on any of them would imply a breakthrough on one of five major open problems in the theory of algorithms: 1) Is the 3SUM problem on n numbers in O(n2−ε) time for some ε> 0? 2) Can one dete ..."
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Abstract—We consider several wellstudied problems in dynamic algorithms and prove that sufficient progress on any of them would imply a breakthrough on one of five major open problems in the theory of algorithms: 1) Is the 3SUM problem on n numbers in O(n2−ε) time for some ε> 0? 2) Can one determine the satisfiability of a CNF formula on n variables and poly n clauses in O((2 − ε)npoly n) time for some ε> 0? 3) Is the All Pairs Shortest Paths problem for graphs on n vertices in O(n3−ε) time for some ε> 0? 4) Is there a linear time algorithm that detects whether a given graph contains a triangle? 5) Is there an O(n3−ε) time combinatorial algorithm for n×n Boolean matrix multiplication? The problems we consider include dynamic versions of bipartite perfect matching, bipartite maximum weight matching, single source reachability, single source shortest paths, strong connectivity, subgraph connectivity, diameter approximation and some nongraph problems such as Pagh’s problem defined in a recent paper by Pǎtraşcu[STOC 2010]. Index Terms—dynamic algorithms; all pairs shortest paths; 3SUM; lower bounds; I.
Decremental SingleSource Shortest Paths on Undirected Graphs in NearLinear Total Update Time
"... AbstractThe decremental singlesource shortest paths (SSSP) problem concerns maintaining the distances between a given source node s to every node in an nnode medge graph G undergoing edge deletions. While its static counterpart can be easily solved in nearlinear time, this decremental problem ..."
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AbstractThe decremental singlesource shortest paths (SSSP) problem concerns maintaining the distances between a given source node s to every node in an nnode medge graph G undergoing edge deletions. While its static counterpart can be easily solved in nearlinear time, this decremental problem is much more challenging even in the undirected unweighted case. In this case, the classic O(mn) total update time of Even and Shiloach (JACM 1981) In contrast to the previous results which rely on maintaining a sparse emulator, our algorithm relies on maintaining a socalled sparse (d, )hop set introduced by Cohen (JACM 2000) in the PRAM literature. A (d, )hop set of a graph G = (V, E) is a set E of weighted edges such that the distance between any pair of nodes in G can be (1 + )approximated by their dhop distance (given by a path containing at most d edges) on G = (V, E ∪E ). Our algorithm can maintain an (n o(1) , )hop set of nearlinear size in nearlinear time under edge deletions. It is the first of its kind to the best of our knowledge. To maintain the distances on this hop set, we develop a monotone boundedhop EvenShiloach tree. It results from extending and combining the monotone EvenShiloach tree of Henzinger, Krinninger, and Nanongkai (FOCS 2013) with the boundedhop SSSP technique of Bernstein (STOC 2013). These two new tools might be of independent interest.