
Multiple Source Replacement Path Problem
One of the classical line of work in graph algorithms has been the Repla...
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Tight Hardness for Shortest Cycles and Paths in Sparse Graphs
Finegrained reductions have established equivalences between many core ...
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Kidney exchange and endless paths: On the optimal use of an altruistic donor
We consider a wellstudied online random graph model for kidney exchange...
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Efficient Exact Paths For Dyck and semiDyck Labeled Path Reachability
The exact path length problem is to determine if there is a path of a gi...
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A* with Perfect Potentials
Quickly determining shortest paths in networks is an important ingredien...
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Distributed Exact Weighted AllPairs Shortest Paths in NearLinear Time
In the distributed allpairs shortest paths problem (APSP), every node ...
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Faster Algorithms for AllPairs Bounded MinCuts
Given a directed graph, the vertex connectivity from u to v is the maxim...
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Near Optimal Algorithm for the Directed Single Source Replacement Paths Problem
In the Single Source Replacement Paths (SSRP) problem we are given a graph G = (V, E), and a shortest paths tree K rooted at a node s, and the goal is to output for every node t ∈ V and for every edge e in K the length of the shortest path from s to t avoiding e. We present an Õ(m√(n) + n^2) time randomized combinatorial algorithm for unweighted directed graphs. Previously such a bound was known in the directed case only for the seemingly easier problem of replacement path where both the source and the target nodes are fixed. Our new upper bound for this problem matches the existing conditional combinatorial lower bounds. Hence, (assuming these conditional lower bounds) our result is essentially optimal and completes the picture of the SSRP problem in the combinatorial setting. Our algorithm extends to the case of small, rational edge weights. We strengthen the existing conditional lower bounds in this case by showing that any O(mn^1/2ϵ) time (combinatorial or algebraic) algorithm for some fixed ϵ >0 yields a truly subcubic algorithm for the weighted All Pairs Shortest Paths problem (previously such a bound was known only for the combinatorial setting).
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