Near Optimal Algorithm for the Directed Single Source Replacement Paths Problem

by   Shiri Chechik, et al.

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^1/2-ϵ) 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).


page 1

page 2

page 3

page 4


Multiple Source Replacement Path Problem

One of the classical line of work in graph algorithms has been the Repla...

Tight Hardness for Shortest Cycles and Paths in Sparse Graphs

Fine-grained reductions have established equivalences between many core ...

Algorithms and Lower Bounds for Replacement Paths under Multiple Edge Failures

This paper considers a natural fault-tolerant shortest paths problem: fo...

Negative-Weight Single-Source Shortest Paths in Near-linear Time

We present a randomized algorithm that computes single-source shortest p...

Deep Distance Sensitivity Oracles

One of the most fundamental graph problems is finding a shortest path fr...

Asymptotic Bounds on the Combinatorial Diameter of Random Polytopes

The combinatorial diameter diam(P) of a polytope P is the maximum shorte...

Kidney exchange and endless paths: On the optimal use of an altruistic donor

We consider a well-studied online random graph model for kidney exchange...

Please sign up or login with your details

Forgot password? Click here to reset