# Multiple Source Replacement Path Problem

One of the classical line of work in graph algorithms has been the Replacement Path Problem: given a graph G, s and t, find shortest paths from s to t avoiding each edge e on the shortest path from s to t. These paths are called replacement paths in literature. For an undirected and unweighted graph, (Malik, Mittal, and Gupta, Operation Research Letters, 1989) and (Hershberger and Suri, FOCS 2001) designed an algorithm that solves the replacement path problem in Õ(m+n) time. It is natural to ask whether we can generalize the replacement path problem: can we find all replacement paths from a source s to all vertices in G? This problem is called the Single Source Replacement Path Problem. Recently (Chechik and Cohen, SODA 2019) designed a randomized combinatorial algorithm that solves the Single Source Replacement Path Problem in Õ(m√(n) + n^2) time. One of the questions left unanswered by their work is the case when there are many sources, not one. When there are n sources, the combinatorial algorithm of (Bernstein and Karger, STOC 2009) can be used to find all pair replacement path in Õ(mn + n^3) time. However, there is no result known for any general σ. Thus, the problem we study is defined as follows: given a set of σ sources, we want to find the replacement path from these sources to all vertices in G. We give a randomized combinatorial algorithm for this problem that takes Õ(m√(n σ) + σ n^2) time. This result generalizes both results known for this problem. Our algorithm is much different and arguably simpler than (Chechik and Cohen, SODA 2019). Like them, we show a matching conditional lower bound using the Boolean Matrix Multiplication conjecture.

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research
04/28/2020

### Near Optimal Algorithm for the Directed Single Source Replacement Paths Problem

In the Single Source Replacement Paths (SSRP) problem we are given a gra...
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05/17/2019

### Deterministic Combinatorial Replacement Paths and Distance Sensitivity Oracles

In this work we derandomize two central results in graph algorithms, rep...
research
06/30/2022

### Near Optimal Algorithm for Fault Tolerant Distance Oracle and Single Source Replacement Path problem

In a graph G with a source s, we design a distance oracle that can answe...
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11/02/2022

### Deep Distance Sensitivity Oracles

One of the most fundamental graph problems is finding a shortest path fr...
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10/21/2017

### Seamless Paxos Coordinators

The Paxos algorithm requires a single correct coordinator process to ope...
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05/01/2018

### Generic Single Edge Fault Tolerant Exact Distance Oracle

Given an undirected unweighted graph G and a source set S of |S| = σ so...
research
08/12/2020

### Deterministic Replacement Path Covering

In this article, we provide a unified and simplified approach to derando...