2026-08-10T22:48:22Z https://drops.dagstuhl.de/oai

oai:drops-oai.dagstuhl.de:12436 2024-03-06T10:50:08Z ddc:004 open_access

Simplifying and Unifying Replacement Paths Algorithms in Weighted Directed Graphs Chechik, Shiri Nechushtan, Moran Fault tolerance Distance oracle Planar graph In the replacement paths (RP) problem we are given a graph G and a shortest path P between two nodes s and t . The goal is to find for every edge e ∈ P, a shortest path from s to t that avoids e. The first result of this paper is a simple reduction from the RP problem to the problem of computing shortest cycles for all nodes on a shortest path. Using this simple reduction we unify and extremely simplify two state of the art solutions for two different well-studied variants of the RP problem. In the first variant (algebraic) we show that by using at most n queries to the Yuster-Zwick distance oracle [FOCS 2005], one can solve the the RP problem for a given directed graph with integer edge weights in the range [-M,M] in Õ(M n^ω) time . This improves the running time of the state of the art algorithm of Vassilevska Williams [SODA 2011] by a factor of log⁶n. In the second variant (planar) we show that by using the algorithm of Klein for the multiple-source shortest paths problem (MSSP) [SODA 2005] one can solve the RP problem for directed planar graph with non negative edge weights in O (n log n) time. This matches the state of the art algorithm of Wulff-Nilsen [SODA 2010], but with arguably much simpler algorithm and analysis. Schloss Dagstuhl – Leibniz-Zentrum für Informatik Shiri Chechik and Moran Nechushtan 2020 Is Part Of LIPIcs, Volume 168, 47th International Colloquium on Automata, Languages, and Programming (ICALP 2020) InProceedings Text doc-type:ResearchArticle publishedVersion application/pdf doi:10.4230/LIPIcs.ICALP.2020.29 urn:nbn:de:0030-drops-124365 https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2020.29 eng https://creativecommons.org/licenses/by/3.0/legalcode