TY - GEN

T1 - Near optimal algorithm for the directed single source replacement paths problem

AU - Chechik, Shiri

AU - Magen, Ofer

N1 - Publisher Copyright:
© Shiri Chechik and Ofer Magen; licensed under Creative Commons License CC-BY 47th International Colloquium on Automata, Languages, and Programming (ICALP 2020).

PY - 2020/6/1

Y1 - 2020/6/1

N2 - In the Single Source Replacement Paths (SSRP) problem we are given a graph G = (V, E), and a shortest paths tree Kb rooted at a node s, and the goal is to output for every node t ∈ V and for every edge e in Kb the length of the shortest path from s to t avoiding e. We present an Õ(m√n + n2) time randomized combinatorial algorithm for unweighted directed graphs1. 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 naturally extends to the case of small, rational edge weights. In the full version of the paper, we strengthen the existing conditional lower bounds in this case by showing that any O(mn1/2−ε) time (combinatorial or algebraic) algorithm for some fixed ε > 0 yields a truly sub-cubic algorithm for the weighted All Pairs Shortest Paths problem (previously such a bound was known only for the combinatorial setting).

AB - In the Single Source Replacement Paths (SSRP) problem we are given a graph G = (V, E), and a shortest paths tree Kb rooted at a node s, and the goal is to output for every node t ∈ V and for every edge e in Kb the length of the shortest path from s to t avoiding e. We present an Õ(m√n + n2) time randomized combinatorial algorithm for unweighted directed graphs1. 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 naturally extends to the case of small, rational edge weights. In the full version of the paper, we strengthen the existing conditional lower bounds in this case by showing that any O(mn1/2−ε) time (combinatorial or algebraic) algorithm for some fixed ε > 0 yields a truly sub-cubic algorithm for the weighted All Pairs Shortest Paths problem (previously such a bound was known only for the combinatorial setting).

KW - Combinatorial algorithms

KW - Conditional lower bounds

KW - Fault tolerance

KW - Replacement Paths

UR - http://www.scopus.com/inward/record.url?scp=85089355382&partnerID=8YFLogxK

U2 - 10.4230/LIPIcs.ICALP.2020.81

DO - 10.4230/LIPIcs.ICALP.2020.81

M3 - ???researchoutput.researchoutputtypes.contributiontobookanthology.conference???

AN - SCOPUS:85089355382

T3 - Leibniz International Proceedings in Informatics, LIPIcs

BT - 47th International Colloquium on Automata, Languages, and Programming, ICALP 2020

A2 - Czumaj, Artur

A2 - Dawar, Anuj

A2 - Merelli, Emanuela

PB - Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing

T2 - 47th International Colloquium on Automata, Languages, and Programming, ICALP 2020

Y2 - 8 July 2020 through 11 July 2020

ER -