Efficient Oracles and Routing Schemes for Replacement Paths

Davide Bilò, Keerti Choudhary, Luciano Gualà, Stefano Leucci, Merav Parter, Guido Proietti

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review

Abstract

Real life graphs and networks are prone to failure of nodes (vertices) and links (edges). In particular, for a pair of nodes s and t and a failing edge e in an n-vertex unweighted graph G = (V (G), E(G)), the replacement path pG-e(s, t) is a shortest s- t path that avoids e. In this paper we present several efficient constructions that, for every (s, t) ? S ×T, where S, T ? V (G), and every e ? E(G), maintain the collection of all pG-e(s, t), either implicitly (i.e., through compact data structures a.k.a. distance sensitivity oracles (DSO)), or explicitly (i.e., through sparse subgraphs a.k.a. fault-tolerant preservers (FTP)). More precisely, we provide the following results: (1) DSO: For every S, T ? V (G), we construct a DSO for maintaining S × T distances under single edge (or vertex) faults. This DSO has size O(n|S||T|) and query time of O(|S||T|). At the expense of having quasi-polynomial query time, the size of the oracle can be improved to O(n|S| + |T||S|n), which is optimal for |T| = (n|S|). When |T| = (n4 3 |S|1 4 ), the construction can be further refined in order to get a polynomial query time. We also consider the approximate additive setting, and show a family of DSOs that exhibits a tradeo between the additive stretch and the size of the oracle. Finally, for the meaningful single-source case, the above result is complemented by a lower bound conditioned on the Set-Intersection conjecture. This lower bound establishes a separation between the oracle and the subgraph settings. (2) FTP: We show the construction of a path-reporting DSO of size O(n4/3(|S||T|)1/3) reporting pG-e(s, t) in O(|pG-e(s, t)| + (n|S||T|)1/3) time. Such a DSO can be transformed into a FTP having the same size, and moreover it can be elaborated in order to make it optimal (up to a poly-logarithmic factor) both in space and query time for the special case in which T = V (G). Our FTP improves over previous constructions when |T| = O(|S|n) (up to inverse poly-logarithmic factors). (3) Routing and Labeling Schemes: For the well-studied single-source setting, we present a novel routing scheme, that allows to route messages on pG-e(s,t) by using edge labels and routing tables of size O(n), and a header message of poly-logarithmic size. We also present a labeling scheme for the setting which is optimal in space up to constant factors.

Original languageEnglish
Title of host publication35th Symposium on Theoretical Aspects of Computer Science, STACS 2018
EditorsRolf Niedermeier, Brigitte Vallee
ISBN (Electronic)9783959770620
DOIs
StatePublished - 2 Feb 2018
Externally publishedYes
Event35th Symposium on Theoretical Aspects of Computer Science, STACS 2018 - Caen, France
Duration: 28 Feb 20183 Mar 2018

Publication series

NameLeibniz International Proceedings in Informatics, LIPIcs
Volume96

Conference

Conference35th Symposium on Theoretical Aspects of Computer Science, STACS 2018
Country/TerritoryFrance
CityCaen
Period28/02/183/03/18

Keywords

  • Fault Tolerant
  • Oracle
  • Routing
  • Shortest Path

All Science Journal Classification (ASJC) codes

  • Software

Fingerprint

Dive into the research topics of 'Efficient Oracles and Routing Schemes for Replacement Paths'. Together they form a unique fingerprint.

Cite this