(1 + ϵ)-Approximate Shortest Paths in Dynamic Streams

Michael Elkin, Chhaya Trehan

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

Abstract

Computing approximate shortest paths in the dynamic streaming setting is a fundamental challenge that has been intensively studied. Currently existing solutions for this problem either build a sparse multiplicative spanner of the input graph and compute shortest paths in the spanner offline, or compute an exact single source BFS tree. Solutions of the first type are doomed to incur a stretch-space tradeoff of 2κ - 1 versus n1+1/κ, for an integer parameter κ. (In fact, existing solutions also incur an extra factor of 1 + ϵ in the stretch for weighted graphs, and an additional factor of logO(1) n in the space.) The only existing solution of the second type uses n1/2-O(1/κ) passes over the stream (for space O(n1+1/κ)), and applies only to unweighted graphs. In this paper we show that (1 + ϵ)-approximate single-source shortest paths can be computed with Õ(n1+1/κ) space using just constantly many passes in unweighted graphs, and polylogarithmically many passes in weighted graphs. Moreover, the same result applies for multi-source shortest paths, as long as the number of sources is O(n1/κ). We achieve these results by devising efficient dynamic streaming constructions of (1 + ϵ, β)-spanners and hopsets. On our way to these results, we also devise a new dynamic streaming algorithm for the 1-sparse recovery problem. Even though our algorithm for this task is slightly inferior to the existing algorithms of [26, 11], we believe that it is of independent interest.

Original languageEnglish
Title of host publicationApproximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques, APPROX/RANDOM 2022
EditorsAmit Chakrabarti, Chaitanya Swamy
PublisherSchloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing
ISBN (Electronic)9783959772495
DOIs
StatePublished - 1 Sep 2022
Event25th International Conference on Approximation Algorithms for Combinatorial Optimization Problems and the 26th International Conference on Randomization and Computation, APPROX/RANDOM 2022 - Virtual, Urbana-Champaign, United States
Duration: 19 Sep 202221 Sep 2022

Publication series

NameLeibniz International Proceedings in Informatics, LIPIcs
Volume245
ISSN (Print)1868-8969

Conference

Conference25th International Conference on Approximation Algorithms for Combinatorial Optimization Problems and the 26th International Conference on Randomization and Computation, APPROX/RANDOM 2022
Country/TerritoryUnited States
CityVirtual, Urbana-Champaign
Period19/09/2221/09/22

Keywords

  • Approximate Distances
  • Dynamic Streams
  • Hopsets
  • Shortest Paths
  • Spanners

ASJC Scopus subject areas

  • Software

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