Probabilistic logarithmic-space algorithms for laplacian solvers

Dean Doron, Francois Le Gall, Amnon Ta-Shma

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

5 Scopus citations

Abstract

A recent series of breakthroughs initiated by Spielman and Teng culminated in the construction of nearly linear time Laplacian solvers, approximating the solution of a linear system Lx = b, where L is the normalized Laplacian of an undirected graph. In this paper we study the space complexity of the problem. Surprisingly we are able to show a probabilistic, logspace algorithm solving the problem. We further extend the algorithm to other families of graphs like Eulerian graphs (and directed regular graphs) and graphs that mix in polynomial time. Our approach is to pseudo-invert the Laplacian, by first "peeling-off" the problematic kernel of the operator, and then to approximate the inverse of the remaining part by using a Taylor series. We approximate the Taylor series using a previous work and the special structure of the problem. For directed graphs we exploit in the analysis the Jordan normal form and results from matrix functions.

Original languageEnglish
Title of host publicationApproximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques - 20th International Workshop, APPROX 2017 and 21st International Workshop, RANDOM 2017
EditorsJose D. P. Rolim, Klaus Jansen, David P. Williamson, Santosh S. Vempala
PublisherSchloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing
ISBN (Electronic)9783959770446
DOIs
StatePublished - 1 Aug 2017
Externally publishedYes
Event20th International Workshop on Approximation Algorithms for Combinatorial Optimization Problems, APPROX 2017 and the 21st International Workshop on Randomization and Computation, RANDOM 2017 - Berkeley, United States
Duration: 16 Aug 201718 Aug 2017

Publication series

NameLeibniz International Proceedings in Informatics, LIPIcs
Volume81
ISSN (Print)1868-8969

Conference

Conference20th International Workshop on Approximation Algorithms for Combinatorial Optimization Problems, APPROX 2017 and the 21st International Workshop on Randomization and Computation, RANDOM 2017
Country/TerritoryUnited States
CityBerkeley
Period16/08/1718/08/17

Keywords

  • Bounded-space complexity classes
  • Laplacian solvers
  • Matrix computation
  • Random walks
  • Randomized logspace

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