How well do random walks parallelize?

Klim Efremenko, Omer Reingold

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

36 Scopus citations


A random walk on a graph is a process that explores the graph in a random way: at each step the walk is at a vertex of the graph, and at each step it moves to a uniformly selected neighbor of this vertex. Random walks are extremely useful in computer science and in other fields. A very natural problem that was recently raised by Alon, Avin, Koucky, Kozma, Lotker, and Tuttle (though it was implicit in several previous papers) is to analyze the behavior of k independent walks in comparison with the behavior of a single walk. In particular, Alon et al. showed that in various settings (e.g., for expander graphs), k random walks cover the graph (i.e., visit all its nodes), Ω(k)-times faster (in expectation) than a single walk. In other words, in such cases k random walks efficiently "parallelize" a single random walk. Alon et al. also demonstrated that, depending on the specific setting, this "speedup" can vary from logarithmic to exponential in k. In this paper we initiate a more systematic study of multiple random walks. We give lower and upper bounds both on the cover time and on the hitting time (the time it takes to hit one specific node) of multiple random walks. Our study revolves over three alternatives for the starting vertices of the random walks: the worst starting vertices (those who maximize the hitting/cover time), the best starting vertices, and starting vertices selected from the stationary distribution. Among our results, we show that the speedup when starting the walks at the worst vertices cannot be too large - the hitting time cannot improve by more than an O(k) factor and the cover time cannot improve by more than min {k logn,k 2} (where n is the number of vertices). These results should be contrasted with the fact that there was no previously known upper-bound on the speedup and that the speedup can even be exponential in k for random starting vertices. Some of these results were independently obtained by Elsässer and Sauerwald (ICALP 2009). We further show that for k that is not too large (as a function of various parameters of the graph), the speedup in cover time is O(k) even for walks that start from the best vertices (those that minimize the cover time). As a rather surprising corollary of our theorems, we obtain a new bound which relates the cover time C and the mixing time mix of a graph. Specifically, we show that C = O (m√mix log2 n) (where m is the number of edges).

Original languageEnglish
Title of host publicationApproximation, Randomization, and Combinatorial Optimization
Subtitle of host publicationAlgorithms and Techniques - 12th International Workshop, APPROX 2009 and 13th International Workshop, RANDOM 2009, Proceedings
Number of pages14
StatePublished - 6 Nov 2009
Externally publishedYes
Event12th International Workshop on Approximation Algorithms for Combinatorial Optimization Problems, APPROX 2009 and 13th International Workshop on Randomization and Computation, RANDOM 2009 - Berkeley, CA, United States
Duration: 21 Aug 200923 Aug 2009

Publication series

NameLecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
Volume5687 LNCS
ISSN (Print)0302-9743
ISSN (Electronic)1611-3349


Conference12th International Workshop on Approximation Algorithms for Combinatorial Optimization Problems, APPROX 2009 and 13th International Workshop on Randomization and Computation, RANDOM 2009
Country/TerritoryUnited States
CityBerkeley, CA


  • Markov chains
  • Random walks

ASJC Scopus subject areas

  • Theoretical Computer Science
  • Computer Science (all)


Dive into the research topics of 'How well do random walks parallelize?'. Together they form a unique fingerprint.

Cite this