Random walks with multiple step lengths

Lucas Boczkowski, Brieuc Guinard, Amos Korman, Zvi Lotker, Marc Renault

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

    5 Scopus citations

    Abstract

    In nature, search processes that use randomly oriented steps of different lengths have been observed at both the microscopic and the macroscopic scales. Physicists have analyzed in depth two such processes on grid topologies: Intermittent Search, which uses two step lengths, and Lévy Walk, which uses many. Taking a computational perspective, this paper considers the number of distinct step lengths k as a complexity measure of the considered process. Our goal is to understand what is the optimal achievable time needed to cover the whole terrain, for any given value of k. Attention is restricted to dimension one, since on higher dimensions, the simple random walk already displays a quasi linear cover time. We say X is a k -intermittent search on the one dimensional n-node cycle if there exists a probability distribution p=(pi)i=1k, and integers L1, L2, …, Lk, such that on each step X makes a jump ± Li with probability pi, where the direction of the jump (+ or −) is chosen independently with probability 1/2. When performing a jump of length Li, the process consumes time Li, and is only considered to visit the last point reached by the jump (and not any other intermediate nodes). This assumption is consistent with biological evidence, in which entities do not search while moving ballistically. We provide upper and lower bounds for the cover time achievable by k-intermittent searches for any integer k. In particular, we prove that in order to reduce the cover time Θ(n2) of a simple random walk to linear in n up to logarithmic factors, roughly lognloglogn step lengths are both necessary and sufficient, and we provide an example where the lengths form an exponential sequence. In addition, inspired by the notion of intermittent search, we introduce the Walk or Probe problem, which can be defined with respect to arbitrary graphs. Here, it is assumed that querying (probing) a node takes significantly more time than moving to a random neighbor. Hence, to efficiently probe all nodes, the goal is to balance the time spent walking randomly and the time spent probing. We provide preliminary results for connected graphs and regular graphs.

    Original languageEnglish
    Title of host publicationLATIN 2018
    Subtitle of host publicationTheoretical Informatics - 13th Latin American Symposium, Proceedings
    EditorsMiguel A. Mosteiro, Michael A. Bender, Martin Farach-Colton
    PublisherSpringer Verlag
    Pages174-186
    Number of pages13
    ISBN (Print)9783319774039
    DOIs
    StatePublished - 1 Jan 2018
    Event13th International Symposium on Latin American Theoretical Informatics, LATIN 2018 - Buenos Aires, Argentina
    Duration: 16 Apr 201819 Apr 2018

    Publication series

    NameLecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
    Volume10807 LNCS
    ISSN (Print)0302-9743
    ISSN (Electronic)1611-3349

    Conference

    Conference13th International Symposium on Latin American Theoretical Informatics, LATIN 2018
    Country/TerritoryArgentina
    CityBuenos Aires
    Period16/04/1819/04/18

    ASJC Scopus subject areas

    • Theoretical Computer Science
    • General Computer Science

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