Random Walks with Multiple Step Lengths

  • Lucas Boczkowski
  • Brieuc Guinard
  • Amos Korman
  • Zvi Lotker
  • Marc Renault
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10807)


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 \(\mathbf{p }=(p_i)_{i=1}^k\), and integers \(L_1,L_2,\ldots , L_k\), such that on each step X makes a jump \(\pm L_i\) with probability \(p_i\), where the direction of the jump (\(+\) or −) is chosen independently with probability 1/2. When performing a jump of length \(L_i\), the process consumes time \(L_i\), 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 \({\varTheta }(n^2)\) of a simple random walk to linear in n up to logarithmic factors, roughly \(\frac{\log n}{\log \log n}\) 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.


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Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.IRIF, CNRS and University Paris DiderotParisFrance
  2. 2.Ben Gurion University of the NegevBeershebaIsrael
  3. 3.Computer Sciences DepartmentUniversity of Wisconsin - MadisonMadisonUSA

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