TY - GEN
T1 - Random walks with multiple step lengths
AU - Boczkowski, Lucas
AU - Guinard, Brieuc
AU - Korman, Amos
AU - Lotker, Zvi
AU - Renault, Marc
N1 - Publisher Copyright:
© Springer International Publishing AG, part of Springer Nature 2018.
PY - 2018/1/1
Y1 - 2018/1/1
N2 - 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.
AB - 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.
UR - http://www.scopus.com/inward/record.url?scp=85045386689&partnerID=8YFLogxK
U2 - 10.1007/978-3-319-77404-6_14
DO - 10.1007/978-3-319-77404-6_14
M3 - Conference contribution
AN - SCOPUS:85045386689
SN - 9783319774039
T3 - Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
SP - 174
EP - 186
BT - LATIN 2018
A2 - Mosteiro, Miguel A.
A2 - Bender, Michael A.
A2 - Farach-Colton, Martin
PB - Springer Verlag
T2 - 13th International Symposium on Latin American Theoretical Informatics, LATIN 2018
Y2 - 16 April 2018 through 19 April 2018
ER -