# Search via Parallel Lévy Walks on ${\mathbb Z}^2$

2 COATI - Combinatorics, Optimization and Algorithms for Telecommunications
CRISAM - Inria Sophia Antipolis - Méditerranée , Laboratoire I3S - COMRED - COMmunications, Réseaux, systèmes Embarqués et Distribués
3 WIDE - the World Is Distributed Exploring the tension between scale and coordination
Inria Rennes – Bretagne Atlantique , IRISA-D1 - SYSTÈMES LARGE ÉCHELLE
Abstract : Motivated by the \emph{Lévy foraging hypothesis} -- the premise that various animal species have adapted to follow \emph{Lévy walks} to optimize their search efficiency -- we study the parallel hitting time of Lévy walks on the infinite two-dimensional grid.We consider $k$ independent discrete-time Lévy walks, with the same exponent $\alpha \in(1,\infty)$, that start from the same node, and analyze the number of steps until the first walk visits a given target at distance $\ell$.We show that for any choice of $k$ and $\ell$ from a large range, there is a unique optimal exponent $\alpha_{k,\ell} \in (2,3)$, for which the hitting time is $\tilde O(\ell^2/k)$ w.h.p., while modifying the exponent by an $\epsilon$ term increases the hitting time by a polynomial factor, or the walks fail to hit the target almost surely.Based on that, we propose a surprisingly simple and effective parallel search strategy, for the setting where $k$ and $\ell$ are unknown:The exponent of each Lévy walk is just chosen independently and uniformly at random from the interval $(2,3)$.This strategy achieves optimal search time (modulo polylogarithmic factors) among all possible algorithms (even centralized ones that know $k$).Our results should be contrasted with a line of previous work showing that the exponent $\alpha = 2$ is optimal for various search problems.In our setting of $k$ parallel walks, we show that the optimal exponent depends on $k$ and $\ell$, and that randomizing the choice of the exponents works simultaneously for all $k$ and $\ell$.
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https://hal.archives-ouvertes.fr/hal-02530253
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Submitted on : Friday, February 19, 2021 - 7:14:23 PM
Last modification on : Monday, September 20, 2021 - 2:01:04 PM

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Andrea Clementi, Francesco d'Amore, George Giakkoupis, Emanuele Natale. Search via Parallel Lévy Walks on ${\mathbb Z}^2$. PODC 2021 - ACM Symposium on Principles of Distributed Computing, Jul 2021, Salerno, Italy. pp.81-91, ⟨10.1145/3465084.3467921⟩. ⟨hal-02530253v4⟩

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