# On the Search Efficiency of 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 flight foraging hypothesis} -- the premise that the movement of various animal species searching for food resembles a \emph{Lévy walk} -- we study the hitting time of parallel Lévy walks on the infinite 2-dimensional grid. Lévy walks are characterized by a parameter $\alpha \in(1,+\infty)$, that is the exponent of the power law distribution of the time intervals at which the moving agent randomly changes direction. In the setting we consider, called the ANTS problem (Feinerman et al. PODC 2012), $k$ independent discrete-time Lévy walks start simultaneously at the origin, and we are interested in the time $h_{k,\ell}$ before some walk visits a given target node on the grid, at distance $\ell$ from the origin. In this setting, we provide a comprehensive analysis of the efficiency of Lévy walks for the complete range of the exponent $\alpha$. For any choice of $\alpha$, we observe that the \emph{total work} until the target is visited, i.e., the product $k \cdot h_{k,\ell}$, is at least $\Omega(\ell^2)$ with constant probability. Our main result is that the right choice for $\alpha$ to get optimal work varies between $1$ and $3$, as a function of the number $k$ of available agents. For instance, when $k = \tilde \Theta(\ell^{1-\epsilon})$, for some positive constant $\epsilon < 1$, then the unique optimal setting for $\alpha$ lies in the \emph{super-diffusive} regime $(2,3)$, namely, $\alpha = 2+\epsilon$. Our results should be contrasted with various previous works in the continuous time-space setting showing that the exponent $\alpha = 2$ is optimal for a wide range of related search problems on the plane.
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https://hal.archives-ouvertes.fr/hal-02530253
Contributor : Francesco d'Amore <>
Submitted on : Thursday, April 16, 2020 - 11:09:10 AM
Last modification on : Tuesday, May 26, 2020 - 6:50:53 PM

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levy_arxiv.pdf
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### Identifiers

• HAL Id : hal-02530253, version 2
• ARXIV : 2004.01562

### Citation

Andrea Clementi, Francesco d'Amore, George Giakkoupis, Emanuele Natale. On the Search Efficiency of Parallel Lévy Walks on ${\mathbb Z}^2$. [Research Report] Inria & Université Cote d'Azur, CNRS, I3S, Sophia Antipolis, France; Università degli Studi di Roma "Tor Vergata"; Univ Rennes, Inria, CNRS, IRISA, France. 2020. ⟨hal-02530253v2⟩

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