Search via Parallel Lévy Walks on $\mathbb{Z}^2$
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| Format: | Preprint |
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2020
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| _version_ | 1866911790581940224 |
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| author | Clementi, Andrea d'Amore, Francesco Giakkoupis, George Natale, Emanuele |
| author_facet | Clementi, Andrea d'Amore, Francesco Giakkoupis, George Natale, Emanuele |
| contents | Motivated by the Lévy foraging hypothesis -- the premise that various animal species have adapted to follow 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 $α\in(1,\infty)$, 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 $α_{k,\ell} \in (2,3)$, for which the hitting time is $\tilde O(\ell^2/k)$ w.h.p., while modifying the exponent by an $ε$ 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 $α= 2$ 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 $\ell$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2004_01562 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | Search via Parallel Lévy Walks on $\mathbb{Z}^2$ Clementi, Andrea d'Amore, Francesco Giakkoupis, George Natale, Emanuele Distributed, Parallel, and Cluster Computing Probability Quantitative Methods Motivated by the Lévy foraging hypothesis -- the premise that various animal species have adapted to follow 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 $α\in(1,\infty)$, 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 $α_{k,\ell} \in (2,3)$, for which the hitting time is $\tilde O(\ell^2/k)$ w.h.p., while modifying the exponent by an $ε$ 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 $α= 2$ 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 $\ell$. |
| title | Search via Parallel Lévy Walks on $\mathbb{Z}^2$ |
| topic | Distributed, Parallel, and Cluster Computing Probability Quantitative Methods |
| url | https://arxiv.org/abs/2004.01562 |