Search via Parallel L{é}vy Walks on ${\mathbb Z}^2$

Motivated by the \emph{L{\'e}vy foraging hypothesis} -- the premise that various animal species have adapted to follow \emph{L{\'e}vy walks} to optimize their search efficiency -- we study the parallel hitting time of L{\'e}vy walks on the infinite two-dimensional grid.We consider $k$ independent discrete-time L{\'e}vy 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 L{\'e}vy 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 $\ell$.