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

翻译：受列维觅食假说（即多种动物已适应遵循列维飞行以优化搜索效率的假设）启发，我们研究无限二维网格上列维飞行的并行命中时间。考虑$k$条独立离散时间列维飞行（具有相同指数$\alpha \in(1,\infty)$）从同一节点出发，分析首个飞行访问到距离$\ell$处给定目标所需的步数。我们证明：对于大范围内任意$k$和$\ell$的取值，存在唯一最优指数$\alpha_{k,\ell} \in (2,3)$，使得命中时间以高概率为$\tilde O(\ell^2/k)$；而将指数修改$\epsilon$量级会导致命中时间增加多项式因子，或使飞行几乎必然无法命中目标。基于此，我们提出一种针对未知$k$和$\ell$场景的惊人简单且高效的并行搜索策略：每条列维飞行的指数独立且均匀随机地从区间$(2,3)$中选取。该策略在所有可能算法（甚至包括已知$k$的集中式算法）中实现了最优搜索时间（对数多项式因子除外）。我们的结果与先前一系列证明指数$\alpha = 2$在各类搜索问题中最优的研究形成对比。在$k$条并行飞行的设定中，我们证明最优指数取决于$k$和$\ell$，且随机化指数选择可同时适用于所有$k$和$\ell$。