在$\mathbb{Z}^2$上通过并行Lévy行走进行搜索
Search via Parallel Lévy Walks on $\mathbb{Z}^2$
AI总结：
该研究针对二维网格上的并行搜索问题，分析了k个并行Lévy行走的击中时间，发现最优指数依赖于k和目标距离ℓ，提出随机选取(2,3)区间内指数的策略，可在未知参数下实现接近最优的搜索效率。
摘要：
受Lévy觅食假说——即多种动物已演化出遵循Lévy行走以优化搜索效率的假设——的启发，我们研究了无限二维网格上Lévy行走的并行击中时间。我们考虑$k$个独立的离散时间Lévy行走，它们具有相同的指数$\alpha\in(1,\infty)$，从同一节点出发，分析首次有行走到达距离为$\ell$的给定目标所需的步数。我们证明，对于很大范围内的任意$k$和$\ell$选择，存在唯一的最优指数$\alpha_{k,\ell} \in (2,3)$，使得击中时间以高概率为$\tilde O(\ell^2/k)$，而将指数改变一个$\varepsilon$项会使击中时间增加多项式因子，或者行走几乎必然无法击中目标。基于此，我们针对$k$和$\ell$未知的场景，提出了一种异常简单且有效的并行搜索策略：每个Lévy行走的指数仅需从区间$(2,3)$中独立均匀随机选取。该策略在所有可能的算法（甚至是知晓$k$的集中式算法）中，实现了最优搜索时间（至多相差多对数因子）。我们的结果与一系列先前的研究形成对比，后者表明指数$\alpha=2$对于各类搜索问题都是最优的。在我们的$k$个并行行走的设定下，我们证明最优指数取决于$k$和$\ell$，并且随机选择指数的方法可同时适用于所有$k$和$\ell$。
英文摘要：
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 $\ell$.
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