Optimal Searcher Distribution for Parallel Random Target Searches

TL;DR

This paper derives the optimal initial distribution of multiple random walkers to minimize search time for a target in a finite domain, revealing different optimal strategies depending on target size and initial overlap considerations.

Contribution

It analytically determines the optimal searcher distribution for large N in a finite domain, including limiting cases and numerical validation.

Findings

Optimal distribution proportional to target^{1/3} when ignoring target volume.

Weak dependence on target distribution when considering finite target volume.

Numerical simulations confirm analytical predictions in 1D and 2D.

Abstract

We consider a problem of finding a target located in a finite $d$-dimensional domain, using $N$ independent random walkers, when partial information on the target location is given as a probability distribution. When $N$ is large, the first-passage time sensitively depends on the initial searcher distribution, which invokes the question of what is the optimal searcher distribution that minimizes the first-passage time. Here, we analytically derive the equation for the optimal distribution and explore its limiting expressions. If the target volume can be ignored, the optimal distribution is proportional to the target distribution to the power of one-third. If we consider a target of a finite volume and the probability of initial overlapping of searchers with the target cannot be ignored in the large $N$ limit, the optimal distribution has a weak dependence on the target distribution, given as a logarithm of the target distribution. Using Langevin dynamics simulations, we numerically demonstrate our predictions in one- and two-dimensions.