Optimal random search for a single hidden target

TL;DR

This paper derives optimal random search strategies for locating a hidden target based on its probability distribution, showing that the best search distribution is proportional to the square root of the target distribution, with specific results for Gaussian targets and network graphs.

Contribution

It introduces a theoretical framework for optimal random search strategies tailored to different target distributions and network structures.

Findings

Optimal search distribution is proportional to the square root of the target distribution.

For Gaussian targets, the optimal search distribution is approximately Gaussian with inverse standard deviation.

In networks, the optimal sampling probability relates to node out-degree.

Abstract

A single target is hidden at a location chosen from a predetermined probability distribution. Then, a searcher must find a second probability distribution from which random search points are sampled such that the target is found in the minimum number of trials. Here it will be shown that if the searcher must get very close to the target to find it, then the best search distribution is proportional to the square root of the target distribution. For a Gaussian target distribution, the optimum search distribution is approximately a Gaussian with a standard deviation that varies inversely with how close the searcher must be to the target to find it. For a network, where the searcher randomly samples nodes and looks for the fixed target along edges, the optimum is to either sample a node with probability proportional to the square root of the out degree plus one or not at all.