Search for an immobile Hider in a known subset of a network

Abstract

A unit speed Searcher, constrained to start in a given closed set S, wishes to quickly find a point x known to be located in a given closed subset H of a metric network Q. This defines a game G=G(Q,H,S), where the payoff to the maximizing Hider is the time for the Searcher path to reach x. Lengths on Q are defined by a measure λ, which then defines distance as least length of connecting path. For trees Q, we find that the minimax search time (value V of G) is given by V=λ(H)−dH(S)/2, where dH(S) is what we call the ‘H-diameter of S’, and equals the usual diameter d(S) of S in the case H=Q. For the classical case of Gal where the S is a singleton and H=Q, our formula reduces to his result V=λ(Q). If S=H=Q, our formula gives Dagan and Gal's result V=λ(Q)−d(Q)/2. In all other cases, our result is new. Optimal searches consist of minimum length paths covering H which start and end at points of S, traversed equiprobably in either direction.