Line-of-sight rendezvous

Abstract

We consider the rendezvous problem faced by two mobile agents, initially placed according to a known distribution on intersections in Manhattan (nodes of the integer lattice Z2). We assume they can distinguish streets from avenues (the two axes) and move along a common axis in each period (both to an adjacent street or both to an adjacent avenue). However they have no common notion of North or East (positive directions along axes). How should they move, from node to adjacent node, so as to minimize the expected time required to ‘see’ each other, to be on a common street or avenue. This is called ‘line-of-sight’ rendezvous. It is equivalent to a rendezvous problem where two rendezvousers attempt to find each other via two means of communication. We show how this problem can be reduced to a double alternating search (DAS) problem in which a single searcher minimizes the time required to find one of two objects hidden according to known distributions in distinct regions (e.g. a datum held on multiple disks), and we develop a theory for solving the latter problem. The DAS problem generalizes a related search problem introduced earlier by the author and J.V. Howard. We solve the original rendezvous problem in the case that the searchers are initially no more than four streets or avenues apart.