Using Time to Break Symmetry

Abstract

Two anonymous mobile agents navigate synchronously in an anonymous graph and have to meet at a node, using a deterministic algorithm. This is a symmetry breaking task called rendezvous, equivalent to the fundamental task of leader election between the agents. When is this feasible in a completely anonymous environment? It is known that agents can always meet if their initial positions are nonsymmetric, and that if they are symmetric and agents start simultaneously then rendezvous is impossible. What happens for symmetric initial positions with non-simultaneous start? Can symmetry between the agents be broken by the delay between their starting times? In order to answer these questions, we consider space-time initial configurations (abbreviated by STIC). A STIC is formalized as [(u,v),δ], where u and v are initial nodes of the agents in some graph and δ is a non-negative integer that represents the difference between their starting times. A STIC is feasible if there exists a deterministic algorithm, even dedicated to this particular STIC, which accomplishes rendezvous for it. Our main result is a characterization of all feasible STICs and the design of a universal deterministic algorithm that accomplishes rendezvous for all of them without any a priori knowledge of the agents. Thus, as far as feasibility is concerned, we completely solve the problem of symmetry breaking between two anonymous agents in anonymous graphs. Moreover, we show that such a universal algorithm cannot work for all feasible STICs in time polynomial in the initial distance between the agents.