Live Exploration of Dynamic Rings

Abstract

In the graph exploration problem, a team of mobile computational entities, called agents, arbitrarily positioned at some nodes of a graph, must cooperate so that each node is eventually visited by at least one agent. In the literature, the main focus has been on graphs that are static; that is, the topology is either invariant in time or subject to localized changes. The few studies on exploration of dynamic graphs have been almost all limited to the centralized case (i.e., assuming complete a priori knowledge of the changes and the times of their occurrence). We investigate the decentralized exploration of dynamic graphs (i.e., when the agents are unaware of the location and timing of the changes) focusing, in this paper, on dynamic systems whose underlying graph is a ring. We first consider the fully-synchronous systems traditionally assumed in the literature; i.e., all agents are active at each time step. We then introduce the notion of semi-synchronous systems, where only a subset of agents might be active at each time step (the choice of the subset is made by an adversary); this model is common in the context of mobile agents in continuous spaces but has never been studied before for agents moving in graphs. Our main focus is on the impact that the level of synchrony as well as other factors such as anonymity, knowledge of the size of the ring, and chirality (i.e., common orientation) have on the solvability of the problem, focusing on the minimum number of agents necessary. We draw an extensive map of feasibility, and of complexity in terms of minimum number of agent movements. All our sufficiency proofs are constructive, and almost all our solution protocols are asymptotically optimal.