Deterministic Rendezvous in Graphs

Abstract

Abstract. Two mobile agents having distinct identifiers and located in nodes of an unknown anonymous connected graph, have to meet at some node of the graph. We seek fast deterministic algorithms for this rendezvous problem, under two scenarios: simultaneous startup, when both agents start executing the algorithm at the same time, and arbitrary startup, when starting times of the agents are arbitrarily decided by an adversary. The measure of performance of a rendezvous algorithm is its cost: for a given initial location of agents in a graph, this is the number of steps since the startup of the later agent until rendezvous is achieved. We first show that rendezvous can be completed at cost O(n + log l) on any n-node tree, where l is the smaller of the two identifiers, even with arbitrary startup. This complexity of the cost cannot be improved for some trees, even with simultaneous startup. Efficient rendezvous in trees relies on fast network exploration and cannot be used when the graph contains cycles. We further study the simplest such network, i.e., the ring. We prove that, with simultaneous startup, optimal cost of rendezvous on any ring is (D log l), where D is the initial distance between agents. We also establish bounds on rendezvous cost in rings with arbitrary startup. For arbitrary connected graphs, our main contribution is a deterministic rendezvous algorithm with cost polynomial in n, τ and log l, where τ is the difference between startup times of the agents. We also show a lower bound (n 2 ) on the cost of rendezvous in some family of graphs. If simultaneous startup is assumed, we construct a generic rendezvous algorithm, working for all connected graphs, which is optimal for the class of graphs of bounded degree, if the initial distance between agents is bounded.

Uniform multi-agent deployment on a ring

“…In their model, either n or k are known to the agents and the agents can place tokens at nodes. Dessmark et al [7] and later Kowalski et al [21] studied the problem of gathering two robots in general graphs, trees and rings. Synchronous agents with distinct identifiers were considered where the identifiers were used in order to break symmetry.…”

How to meet when you forget: log-space rendezvous in arbitrary graphs

“…Authors investigating rendezvous (cf. [4] for an extensive survey) considered either the geometric scenario (rendezvous in an interval of the real line, see, e.g., [11,12,26], or in the plane, see, e.g., [7,8]) or the graph scenario (see, e.g., [16,23,29]). Many papers, e.g., [1,2,6,11,27] study the probabilistic setting: inputs and/or rendezvous strategies are random.…”

“…Under the deterministic scenario two alternative assumptions were used for synchronous rendezvous: either agents are labeled but marking of nodes is not permitted [16,29] or agents are anonymous but they can mark nodes using tokens [18,32]. Labeled agents allowed to mark nodes using whiteboards were considered in [41].…”

“…The network is modeled as an undirected connected graph, and agents traverse links in synchronous rounds. Agents are anonymous, i.e., they are identical and follow the same protocol, as opposed to labeled agents (see, e.g., [16,37]) that have distinct labels and each agent can use its label as a parameter in its protocol. In this paper we consider deterministic rendezvous in arbitrary undirected networks, and seek rendezvous protocols that do not rely on the knowledge of node labels, and can work in anonymous graphs as well.…”

“…A fundamental problem when studying feasibility and efficiency of deterministic rendezvous is that of breaking symmetry. This can be done by using tokens to mark nodes [32], or using different labels of agents [15,16,29]. Under these stronger scenarios rendezvous is sometimes possible even for symmetric initial positions of the agents.…”