Deterministic Rendezvous at a Node of Agents with Arbitrary Velocities

TL;DR

This paper presents a deterministic rendezvous algorithm for asynchronous agents with arbitrary edge traversal speeds in undirected graphs, enabling agents to meet at nodes despite asynchrony and adversarial edge traversal times.

Contribution

It introduces a novel rendezvous algorithm for agents with restricted asynchrony, where the adversary controls edge traversal speeds, extending previous results to more general asynchronous settings.

Findings

Rendezvous is achievable under restricted asynchrony with arbitrary edge speeds.

The algorithm operates in polynomial time relative to graph size, label length, and maximum edge traversal time.

The approach generalizes previous synchronous and weak asynchronous models to a broader asynchronous context.

Abstract

We consider the task of rendezvous in networks modeled as undirected graphs. Two mobile agents with different labels, starting at different nodes of an anonymous graph, have to meet. This task has been considered in the literature under two alternative scenarios: weak and strong. Under the weak scenario, agents may meet either at a node or inside an edge. Under the strong scenario, they have to meet at a node, and they do not even notice meetings inside an edge. Rendezvous algorithms under the strong scenario are known for synchronous agents. For asynchronous agents, rendezvous under the strong scenario is impossible even in the two-node graph, and hence only algorithms under the weak scenario were constructed. In this paper we show that rendezvous under the strong scenario is possible for agents with restricted asynchrony: agents have the same measure of time but the adversary can arbitrarily impose the speed of traversing each edge by each of the agents. We construct a deterministic rendezvous algorithm for such agents, working in time polynomial in the size of the graph, in the length of the smaller label, and in the largest edge traversal time.