Asynchronous deterministic rendezvous in bounded terrains

TL;DR

This paper studies deterministic rendezvous of two mobile agents in unknown bounded polygonal terrains with obstacles, analyzing various scenarios and providing optimal algorithms and lower bounds on their trajectory costs.

Contribution

It introduces a comprehensive framework for rendezvous in polygonal terrains with multiple scenarios, offering tight bounds and optimal algorithms for each case.

Findings

Optimal rendezvous algorithms are designed for all scenarios.

Matching lower bounds prove the optimality of these algorithms.

The cost of rendezvous depends on terrain complexity and agent capabilities.

Abstract

Two mobile agents (robots) have to meet in an a priori unknown bounded terrain modeled as a polygon, possibly with polygonal obstacles. Agents are modeled as points, and each of them is equipped with a compass. Compasses of agents may be incoherent. Agents construct their routes, but the actual walk of each agent is decided by the adversary: the movement of the agent can be at arbitrary speed, the agent may sometimes stop or go back and forth, as long as the walk of the agent in each segment of its route is continuous, does not leave it and covers all of it. We consider several scenarios, depending on three factors: (1) obstacles in the terrain are present, or not, (2) compasses of both agents agree, or not, (3) agents have or do not have a map of the terrain with their positions marked. The cost of a rendezvous algorithm is the worst-case sum of lengths of the agents' trajectories until their meeting. For each scenario we design a deterministic rendezvous algorithm and analyze its cost. We also prove lower bounds on the cost of any deterministic rendezvous algorithm in each case. For all scenarios these bounds are tight.