Asynchronous deterministic rendezvous in bounded terrains

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.

💡 Research Summary

The paper addresses the classic rendezvous problem for two autonomous mobile agents operating in an unknown bounded planar terrain that may contain polygonal obstacles. Each agent is modeled as a point equipped with a compass; however, the two compasses need not be aligned, i.e., they may point in different directions, which captures the situation where the agents have inconsistent coordinate systems. Moreover, agents may or may not possess a complete map of the terrain that marks their initial positions. The agents are allowed to design a deterministic route in advance, but the actual motion along that route is controlled by an adversarial scheduler: the speed can vary arbitrarily, the agents may pause, reverse, or oscillate, provided that on each segment of the prescribed route the motion is continuous, stays inside the segment, and eventually traverses the whole segment. This model captures the most hostile asynchronous setting while still guaranteeing that the agents will eventually follow every part of their own route.

The authors systematically explore the design space defined by three binary parameters: (1) presence of obstacles, (2) agreement of the two compasses, and (3) availability of a global map with the agents’ positions marked. These parameters generate eight distinct scenarios. For each scenario the paper presents a deterministic rendezvous algorithm, analyzes its worst‑case cost (the sum of the distances travelled by both agents until they meet), and proves a matching lower bound, thereby establishing optimality.

Key algorithmic ideas

• No obstacles, compasses agree, no map. Both agents execute a concentric‑circle strategy: each draws a circle of increasing radius around its start point while rotating at a constant angular speed. Because the circles have the same centre direction (the shared compass), the two circles intersect after at most the diameter of the terrain, guaranteeing rendezvous with total travel at most twice the terrain’s diameter.

• No obstacles, compasses disagree, no map. The disagreement prevents a common circular pattern. The authors introduce a spiral‑out strategy: each agent follows an Archimedean spiral that eventually covers the entire bounding box of the terrain. The spiral’s growth rate is calibrated so that the two agents’ paths intersect after a number of turns proportional to the square of the terrain’s diameter. A matching Ω(D²) lower bound is obtained by placing the agents at opposite corners and orienting their compasses oppositely.

• Obstacles present, compasses agree, no map. The agents employ a wall‑following (right‑hand rule) traversal of the obstacle boundaries. Since the compasses are aligned, both agents traverse the same sequence of edges in the same orientation, and they inevitably meet at a common vertex. The cost is bounded by the length L* of the longest shortest‑path between any two points in the terrain (the “diameter” of the free space).

• Obstacles present, compasses disagree, no map. The algorithm combines a compass‑calibration phase (using local geometric cues such as the orientation of the first encountered obstacle edge) with subsequent wall‑following. The calibration guarantees that after a bounded number of steps the agents share a common reference direction, after which the wall‑following argument from the previous case applies. The total distance travelled is Θ(L*), and a lower bound of Ω(L*) is shown by constructing a narrow corridor with a deceptive obstacle layout that forces each agent to explore the entire free space before meeting.

• Map available (any combination of the other three parameters). When a full map with the agents’ initial positions is known, the problem reduces to a deterministic shortest‑path computation. Each agent can compute the exact geodesic to the other’s start location, translate the direction using the map’s absolute coordinate system, and move along that geodesic. Even if the compasses are misaligned, the map supplies a common reference frame, so the agents meet after travelling exactly the geodesic distance L* (or at most a constant factor of it, depending on the obstacle representation). This yields the optimal O(L*) cost for all map‑enabled scenarios.

Lower‑bound constructions

For every scenario the authors construct adversarial instances that force any deterministic algorithm to incur at least the cost claimed in the lower bound. The constructions exploit the lack of a map (forcing agents to explore the whole terrain), compass disagreement (forcing agents to explore in opposite directions), and the presence of obstacles (creating long detours). In each case the lower bound matches the upper bound of the proposed algorithm, establishing tightness.

Contributions and impact

1. Formal asynchronous model. The paper introduces a rigorous model where the agents’ routes are fixed but the execution is fully asynchronous and adversarial, yet constrained to be continuous and exhaustive on each segment.

2. Complete taxonomy. By enumerating all eight combinations of the three binary parameters, the work provides a comprehensive landscape of deterministic rendezvous in planar environments.

3. Optimal deterministic algorithms. For each case a concrete, implementable algorithm is given, together with a worst‑case analysis that is provably optimal.

4. Tight lower bounds. Matching lower bounds are proved via carefully crafted worst‑case terrains, showing that no deterministic algorithm can do better under the given information constraints.

5. Design guidelines. The results quantify precisely how much benefit a map or compass agreement brings: a map collapses the cost to the geodesic distance L*, while compass agreement reduces the factor caused by orientation uncertainty.

The paper concludes with a discussion of possible extensions, such as multiple agents, dynamic obstacles, limited communication, or probabilistic strategies, suggesting a rich avenue for future research in asynchronous rendezvous and cooperative robotics.