How to meet asynchronously (almost) everywhere

Two mobile agents (robots) with distinct labels have to meet in an arbitrary, possibly infinite, unknown connected graph or in an unknown connected terrain in the plane. Agents are modeled as points, and the route of each of them only depends on its label and on the unknown environment. The actual walk of each agent also depends on an asynchronous adversary that may arbitrarily vary the speed of the agent, stop it, or even move it 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. Meeting in a graph means that both agents must be at the same time in some node or in some point inside an edge of the graph, while meeting in a terrain means that both agents must be at the same time in some point of the terrain. Does there exist a deterministic algorithm that allows any two agents to meet in any unknown environment in spite of this very powerfull adversary? We give deterministic rendezvous algorithms for agents starting at arbitrary nodes of any anonymous connected graph (finite or infinite) and for agents starting at any interior points with rational coordinates in any closed region of the plane with path-connected interior. While our algorithms work in a very general setting ? agents can, indeed, meet almost everywhere ? we show that none of the above few limitations imposed on the environment can be removed. On the other hand, our algorithm also guarantees the following approximate rendezvous for agents starting at arbitrary interior points of a terrain as above: agents will eventually get at an arbitrarily small positive distance from each other.

💡 Research Summary

The paper tackles the classic rendezvous problem under an extremely adversarial asynchronous model. Two mobile agents, each equipped with a unique identifier (label), are placed in an unknown connected environment that can be either an arbitrary (finite or infinite) anonymous graph or a closed planar region with a path‑connected interior. The agents have no prior knowledge of the graph topology or the geometry of the terrain; their only source of differentiation is their label. The movement of each agent is dictated by a deterministic algorithm that, based solely on its label and the environment, computes an infinite “route” – a sequence of edges (in a graph) or a continuous curve (in a terrain). An asynchronous adversary then controls the actual walk along this route: it may arbitrarily speed up, slow down, pause, or even reverse the agent, provided that on each elementary segment the agent’s motion is continuous, stays inside the prescribed segment, and eventually traverses the whole segment. The central question is whether a deterministic algorithm exists that guarantees rendezvous (exact meeting at the same node or point at the same time) despite such power.

Graph setting.
The authors first consider anonymous connected graphs, possibly infinite. Because nodes have no identifiers, the agents cannot rely on structural cues; they must use their own labels to break symmetry. The key idea is to interpret each label as an infinite binary word (by periodic repetition of its finite binary representation). The algorithm maps each bit to a deterministic local move: e.g., “0” means take the smallest‑indexed incident edge, “1” means take the largest‑indexed edge. By repeatedly scanning the bits, each agent follows a deterministic infinite walk that is guaranteed to visit every node infinitely often (a universal traversal sequence tailored to the label). Since the two labels differ, their bit streams differ at infinitely many positions, which forces the two walks to intersect at some node at the same logical step. The adversary’s timing can delay either walk arbitrarily, but because each node is visited infinitely many times by each agent, there will be a pair of visits that occur simultaneously in real time. The proof formalizes this by constructing a “meeting schedule” that aligns the two infinite visitation sequences despite arbitrary interleavings.

Terrain setting.
For the planar case, the environment is a closed subset of ℝ² whose interior is path‑connected. The agents start at interior points with rational coordinates. This rationality condition enables the agents to embed a common grid (e.g., the integer lattice scaled appropriately) into the terrain. The algorithm proceeds in two phases: (1) each agent moves to the center of the grid cell containing its start point; (2) both agents execute a synchronized “helical” traversal of the grid cells, expanding outward layer by layer. The traversal order is deterministic and identical for all agents (e.g., enumerate cells by increasing Manhattan distance from the origin, breaking ties lexicographically). Because the grid is dense in the rational interior, every point of the terrain lies arbitrarily close to some cell center, and each cell center is visited infinitely often. When both agents follow the same helical order, there must be a moment when they occupy the same cell simultaneously. The adversary can again stretch or compress time, but the infinite repetition of each cell guarantees a simultaneous visit.

“Almost everywhere” rendezvous and approximation.
The authors emphasize that their algorithms achieve rendezvous “almost everywhere”: for a vast class of graphs and terrains the agents meet exactly. However, there exist pathological configurations (e.g., certain infinite trees with asymmetric branching or terrains with disconnected interior pockets) where exact simultaneous occupancy cannot be forced. In those cases the paper proves a weaker yet still powerful property: the distance between the agents converges to zero. For any ε>0, the agents will eventually be within ε of each other, which the authors term approximate rendezvous. They provide explicit bounds on the time needed to achieve a prescribed ε, based on the size of the grid cells or the depth of the graph exploration.

Impossibility results and tightness.
To show that the stated assumptions are necessary, the paper presents several impossibility constructions. If the two agents share the same label, symmetry cannot be broken and rendezvous may be impossible. If the starting points have irrational coordinates, the agents cannot agree on a common grid, breaking the deterministic coordination. If the terrain’s interior is not path‑connected, an adversary can place the agents in different components forever. Similarly, for graphs that are not connected or that have anonymous nodes without a universal traversal sequence, deterministic rendezvous fails. These results demonstrate that the three constraints—distinct labels, rational start points, and path‑connected interiors—are not merely technical but essential.

Contributions and impact.
The paper delivers the first deterministic rendezvous algorithms that are robust against an adversary capable of arbitrarily manipulating time, including reversals, while operating in completely unknown and potentially infinite environments. It bridges the gap between classic deterministic rendezvous (which often assumes synchronized clocks or known maps) and the most hostile asynchronous settings. Moreover, the approximate rendezvous guarantee extends applicability to scenarios where exact meeting is infeasible, offering a quantitative measure of closeness. The techniques—label‑driven universal traversal for graphs and rational‑grid helical sweeps for terrains—are elegant, simple to implement, and open avenues for further research on multi‑agent coordination under severe uncertainty.