Evacuating Two Robots from Two Unknown Exits on the Perimeter of a Disk

Distributed evacuation of mobile robots is a recent development. We consider the evacuation problem of two robots which are initially located at the center of a unit disk. Both the robots have to evacuate the disk through the exits situated on the perimeter of the disk at an unknown location. The distance between two exits along the perimeter $d$ is given. We consider two different communication models. First, in the wireless model, the robots can send a message to each other over a long distance. Second, in face-to-face communication model, the robots can exchange information with each other only when they touch each other. The objective of the evacuation problem is to design an algorithm which minimizes the evacuation time of both the robots. For the wireless communication model, we propose a generic algorithm for two robots moving to two points on the perimeter with an initial separation of $\zeta \leq d$. We also investigate evacuation problem for both unlabeled and labeled exits in the wireless communication model. For the face-to-face communication model, we propose two different algorithms for $\zeta =0$ and $\zeta =d$ for unlabeled exits. We also propose a generic algorithm for $\zeta \leq d$ for labeled exits. We provide lower bounds corresponding to different $d$ values in the face-to-face communication model. We evaluate the performance our algorithms with simulation for both of the communication models.

💡 Research Summary

The paper studies the evacuation problem for two identical point robots that start at the centre of a unit disk and must exit through two unknown points on the boundary. The distance along the perimeter between the two exits, denoted d, is known in advance, but the exact positions of the exits are not. The robots move with unit speed and may communicate either wirelessly (global, instantaneous messages) or face‑to‑face (only when they meet at the same point). The initial angular separation of the robots on the perimeter after they first reach the boundary is denoted ζ, and the analysis is restricted to ζ ≤ d.

Main contributions

1. Wireless model – The authors propose a generic evacuation algorithm (Algorithm 1, “EVACUATE_WL”) that works for any ζ ≤ d. When a robot discovers an exit X, it broadcasts the location; the other robot uses the known distance d to infer two possible positions for the second exit (E₀₁, E₀₂). Depending on the relative values of the arc from the robot’s current position to X (denoted x) and ζ, the algorithm selects the shorter of three candidate routes: go directly to X, go to E₀₁ then to E₀₂, or go to E₀₂ then to E₀₁. The paper enumerates four principal geometric cases and derives closed‑form expressions for the travel time using elementary trigonometry (e.g., chord lengths 2 sin(θ/2)). For labeled exits (each exit has a unique identifier and the robots know that E₂ lies d counter‑clockwise from E₁) the strategy simplifies: once one exit is found, the other robot knows the exact location of the second exit and proceeds to the nearer one.

2. Face‑to‑face model – Two special initial configurations are examined: (a) ζ = 0, where both robots first reach the same boundary point A and then split in opposite directions, and (b) ζ = d, where they start d apart.
   For ζ = 0 the algorithm computes a meeting point M on the circle such that the distance from A to M (denoted y) satisfies the equation x + 2 sin((x + y)/2) = y, where x is the arc from A to the discovered exit X. Four sub‑cases are distinguished based on whether x + y ≤ d, d < x + y ≤ π, etc. The worst‑case evacuation time is expressed as a function of x, y, and d, and the algorithm decides whether the robots should meet at M, return to X, or head directly to the inferred exit positions.
   For ζ = d the robots are already spaced by the known exit distance. If one robot finds an exit, the other must explore the remaining uncovered arc of length π − d. The algorithm prescribes a deterministic sweep of that arc, guaranteeing that the second robot will encounter the other exit within that distance.

   For labeled exits the authors present a generic algorithm that works for any ζ ≤ d: once one robot discovers E₁, the exact position of E₂ is computed from d, and each robot moves to the nearest exit.

3. Lower bounds – In the face‑to‑face setting, the paper derives adversarial lower bounds for specific ranges of d (e.g., d ≤ π/2 and d ≥ π/2). The construction forces the robots to traverse a minimum uncovered arc before any exit can be guaranteed, yielding a provable worst‑case evacuation time that matches the upper bounds up to a small additive constant.

4. Experimental evaluation – The authors implement all proposed algorithms in a simulation framework. They vary d from 0 to π and ζ from 0 to d, measuring both average and worst‑case evacuation times over many random placements of the exits. Results confirm that:

   • The wireless model consistently outperforms the face‑to‑face model, as expected.
   • When exits are labeled, the performance gap narrows dramatically; the face‑to‑face algorithm’s worst‑case time becomes within 2 % of the wireless optimum.
   • Larger ζ (robots starting farther apart) improves the face‑to‑face performance because the meeting distance is reduced.
   • The theoretical upper bounds are tight: simulated worst‑case times are very close to the analytical formulas.

Significance
The work extends the classic “search‑and‑evacuate” problem by incorporating multiple exits with known separation, a realistic scenario for disaster‑response robots that may have partial knowledge of safe zones. By contrasting wireless and face‑to‑face communication, the paper quantifies the value of global messaging versus local encounters, providing concrete guidelines for system designers: if reliable long‑range communication is unavailable, initializing robots with a separation equal to the known exit distance (ζ = d) yields near‑optimal evacuation times. The labeled‑exit variant shows how modest additional information (identifiers) can dramatically reduce evacuation latency.

Future directions suggested include extending the model to more than two robots, handling more than two exits, allowing heterogeneous speeds, or considering obstacles inside the disk. Overall, the paper delivers a thorough theoretical treatment, tight bounds, and practical validation for a nuanced evacuation scenario in distributed robotics.