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Ariadne and Theseus: Exploration and Rendezvous with Two Mobile Agents in an Unknown Graph

Romain Cosson

TL;DR

This work studies two core problems—exploration and rendezvous—for two mobile agents in an unknown undirected graph. It introduces a two-agent extension of Trémaux's DFS via navigation-table formalism and shows that collective exploration can be achieved in exactly $m$ steps, while rendezvous can be guaranteed in at most $\lceil \tfrac{3}{2}m \rceil$ steps, under asynchronous adversarial speeds. The results extend naturally to weighted graphs with total edge length $L$ and include matching lower bounds, strengthening the theoretical limits of multi-agent exploration and coordination. Practically, the methods reduce the runtime compared to classic DFS-based strategies and provide robust, generalizable algorithms for distributed mobile systems with limited communication and adversarial timing constraints.

Abstract

We investigate two fundamental problems in mobile computing: exploration and rendezvous, with two distinct mobile agents in an unknown graph. The agents may communicate by reading and writing information on whiteboards that are located at all nodes. They both move along one adjacent edge at every time-step. In the exploration problem, the agents start from the same arbitrary node and must traverse all the edges. We present an algorithm achieving collective exploration in $m$ time-steps, where $m$ is the number of edges of the graph. This improves over the guarantee of depth-first search, which requires $2m$ time-steps. In the rendezvous problem, the agents start from different nodes of the graph and must meet as fast as possible. We present an algorithm guaranteeing rendezvous in at most $\frac{3}{2}m$ time-steps. This improves over the so-called `wait for Mommy' algorithm which is based on depth-first search and which also requires $2m$ time-steps. Importantly, all our guarantees are derived from a more general asynchronous setting in which the speeds of the agents are controlled by an adversary at all times. Our guarantees generalize to weighted graphs, when replacing the number of edges $m$ with the sum of all edge lengths. We show that our guarantees are met with matching lower-bounds in the asynchronous setting.

Ariadne and Theseus: Exploration and Rendezvous with Two Mobile Agents in an Unknown Graph

TL;DR

This work studies two core problems—exploration and rendezvous—for two mobile agents in an unknown undirected graph. It introduces a two-agent extension of Trémaux's DFS via navigation-table formalism and shows that collective exploration can be achieved in exactly steps, while rendezvous can be guaranteed in at most steps, under asynchronous adversarial speeds. The results extend naturally to weighted graphs with total edge length and include matching lower bounds, strengthening the theoretical limits of multi-agent exploration and coordination. Practically, the methods reduce the runtime compared to classic DFS-based strategies and provide robust, generalizable algorithms for distributed mobile systems with limited communication and adversarial timing constraints.

Abstract

We investigate two fundamental problems in mobile computing: exploration and rendezvous, with two distinct mobile agents in an unknown graph. The agents may communicate by reading and writing information on whiteboards that are located at all nodes. They both move along one adjacent edge at every time-step. In the exploration problem, the agents start from the same arbitrary node and must traverse all the edges. We present an algorithm achieving collective exploration in time-steps, where is the number of edges of the graph. This improves over the guarantee of depth-first search, which requires time-steps. In the rendezvous problem, the agents start from different nodes of the graph and must meet as fast as possible. We present an algorithm guaranteeing rendezvous in at most time-steps. This improves over the so-called `wait for Mommy' algorithm which is based on depth-first search and which also requires time-steps. Importantly, all our guarantees are derived from a more general asynchronous setting in which the speeds of the agents are controlled by an adversary at all times. Our guarantees generalize to weighted graphs, when replacing the number of edges with the sum of all edge lengths. We show that our guarantees are met with matching lower-bounds in the asynchronous setting.
Paper Structure (28 sections, 9 theorems, 1 equation, 3 figures, 4 tables, 1 algorithm)

This paper contains 28 sections, 9 theorems, 1 equation, 3 figures, 4 tables, 1 algorithm.

Table of Contents

  1. Keywords.
  2. Introduction
  3. Main results.
  4. Related works
  5. History of Depth-First Search.
  6. Single mobile agent in an unknown undirected graph.
  7. Collective exploration.
  8. Rendezvous of two mobile agents.
  9. Problem setting and definitions
  10. Navigation model
  11. Move of an agent.
  12. Synchronous and asynchronous models.
  13. Trémaux's algorithm
  14. Navigation table
  15. Exploration by two agents
  16. ...and 13 more sections

Key Result

Proposition 2.1

For any graph $G$ with $m$ edges, Trémaux's algorithm (Algorithm alg:tremaux) has the agent traverse all edges once in each direction and stop at the origin.

Figures (3)

  • Figure 1: Example of execution of Trémaux's algorithm, starting from node A, for a graph with $m=5$ edges.
  • Figure 2: Ariadne (blue) and Theseus (red), exploring a graph with $m=5$ (asynchronous setting).
  • Figure 3: The cycle and the broken cycle, edge lengths are adversarial and unknown.

Theorems & Definitions (43)

  • Proposition 2.1: lucas1883recreations
  • proof
  • Claim 2.2
  • proof
  • Claim 2.3
  • proof
  • Theorem 3.1
  • Claim 3.2
  • proof
  • Claim 3.3
  • ...and 33 more