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Deterministic Collision-Free Exploration of Unknown Anonymous Graphs

Subhash Bhagat, Andrzej Pelc

TL;DR

The paper tackles collision-free exploration of unknown anonymous graphs by two identical agents. It introduces EXP(n), based on universal exploration sequences, to enable a single agent to visit all nodes in any graph of size at most $n$ in $R(n)$ edge traversals, and constructs Explo-without-Collisions by interleaving EXP(n) with mechanisms for approaching the other agent and resolving symmetry. The main contribution is a collision-free two-agent exploration algorithm that runs in $O(R(n)\log n)$ rounds for any graph with $|V|>2$ under radius-$2$ vision, and it proves impossibility with radius-$1$ in diameter-$2$ trees while showing two-node graphs cannot be handled. The approach is generic: from any one-agent exploration procedure on graphs of size at most $n$, it yields a two-agent collision-free exploration with only a logarithmic slowdown. This advances understanding of distributed exploration under anonymity and collision constraints, with implications for multi-robot coordination and distributed data collection.

Abstract

We consider the fundamental task of network exploration. A network is modeled as a simple connected undirected n-node graph with unlabeled nodes, and all ports at any node of degree d are arbitrarily numbered 0,.....,d-1. Each of two identical mobile agents, initially situated at distinct nodes, has to visit all nodes and stop. Agents execute the same deterministic algorithm and move in synchronous rounds: in each round, an agent can either remain at the same node or move to an adjacent node. Exploration must be collision-free: in every round at most one agent can be at any node. We assume that agents have vision of radius 2: an awake agent situated at a node v can see the subgraph induced by all nodes at a distance at most 2 from v, sees all port numbers in this subgraph, and the agents located at these nodes. Agents do not know the entire graph but they know an upper bound n on its size. The time of an exploration is the number of rounds since the wakeup of the later agent to the termination by both agents. We show a collision-free exploration algorithm working in time polynomial in n, for arbitrary graphs of size larger than 2. Moreover, we show that if agents have only vision of radius 1, then collision-free exploration is impossible, e.g., in any tree of diameter 2.

Deterministic Collision-Free Exploration of Unknown Anonymous Graphs

TL;DR

The paper tackles collision-free exploration of unknown anonymous graphs by two identical agents. It introduces EXP(n), based on universal exploration sequences, to enable a single agent to visit all nodes in any graph of size at most in edge traversals, and constructs Explo-without-Collisions by interleaving EXP(n) with mechanisms for approaching the other agent and resolving symmetry. The main contribution is a collision-free two-agent exploration algorithm that runs in rounds for any graph with under radius- vision, and it proves impossibility with radius- in diameter- trees while showing two-node graphs cannot be handled. The approach is generic: from any one-agent exploration procedure on graphs of size at most , it yields a two-agent collision-free exploration with only a logarithmic slowdown. This advances understanding of distributed exploration under anonymity and collision constraints, with implications for multi-robot coordination and distributed data collection.

Abstract

We consider the fundamental task of network exploration. A network is modeled as a simple connected undirected n-node graph with unlabeled nodes, and all ports at any node of degree d are arbitrarily numbered 0,.....,d-1. Each of two identical mobile agents, initially situated at distinct nodes, has to visit all nodes and stop. Agents execute the same deterministic algorithm and move in synchronous rounds: in each round, an agent can either remain at the same node or move to an adjacent node. Exploration must be collision-free: in every round at most one agent can be at any node. We assume that agents have vision of radius 2: an awake agent situated at a node v can see the subgraph induced by all nodes at a distance at most 2 from v, sees all port numbers in this subgraph, and the agents located at these nodes. Agents do not know the entire graph but they know an upper bound n on its size. The time of an exploration is the number of rounds since the wakeup of the later agent to the termination by both agents. We show a collision-free exploration algorithm working in time polynomial in n, for arbitrary graphs of size larger than 2. Moreover, we show that if agents have only vision of radius 1, then collision-free exploration is impossible, e.g., in any tree of diameter 2.
Paper Structure (8 sections, 2 theorems, 1 figure)

This paper contains 8 sections, 2 theorems, 1 figure.

Table of Contents

  1. Introduction
  2. Our results
  3. Related work
  4. Preliminaries
  5. The algorithm
  6. Discussion
  7. Conclusion
  8. Acknowledgment

Key Result

Lemma 3.1

Upon completion of subroutine Synchronization both agents establish the same round as red, in this round they know that they are awake and they are at the positions they occupied at the start of the subroutine.

Figures (1)

  • Figure 1: An illustration for the Case A: In round $\hat{t}-1$, agents $a$ and $\hat{a}$ were at nodes $u_1$ and $v_1$ respectively. During the round $\hat{t}$, both agents $a$ and $\hat{a}$ move concurrently to the nodes $u_2$ and $v_2$ respectively.

Theorems & Definitions (2)

  • Lemma 3.1
  • Theorem 3.1