Self-stabilizing Graph Exploration by a Single Agent

TL;DR

This paper introduces two self-stabilizing algorithms enabling a single mobile agent to explore any graph from any initial state, with optimized cover times and minimal memory usage, applicable to various graph sizes and structures.

Contribution

The paper presents novel randomized and deterministic self-stabilizing algorithms for graph exploration by a single agent, with proven optimal or near-optimal cover times and low memory requirements.

Findings

Randomized algorithm achieves expected $O(m)$ cover time for large $c$

Deterministic algorithm has $O(m + nD)$ cover time with minimal memory

Algorithms work from any initial configuration, ensuring full graph exploration

Abstract

In this paper, we present two self-stabilizing algorithms that enable a single (mobile) agent to explore graphs. Starting from any initial configuration, \ie regardless of the initial states of the agent and all nodes, as well as the initial location of the agent, the algorithms ensure the agent visits all nodes. We evaluate the algorithms based on two metrics: the \emph{cover time}, defined as the number of moves required to visit all nodes, and \emph{memory usage}, defined as the storage needed for maintaining the states of the agent and each node. The first algorithm is randomized. Given an integer $c = \Omega(n)$, its cover time is optimal, \ie $O(m)$ in expectation, and its memory requirements are $O(\log c)$ bits for the agent and $O(\log (c+\delta_v))$ bits for each node $v$, where $n$ and $m$ are the numbers of nodes and edges, respectively, and $\delta_v$ is the degree of node $v$. For general $c \ge 2$, its cover time is $O( m \cdot \min(D, \frac{n}{c}+1, \frac{D}{c} + \log n))$, where $D$ is the diameter of a graph. The second algorithm is deterministic. It requires an input integer $k \ge \max(D, \dmax)$, where $\dmax$ is the maximum degree of the graph. The cover time of this algorithm is $O(m + nD)$, and it uses $O(\log k)$ bits of memory for both the agent and each node.