Exploration of Faulty Hamiltonian Graphs

TL;DR

This paper studies fault-tolerant network exploration, designing optimal algorithms for rings and analyzing the efficiency of DFS on Hamiltonian graphs, showing near-optimal performance bounds.

Contribution

It introduces a perfectly competitive exploration algorithm for rings and provides bounds on DFS efficiency in Hamiltonian graphs, advancing fault-tolerant exploration strategies.

Findings

Designed a perfectly competitive exploration algorithm for rings.

Proved DFS overhead is at most 10/9 times that of optimal on Hamiltonian graphs.

For large Hamiltonian graphs, DFS overhead is within 6% of optimal.

Abstract

We consider the problem of exploration of networks, some of whose edges are faulty. A mobile agent, situated at a starting node and unaware of which edges are faulty, has to explore the connected fault-free component of this node by visiting all of its nodes. The cost of the exploration is the number of edge traversals. For a given network and given starting node, the overhead of an exploration algorithm is the worst-case ratio (taken over all fault configurations) of its cost to the cost of an optimal algorithm which knows where faults are situated. An exploration algorithm, for a given network and given starting node, is called perfectly competitive if its overhead is the smallest among all exploration algorithms not knowing the location of faults. We design a perfectly competitive exploration algorithm for any ring, and show that, for networks modeled by hamiltonian graphs, the overhead of any DFS exploration is at most 10/9 times larger than that of a perfectly competitive algorithm. Moreover, for hamiltonian graphs of size at least 24, this overhead is less than 6% larger than that of a perfectly competitive algorithm.