More efficient periodic traversal in anonymous undirected graphs

TL;DR

This paper improves bounds on the periodic exploration of anonymous undirected graphs by agents with limited memory, showing shorter exploration periods are achievable with careful port number assignments.

Contribution

It presents improved upper bounds for exploration periods for agents with and without memory, and introduces the first non-trivial lower bound for oblivious agents.

Findings

Oblivious agents can explore in at most 4 1/3 n period.

Agents with constant memory can explore in at most 3.5 n period.

A lower bound of 2.8 n period is established for oblivious agents.

Abstract

We consider the problem of periodic graph exploration in which a mobile entity with constant memory, an agent, has to visit all n nodes of an arbitrary undirected graph G in a periodic manner. Graphs are supposed to be anonymous, that is, nodes are unlabeled. However, while visiting a node, the robot has to distinguish between edges incident to it. For each node v the endpoints of the edges incident to v are uniquely identified by different integer labels called port numbers. We are interested in minimisation of the length of the exploration period. This problem is unsolvable if the local port numbers are set arbitrarily. However, surprisingly small periods can be achieved when assigning carefully the local port numbers. Dobrev et al. described an algorithm for assigning port numbers, and an oblivious agent (i.e. agent with no memory) using it, such that the agent explores all graphs of size n within period 10n. Providing the agent with a constant number of memory bits, the optimal length of the period was previously proved to be no more than 3.75n (using a different assignment of the port numbers). In this paper, we improve both these bounds. More precisely, we show a period of length at most 4 1/3 n for oblivious agents, and a period of length at most 3.5n for agents with constant memory. Moreover, we give the first non-trivial lower bound, 2.8n, on the period length for the oblivious case.