Graph Decomposition for Improving Memoryless Periodic Exploration

Abstract

We consider a general framework in which a memoryless robot periodically explores all the nodes of a connected anonymous graph by following local information available at each vertex. For each vertex v, the endpoints of all edges adjacent to v are assigned unique labels from the range 1 to deg(v) (the degree of v). The generic exploration strategy is implemented using a right-hand-rule transition function: after entering vertex v via the edge labeled i, the robot proceeds with its exploration, leaving via the edge having label [i mod deg(v)]+1 at v.

A lot of attention has been given to the problem of labeling the graph so as to achieve a periodic exploration having the minimum possible length π. It has recently been proved [Czyzowicz et al., Proc. SIROCCO’09 [1]] that $\pi \leq 4\frac13 n$ holds for all graphs of n vertices. Herein, we provide a new labeling scheme which leads to shorter exploration cycles, improving the general bound to π ≤ 4 n − 2. This main result is shown to be tight with respect to the class of labelings admitting certain connectivity properties. The labeling scheme is based on a new graph decomposition which may be of independent interest.

The research was partially funded by the State Committee for Scientific Research (Poland) Grant 4 T11C 047 25, by the ANR-project “ALADDIN” (France), and by the project “CEPAGE” of INRIA (France).