Setting port numbers for fast graph exploration

International audienceWe consider the problem of periodic graph exploration by a finite automaton in which an automaton with a constant number of states has to explore all unknown anonymous graphs of arbitrary size and arbitrary maximum degree. In anonymous graphs, nodes are not labeled but edges are labeled in a local manner (called {\em local orientation}) so that the automaton is able to distinguish them. Precisely, the edges incident to a node $v$ are given port numbers from $1$ to $d_v$, where $d_v$ is the degree of~$v$. Periodic graph exploration means visiting every node infinitely often. We are interested in the length of the period, i.e., the maximum number of edge traversals between two consecutive visits of any node by the automaton in the same state and entering the node by the same port. This problem is unsolvable if local orientations are set arbitrarily. Given this impossibility result, we address the following problem: what is the mimimum function $\pi(n)$ such that there exist an algorithm for setting the local orientation, and a finite automaton using it, such that the automaton explores all graphs of size $n$ within the period $\pi(n)$? The best result so far is the upper bound $\pi(n)\leq 10n$, by Dobrev et al. [SIROCCO 2005], using an automaton with no memory (i.e. only one state). In this paper we prove a better upper bound $\pi(n)\leq 4n$. Our automaton uses three states but performs periodic exploration independently of its starting position and initial state

Interval Routing Schemes allow Broadcasting with Linear Message-Complexity

The purpose of compact routing is to provide a labeling of the nodes of a network, and a way to encode the routing tables so that routing can be performed eciently (e.g., on shortest paths) while keeping the memory-space required to store the routing tables as small as possible. In this paper, we answer a long-standing conjecture by showing that compact routing can also help to perform distributed computations. In particular, we show that a network supporting a shortest path interval routing scheme allows to broadcast with an O(n) message-complexity, where n is the number of nodes of the network. As a consequence, we prove that O(n) messages suce to solve leader-election for any graph labeled by a shortest path interval routing scheme, improving therefore the O(m + n) previous known bound. A general consequence of our result is that a shortest path interval routing scheme contains ample structural information to avoid developing ad-hoc or network-specic solutions for basic p..