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

Time Versus Cost Tradeoffs for Deterministic Rendezvous in Networks

Two mobile agents, starting from different nodes of a network at possibly different times, have to meet at the same node. This problem is known as $\mathit{rendezvous}$. Agents move in synchronous rounds. Each agent has a distinct integer label from the set $\{1,\dots,L\}$. Two main efficiency measures of rendezvous are its $\mathit{time}$ (the number of rounds until the meeting) and its $\mathit{cost}$ (the total number of edge traversals). We investigate tradeoffs between these two measures. A natural benchmark for both time and cost of rendezvous in a network is the number of edge traversals needed for visiting all nodes of the network, called the exploration time. Hence we express the time and cost of rendezvous as functions of an upper bound $E$ on the time of exploration (where $E$ and a corresponding exploration procedure are known to both agents) and of the size $L$ of the label space. We present two natural rendezvous algorithms. Algorithm $\mathtt{Cheap}$ has cost $O(E)$ (and, in fact, a version of this algorithm for the model where the agents start simultaneously has cost exactly $E$) and time $O(EL)$. Algorithm $\mathtt{Fast}$ has both time and cost $O(E\log L)$. Our main contributions are lower bounds showing that, perhaps surprisingly, these two algorithms capture the tradeoffs between time and cost of rendezvous almost tightly. We show that any deterministic rendezvous algorithm of cost asymptotically $E$ (i.e., of cost $E+o(E)$) must have time $\Omega(EL)$. On the other hand, we show that any deterministic rendezvous algorithm with time complexity $O(E\log L)$ must have cost $\Omega (E\log L)$