Building a Nest by an Automaton

A robot modeled as a deterministic finite automaton has to build a structure from material available to it. The robot navigates in the infinite oriented grid Z x Z. Some cells of the grid are full (contain a brick) and others are empty. The subgraph of the grid induced by full cells, called the field, is initially connected. The (Manhattan) distance between the farthest cells of the field is called its span. The robot starts at a full cell. It can carry at most one brick at a time. At each step it can pick a brick from a full cell, move to an adjacent cell and drop a brick at an empty cell. The aim of the robot is to construct the most compact possible structure composed of all bricks, i.e., a nest. That is, the robot has to move all bricks in such a way that the span of the resulting field be the smallest. Our main result is the design of a deterministic finite automaton that accomplishes this task and subsequently stops, for every initially connected field, in time O(sz), where s is the span of the initial field and z is the number of bricks. We show that this complexity is optimal

Exploring an Infinite Space with Finite Memory Scouts

Consider a small number of scouts exploring the infinite $d$-dimensional grid with the aim of hitting a hidden target point. Each scout is controlled by a probabilistic finite automaton that determines its movement (to a neighboring grid point) based on its current state. The scouts, that operate under a fully synchronous schedule, communicate with each other (in a way that affects their respective states) when they share the same grid point and operate independently otherwise. Our main research question is: How many scouts are required to guarantee that the target admits a finite mean hitting time? Recently, it was shown that $d + 1$ is an upper bound on the answer to this question for any dimension $d \geq 1$ and the main contribution of this paper comes in the form of proving that this bound is tight for $d \in \{ 1, 2 \}$.Comment: Added (forgotten) acknowledgement

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