More efficient periodic traversal in anonymous undirected graphs

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

On the Solvability of Anonymous Partial Grids Exploration by Mobile Robots

Given an arbitrary partial anonymous grid (a finite grid with possibly missing vertices or edges), this paper focuses on the exploration of such a grid by a set of mobile anonymous agents (called robots). Assuming that the robots can move synchronously, but cannot communicate with each other, the aim is to design an algorithm executed by each robot that allows, as many robots as possible (let k be this maximal number), to visit infinitely often all the vertices of the grid, in such a way that no vertex hosts more than one robot at a time, and each edge is traversed by at most one robot at a time. The paper addresses this problem by considering a central parameter, denoted p, that captures the view of each robot. More precisely, it is assumed that each robot sees the part of the grid (and its current occupation by other robots, if any) centered at the vertex it currently occupies and delimited by the radius p. Based on such a radius notion, the paper investigates the following cases and presents associated results. It first shows that there is no solution (i.e., k = 0) when = 0. It then shows that k

Fast Periodic Graph Exploration with Constant Memory

International audienc

Graph Explorations with Mobile Agents

International audienceThe basic primitive for a mobile agent is the ability to visit all the nodes of the graph in a systematic manner. This chapter considers the exploration of unknown graphs in full detail, for the specific mobile agent model considered in this book. The graph is considered to be finite, undirected and connected. Other than this fact, no prior knowledge of the graph is assumed. Several exploration techniques are introduced and explained for either a single agent, or multiple agents, exploring either labelled or unlabelled graphs. We focus on the efficiency of exploration and consider three different complexity measures, the time taken, the amount of memory used by the agents and the storage needed at each node of the graph. For exploration by multiple agents, we consider collaborative exploration by a team of colocated agents as well as distributed exploration by agents scattered in a graph. The concluding section presents some brief ideas and references on more advanced topics on graph exploration that are not covered in this chapter

Moving and Computing Models: Agents

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Dangerous graphs

Anomalies and faults are inevitable in computer networks, today more than ever before. This is due to the large scale and dynamic nature of the networks used to process big data and to the ever-increasing number of ad-hoc devices. Beyond natural faults and anomalies occurring in a network, threats proceeding from attacks conducted by malicious intruders must be considered. Consequently, there is often a need to quickly isolate and even repair a fault in a network when it appears. Furthermore, despite the presence in a network of faults stemming from malicious entities, we need to identify the latter and their behaviours, and develop protocols resilient to their attacks. Thus, defining models to capture the dangers inherent to various faults, anomalies and threats in a network and studying such threats, has become increasingly important and popular. Threats in networks can be of two kinds: either mobile or stationary. A malicious mobile process can move along the network, whereas a stationary harmful process resides in a host. One of the most studied models for stationary harmful processes is the black hole, which was introduced by Dobrev, Flocchini, Prencipe and Santoro in 2001. A black hole models a network node in which a destructive process deletes any visiting agent or incoming data upon arrival, without leaving any observable trace. Conversely, a network may face one or more malicious mobile processes infecting one or more nodes. Given both kinds of threats, a first crucial task consists in searching for and reporting as quickly as possible the location all faulty nodes while using a minimum number of mobile agents. In general, the main issue is to identify the minimal hypotheses under which faulty nodes can be found. This problem has been investigated in both asynchronous and synchronous networks. A corollary task is to make sure that the protocols designed for solving problems such as gathering and transferring data still work despite the presence of one or more faulty nodes. In this chapter, we review the state-of-the-art of research pertaining to the presence of faulty nodes in a network. We discuss different models in synchronous and asynchronous networks and for different communication and computation capabilities of the agents. We also address relevant computational issues and present algorithmic techniques and impossibility results