A set of identical, mobile agents is deployed in a weighted network. Each agent has a battery--a power source allowing it to move along network edges. An agent uses its battery proportionally to the distance traveled. We consider two tasks: convergecast, in which at the beginning, each agent has some initial piece of information, and information of all agents has to be collected by some agent; and broadcast in which information of one specified agent has to be made available to all other agents. In both tasks, the agents exchange the currently possessed information when they meet. The objective of this paper is to investigate what is the minimal value of power, initially available to all agents, so that convergecast or broadcast can be achieved. We study this question in the centralized and the distributed settings. In the centralized setting, there is a central monitor that schedules the moves of all agents. In the distributed setting every agent has to perform an algorithm being unaware of the network. In the centralized setting, we give a linear-time algorithm to compute the optimal battery power and the strategy using it, both for convergecast and for broadcast, when agents are on the line. We also show that finding the optimal battery power for convergecast or for broadcast is NP-hard for the class of trees. On the other hand, we give a polynomial algorithm that finds a 2-approximation for convergecast and a 4-approximation for broadcast, for arbitrary graphs.In the distributed setting, we give a 2-competitive algorithm for convergecast in trees and a 4-competitive algorithm for broadcast in trees. The competitive ratio of 2 is proved to be the best for the problem of convergecast, even if we only consider line networks. Indeed, we show that there is no ( @math 2-∈)-competitive algorithm for convergecast or for broadcast in the class of lines, for any @math ∈>0.
While a single agent cannot explore a graph of unknown size unless pebble (landmark) usage is permitted (see @cite_20 ), a pair of robots are able to explore and map a directed graph of maximal degree @math in @math time with high probability (cf. @cite_24 ). In the case of a team of collaborating mobile agents, the challenge is to balance the workload among the agents so that the time to achieve the required goal is minimized. However this task is often hard (cf. @cite_13 ), even in the case of two agents in a tree, @cite_21 . On the other hand, the authors of @cite_15 study the problem of agents exploring a tree, showing @math competitive ratio of their distributed algorithm provided that writing (and reading) at tree nodes is permitted.
@cite_21 @cite_24 @cite_15 @cite_13 @cite_20
2027056835 2154255883 2137561762 2143987816 2149888497
Abstract Suppose two travelling salesmen must visit together all points nodes of a tree, and the objective is to minimize the maximal length of their tours. Home locations of the salesmen are given, and it is required to find optimal tours. For this NP-hard problem a heuristic with complexity O ( n ) is presented. The worst-case relative error for the heuristic performance is 1 3 for the case of equal home locations for both servers and 1 2 for the case of different home locations. We show that two cooperating robots can learn exactly any strongly-connected directed graph with n indistinguishable nodes in expected time polynomial in n. We introduce a new type of homing sequence for two robots which helps the robots recognize certain previously-seen nodes. We then present an algorithm in which the robots learn the graph and the homing sequence simultaneously by wandering actively through the graph. Unlike most previous learning results using homing sequences, our algorithm does not require a teacher to provide counterexamples. Furthermore, the algorithm can use efficiently any additional information available that distinguishes nodes. We also present an algorithm in which the robots learn by taking random walks. The rate at which a random walk converges to the stationary distribution is characterized by the conductance of the graph. Our random-walk algorithm learns in expected time polynomial in n and in the inverse of the conductance and is more efficient than the homing-sequence algorithm for high-conductance graphs. > An n-node tree has to be explored by k mobile agents (robots), starting in its root. Every edge of the tree must be traversed by at least one robot, and exploration must be completed as fast as possible. Even when the tree is known in advance, scheduling optimal collective exploration turns out to be NP-hard. We investigate the problem of distributed collective exploration of unknown trees. Not surprisingly, communication between robots influences the time of exploration. Our main communication scenario is the following: robots can communicate by writing at the currently visited node previously acquired information, and reading information available at this node. We construct an exploration algorithm whose running time for any tree is only O(k log k) larger than optimal exploration time with full knowledge of the tree. (We say that the algorithm has overheadO(k log k)). On the other hand we show that, in order to get overhead sublinear in the number of robots, some communication is necessary. Indeed, we prove that if robots cannot communicate at all, then every distributed exploration algorithm works in time Ω (k) larger than optimal exploration time with full knowledge, for some trees. Several polynomial time approximation algorithms for some @math -complete routing problems are presented, and the worst-case ratios of the cost of the obtained route to that of an optimal are determined. A mixed-strategy heuristic with a bound of 9 5 is presented for the stacker-crane problem (a modified traveling salesman problem). A tour-splitting heuristic is given for k-person variants of the traveling salesman problem, the Chinese postman problem, and the stacker-crane problem, for which a minimax solution is sought. This heuristic has a bound of @math , where e is the bound for the corresponding 1-person algorithm. Exploring and mapping an unknown environment is a fundamental problem that is studied in a variety of contexts. Many results have focused on finding efficient solutions to restricted versions of the problem. In this paper, we consider a model that makes very limited assumptions about the environment and solve the mapping problem in this general setting. We model the environment by an unknown directed graph G, and consider the problem of a robot exploring and mapping G. The edges emanating from each vertex are numbered from ‘1’ to ‘d’, but we do not assume that the vertices of G are labeled. Since the robot has no way of distinguishing between vertices, it has no hope of succeeding unless it is given some means of distinguishing between vertices. For this reason we provide the robot with a “pebble”—a device that it can place on a vertex and use to identify the vertex later. In this paper we show: (1) If the robot knows an upper bound on the number of vertices then it can learn the graph efficiently with only one pebble. (2) If the robot does not know an upper bound on the number of vertices n, then (log log n) pebbles are both necessary and sufficient. In both cases our algorithms are deterministic. C © 2002 Elsevier Science (USA)