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.
We study the problem of exploring an unknown undirected connected graph. Beginning in some start vertex, a searcher must visit each node of the graph by traversing edges. Upon visiting a vertex for the first time, the searcher learns all incident edges and their respective traversal costs. The goal is to find a tour of minimum total cost. Kalyanasundaram and Pruhs (Constructing competitive tours from local information, Theoretical Computer Science 130, pp. 125-138, 1994) proposed a sophisticated generalization of a Depth First Search that is 16-competitive on planar graphs. While the algorithm is feasible on arbitrary graphs, the question whether it has constant competitive ratio in general has remained open. Our main result is an involved lower bound construction that answers this question negatively. On the positive side, we prove that the algorithm has constant competitive ratio on any class of graphs with bounded genus. Furthermore, we provide a constant competitive algorithm for general graphs with a bounded number of distinct weights. Computing energy efficient broadcast trees is one of the most prominent operations in wireless networks. For stations embedded in the Euclidean plane, the best analytic result known to date is a 6.33-approximation algorithm based on computing an Euclidean minimum spanning tree. We improve the analysis of this algorithm and show that its approximation ratio is 6, which matches a previously known lower bound for this algorithm.  We wish to explore all edges of an unknown directed, strongly connected graph. At each point, we have a map of all nodes and edges we have visited, we can recognize these nodes and edges if we see them again, and we know how many unexplored edges emanate from each node we have visited, but we cannot tell where each leads until we traverse it. We wish to minimize the ratio of the total number of edges traversed divided by the optimum number of traversals, had we known the graph. For Eulerian graphs, this ratio cannot be better than two, and two is achievable by a simple algorithm. In contrast, the ratio is unbounded when the deficiency of the graph (the number of edges that have to be added to make it Eulerian) is unbounded. Our main result is an algorithm that achieves a bounded ratio when the deficiency is bounded. © 1999 John Wiley & Sons, Inc. J Graph Theory 32: 265–297, 1999 In this paper we initiate a new area of study dealing with the best way to search a possibly unbounded region for an object. The model for our search algorithms is that we must pay costs proportional to the distance of the next probe position relative to our current position. This model is meant to give a realistic cost measure for a robot moving in the plane. We also examine the effect of decreasing the amount of a priori information given to search problems. Problems of this type are very simple analogues of non-trivial problems on searching an unbounded region, processing digitized images, and robot navigation. We show that for some simple search problems, knowing the general direction of the goal is much more informative than knowing the distance to the goal. Consider a robot that has to travel from a start location @math to a target @math in an environment with opaque obstacles that lie in its way. The robot always knows its current absolute position and that of the target. It does not, however, know the positions and extents of the obstacles in advance; rather, it finds out about obstacles as it encounters them. We compare the distance walked by the robot in going from @math to @math to the length of the shortest (obstacle-free) path between @math and @math in the scene. We describe and analyze robot strategies that minimize this ratio for different kinds of scenes. In particular, we consider the cases of rectangular obstacles aligned with the axes, rectangular obstacles in more general orientations, and wider classes of convex bodies both in two and three dimensions. For many of these situations, our algorithms are optimal up to constant factors. We study scenes with nonconvex obstacles, which are related to the study of maze traversal. We also show scenes where randomized algorithms are provably better than deterministic algorithms. We study how a mobile robot can learn an unknown environment in a piecemeal manner. The robot's goal is to learn a complete map of its environment, while satisfying the constraint that it must return every so often to its starting position (for refueling, say). The environment is modeled as an arbitrary, undirected graph, which is initially unknown to the robot. We assume that the robot can distinguish vertices and edges that it has already explored. We present a surprisingly efficient algorithm for piecemeal learning an unknown undirected graph G=(V, E) in which the robot explores every vertex and edge in the graph by traversing at most O(E+V1+o(1)) edges. This nearly linear algorithm improves on the best previous algorithm, in which the robot traverses at most O(E+V2) edges. We also give an application of piecemeal learning to the problem of searching a graph for a “treasure.” Search Theory is one of the original disciplines within the field of Operations Research. It deals with the problem faced by a Searcher who wishes to minimize the time required to find a hidden object, or “target. ” The Searcher chooses a path in the “search space” and finds the target when he is sufficiently close to it. Traditionally, the target is assumed to have no motives of its own regarding when it is found; it is simply stationary and hidden according to a known distribution (e. g. , oil), or its motion is determined stochastically by known rules (e. g. , a fox in a forest). The problems dealt with in this book assume, on the contrary, that the “target” is an independent player of equal status to the Searcher, who cares about when he is found. We consider two possible motives of the target, and divide the book accordingly. Book I considers the zero-sum game that results when the target (here called the Hider) does not want to be found. Such problems have been called Search Games (with the “ze- sum” qualifier understood). Book II considers the opposite motive of the target, namely, that he wants to be found. In this case the Searcher and the Hider can be thought of as a team of agents (simply called Player I and Player II) with identical aims, and the coordination problem they jointly face is called the Rendezvous Search Problem. A cost aware metric for wireless networks based on remaining battery power at nodes was proposed for shortest-cost routing algorithms, assuming constant transmission power. Power-aware metrics, where transmission power depends on distance between nodes and corresponding shortest power algorithms were also proposed. We define a power-cost metric based on the combination of both node's lifetime and distance-based power metrics. We investigate some properties of power adjusted transmissions and show that, if additional nodes can be placed at desired locations between two nodes at distance d, the transmission power can be made linear in d as opposed to d sup spl alpha dependence for spl alpha spl ges 2. This provides basis for power, cost, and power-cost localized routing algorithms where nodes make routing decisions solely on the basis, of location of their neighbors and destination. The power-aware routing algorithm attempts to minimize the total power needed to route a message between a source and a destination. The cost-aware routing algorithm is aimed at extending the battery's worst-case lifetime at each node. The combined power-cost localized routing algorithm attempts to minimize the total power needed and to avoid nodes with a short battery's remaining lifetime. We prove that the proposed localized power, cost, and power-cost efficient routing algorithms are loop-free and show their efficiency by experiments. 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. We consider exploration problems where a robot has to construct a complete map of an unknown environment. We assume that the environment is modeled by a directed, strongly connected graph. The robot's task is to visit all nodes and edges of the graph using the minimum number R of edge traversals. Deng and Papadimitriou [ Proceedings of the 31st Symposium on the Foundations of Computer Science, 1990, pp. 356--361] showed an upper bound for R of dO(d) m and Koutsoupias (reported by Deng and Papadimitriou) gave a lower bound of @math , where m is the number of edges in the graph and d is the minimum number of edges that have to be added to make the graph Eulerian. We give the first subexponential algorithm for this exploration problem, which achieves an upper bound of dO(log d) m. We also show a matching lower bound of @math for our algorithm. Additionally, we give lower bounds of @math , respectively, @math for various other natural exploration algorithms. We introduce a new learning problem: learning a graph by piecemeal search, in which the learner must return every so often to its starting point (for refueling, say). We present two linear-time piecemeal-search algorithms for learning city-block graphs: grid graphs with rectangular obstacles.