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 computational power of a distributed system consisting of simple autonomous robots moving on the plane. The robots are endowed with visual perception but do not have any means of explicit communication with each other, and have no memory of the past. In the extensive literature it has been shown how such simple robots can form a single geometric pattern (e.g., a line, a circle, etc), however arbitrary, in spite of their obliviousness. This brings to the front the natural research question: what are the real computational limits imposed by the robots being oblivious? In particular, since obliviousness limits what can be remembered, under what conditions can oblivious robots form a series of geometric patterns? Notice that a series of patterns would create some form of memory in an otherwise memory-less system. In this paper we examine and answer this question showing that, under particular conditions, oblivious robot systems can indeed form series of geometric patterns starting from any arbitrary configuration. More precisely, we study the series of patterns that can be formed by robot systems under various restrictions such as anonymity, asynchrony and lack of common orientation. These results are the first strong indication that oblivious solutions may be obtained also for tasks that intuitively seem to require memory. In this paper we study the problem of gathering a collection of identical oblivious mobile robots in the same location of the plane. Previous investigations have focused mostly on the unlimited visibility setting, where each robot can always see all the others regardless of their distance.In the more difficult and realistic setting where the robots have limited visibility, the existing algorithmic results are only for convergence (towards a common point, without ever reaching it) and only for semi-synchronous environments, where robots' movements are assumed to be performed instantaneously.In contrast, we study this problem in a totally asynchronous setting, where robots' actions, computations, and movements require a finite but otherwise unpredictable amount of time. We present a protocol that allows anonymous oblivious robots with limited visibility to gather in the same location in finite time, provided they have orientation (i.e., agreement on a coordinate system).Our result indicates that, with respect to gathering, orientation is at least as powerful as instantaneous movements. Consider a set of n > 2 simple autonomous mobile robots (decentralized, asynchronous, no common coordinate system, no identities, no central coordination, no direct communication, no memory of the past, deterministic) moving freely in the plane and able to sense the positions of the other robots. We study the primitive task of gathering them at a point not fixed in advance (GATHERING PROBLEM). In the literature, most contributions are simulation-validated heuristics. The existing algorithmic contributions for such robots are limited to solutions for n ≤ 4 or for restricted sets of initial configurations of the robots. In this paper, we present the first algorithm that solves the GATHERING PROBLEM for any initial configuration of the robots. The computational power of networks of small resource-limited mobile agents is explored. Two new models of computation based on pairwise interactions of finite-state agents in populations of finite but unbounded size are defined. With a fairness condition on interactions, the concept of stable computation of a function or predicate is defined. Protocols are given that stably compute any predicate in the class definable by formulas of Presburger arithmetic, which includes Boolean combinations of threshold-k, majority, and equivalence modulo m. All stably computable predicates are shown to be in NL. Assuming uniform random sampling of interacting pairs yields the model of conjugating automata. Any counter machine with O (1) counters of capacity O (n) can be simulated with high probability by a conjugating automaton in a population of size n. All predicates computable with high probability in this model are shown to be in P; they can also be computed by a randomized logspace machine in exponential time. Several open problems and promising future directions are discussed. In a system in which anonymous mobile robots repeatedly execute a ''Look-Compute-Move'' cycle, a robot is said to be oblivious if it has no memory to store its observations in the past, and hence its move depends only on the current observation. This paper considers the pattern formation problem in such a system, and shows that oblivious robots can form any pattern that non-oblivious robots can form, except that two oblivious robots cannot form a point while two non-oblivious robots can. Therefore, memory does not help in forming a pattern, except for the case in which two robots attempt to form a point. Related results on the pattern convergence problem are also presented. In this note we make a minor correction to a scheme for robots to broadcast their private information. All major results of the paper [I. Suzuki and M. Yamashita, SIAM J. Comput., 28 (1999), pp. 1347-1363] hold with this correction. Given a set of n mobile robots in the d-dimensional Euclidean space, the goal is to let them converge to a single not predefined point. The challenge is that the robots are limited in their capabilities. Robots can, upon activation, compute the positions of all other robots using an individual affine coordinate system. The robots are indistinguishable, oblivious and may have different affine coordinate systems. A very general discrete time model assumes that robots are activated in arbitrary order. Further, the computation of a new target point may happen much earlier than the movement, so that the movement is based on outdated information about other robot's positions. Time is measured as the number of rounds, where a round ends as soon as each robot has moved at least once. In [6], the Center of Gravity is considered as target function, convergence was proven, and the number of rounds needed for halving the diameter of the convex hull of the robot's positions was shown to be O(n2) and Ω(n). We present an easy-to-check property of target functions that guarantee convergence and yields upper time bounds. This property intuitively says that when a robot computes a new target point, this point is significantly within the current axes aligned minimal box containing all robots. This property holds, e.g., for the above-mentioned target function, and improves the above O(n2) to an asymptotically optimal O(n) upper bound. Our technique also yields a constant time bound for a target function that requires all robots having identical coordinate axes. This paper considers the convergence problem in autonomous mobile robot systems. A natural algorithm for the problem requires the robots to move towards their center of gravity. This paper proves the correctness of the gravitational algorithm in the fully asynchronous model. It also analyzes its convergence rate and establishes its convergence in the presence of crash faults. We present a distributed algorithm for converging autonomous mobile robots with limited visibility toward a single point. Each robot is an omnidirectional mobile processor that repeatedly: 1) observes the relative positions of those robots that are visible; 2) computes its new position based on the observation using the given algorithm; 3) moves to that position. The robots' visibility is limited so that two robots can see each other if and only if they are within distance V of each other and there are no other robots between them. Our algorithm is memoryless in the sense that the next position of a robot is determined entirely from the positions of the robots that it can see at that moment. The correctness of the algorithm is proved formally under an abstract model of the robot system in which: 1) each robot is represented by a point that does not obstruct the view of other robots; 2) the robots' motion is instantaneous; 3) there are no sensor and control error; 4) the issue of collision is ignored. The results of computer simulation under a more realistic model give convincing indication that the algorithm, if implemented on physical robots, will be robust against sensor and control error.