The past few years have seen intensive research efforts carried out in some apparently unrelated areas of dynamic systems -delay-tolerant networks, opportunistic-mobility networks, social networksobtaining closely related insights. Indeed, the concepts discovered in these investigations can be viewed as parts of the same conceptual universe; and the formal models proposed so far to express some specific concepts are components of a larger formal description of this universe. The main contribution of this paper is to integrate the vast collection of concepts, formalisms, and results found in the literature into a unified framework, which we call TVG (for time-varying graphs). Using this framework, it is possible to express directly in the same formalism not only the concepts common to all those different areas, but also those specific to each. Based on this definitional work, employing both existing results and original observations, we present a hierarchical classification of TVGs; each class corresponds to a significant property examined in the distributed computing literature. We then examine how TVGs can be used to study the evolution of network properties, and propose different techniques, depending on whether the indicators for these properties are a-temporal (as in the majority of existing studies) or temporal. Finally, we briefly discuss the introduction of randomness in TVGs.

In this paper we study the problem of gathering in the same location of the plane a collection of identical oblivious mobile robots. Previous investigations have focused mostly on the unlimited visibility setting, where each robot can always see all the other ones, 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 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 team of mobile software agents deployed to capture a (possibly hostile) intruder in a network. All agents, including the intruder move along the network links; the intruder could be arbitrarily fast, and aware of the positions of all the agents. The problem is to design the agents' strategy for capturing the intruder. The main efficiency parameter is the size of the team. This is an instance of the well known graph-searching problem whose many variants have been extensively studied in the literature. In all existing solutions, and in all the variants of the problem, it is assumed that agents can be removed from their current location and placed in another network site arbitrarily and at any time. As a consequence, the existing optimal strategies cannot be employed in situations for which agents cannot access the network at any point, or cannot "jump" across the network, or cannot reach an arbitrary point of the network via an internal travel through insecure zones. This motivates the contiguous search problem in which agents cannot be removed from the network, and clear links must form a connected sub-network at any time, providing safety of movements. This new problem is NP-complete in general. We study it for tree networks, and we consider its more general version, the weighted case, which arises naturally when considering networks whose nodes and links are of different nature and thus require a different number of agents to be explored. We give a linear-time algorithm that computes, for any tree T , the minimum number of agents to capture the intruder, and the corresponding search strategy. Beside its optimality in time, our algorithm is naturally distributed:

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The study of what can be computed by a team of autonomous mobile robots, originally started in robotics and AI, has become increasingly popular in theoretical computer science (especially in distributed computing), where it is now an integral part of the investigations on computability by mobile entities. The robots are identical computational entities located and able to move in a spatial universe; they operate without explicit communication and are usually unable to remember the past; they are extremely simple, with limited resources, and individually quite weak. However, collectively the robots are capable of performing complex tasks, and form a system with desirable fault-tolerant and self-stabilizing properties. The research has been concerned with the computational aspects of such systems. In particular, the focus has been on the minimal capabilities that the robots should have in order to solve a problem.\ud \ud \ud \ud This book focuses on the recent algorithmic results in the field of distributed computing by oblivious mobile robots (unable to remember the past). After introducing the computational model with its nuances, we focus on basic coordination problems: pattern formation, gathering, scattering, leader election, as well as on dynamic tasks such as flocking. For each of these problems, we provide a snapshot of the state of the art, reviewing the existing algorithmic results. In doing so, we outline solution techniques, and we analyze the impact of the different assumptions on the robots' computability power

Abstract. Consider a set of n > 2 identical mobile computational entities in the plane, called robots, operating in Look-Compute-Move cycles, without any means of direct communication. The Gathering Problem is the primitive task of all entities gathering in finite time at a point not fixed in advance, without any external control. The problem has been extensively studied in the literature under a variety of strong assumptions (e.g., synchronicity of the cycles, instantaneous movements, complete memory of the past, common coordinate system, etc.). In this paper we consider the setting without those assumptions, that is, when the entities are oblivious (i.e., they do not remember results and observations from previous cycles), disoriented (i.e., have no common coordinate system), and fully asynchronous (i.e., no assumptions exist on timing of cycles and activities within a cycle). The existing algorithmic contributions for such robots are limited to solutions for n ≤ 4 or for restricted sets of initial configurations of the robots; the question of whether such weak robots could deterministically gather has remained open. In this paper, we prove that indeed the Gathering Problem is solvable, for any n > 2 and any initial configuration, even under such restrictive conditions.

In this paper we address the problem of mobile agents searching for a highly harmful item (called black hole) in a ring network. The black hole is a stationary process that destroys visiting agents upon their arrival without leaving any observable trace of such a destruction. The task is to have at least one surviving agent able to unambiguously report the location of the black hole.We consider different scenarios and in each situation we answer some computational as well as complexity questions. We first consider agents that start from the same homebase (co-located). We prove that two such agents are necessary and sufficient to locate the black hole; in our algorithm the agents perform O(nlogn) moves (where n is the size of the ring) and we show that such a bound is optimal. We also consider time complexity and we show how to achieve the optimal bound of 2n − 4 units of time using n − 1 agents. We generalize our technique to establish a trade-off between time and number of agents. We then consider the case of agents that start from different homebases (dispersed) and we show that, if the ring is oriented, two dispersed agents * A preliminary version of this paper appeared in the Proceedings of the 15th International Symposium on Distributed Computing [8].† University of Ottawa, email:sdobrev@site.uottawa.ca ‡ University of Ottawa, email:flocchin@site.uottawa.ca § contact author: Università di Pisa, Dipartimento di Informatica, Via Buonarroti, 2 -56100, Pisa, tel.+39 050 2213148, fax. +39 050 2212726, email:prencipe@di.unipi.it ¶ Carleton University, email: santoro@scs.carleton.ca 1 are still necessary and sufficient. Also in this case our algorithm is optimal in terms of number of moves (Θ(nlogn)). We finally show that, if the ring is unoriented, three agents are necessary and sufficient; an optimal algorithm follows from the oriented case.

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