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The research areas of mobile robotic sensors lie in the intersection of two major fields of investigations carried out by quite distinct communities of researchers: autonomous robots and mobile sensor networks. Robotic sensors are micro-robots capable of locomotion and sensing. Like the sensors in wireless sensor networks, they are myopic: their sensing range is limited. Unlike the sensors in wireless sensor networks, robotic sensors are silent: they have no direct communication capabilities. This means that synchronization, interaction, and communication of information among the robotic sensors can be achieved solely by means of their sensing capability, usually called vision. In this chapter, we review the results of the investigations on the computability and complexity aspects of systems formed by these myopic and silent mobile sensors. [ABSTRACT FROM AUTHOR]

Most highly dynamic infrastructure-less networks have in common that the assumption of connectivity does not necessarily hold at a given instant. Still, communication routes can be available between any pair of nodes over time and space. These networks (variously called delay-tolerant, disruptive-tolerant, challenged) are naturally modeled as time-varying graphs (or evolving graphs), where the existence of an edge is a function of time. In this paper we study deterministic computations under unstructured mobility, that is when the edges of the graph appear infinitely often but without any (known) pattern. In particular, we focus on the problem of broadcasting with termination detection. We explore the problem with respect to three possible metrics: the date of message arrival (foremost), the time spent doing the broadcast (fastest), and the number of hops used by the broadcast (shortest). We prove that the solvability and complexity of this problem vary with the metric considered, as well as with the type of knowledge a priori available to the entities. These results draw a complete computability map for this problem when mobility is unstructured. [ABSTRACT FROM AUTHOR]

We consider a class of highly dynamic networks modelled on an urban subway system. We examine the problem of creating a map of such a subway in less than ideal conditions, where the local residents are not enthusiastic about the process and there is a limited ability to communicate amongst the mappers. More precisely, we study the problem of a team of asynchronous computational entities (the mapping agents) determining the location of black holes in a highly dynamic graph, whose edges are defined by the asynchronous movements of mobile entities (the subway carriers). We present and analyze a solution protocol. The algorithm solves the problem with the minimum number of agents possible. We also establish lower bounds on the number of carrier moves in the worst case, showing that our protocol is also move-optimal. [ABSTRACT FROM AUTHOR]

In a network environments supporting mobile entities (called robots or agents), a black hole is harmful site that destroys any incoming entity without leaving any visible trace. The black-hole search problem is the task of a team of k > 1 mobile entities, starting from the same safe location and executing the same algorithm, to determine within finite time the location of the black hole. In this paper we consider the black hole search problem in asynchronous ring networks of n nodes, and focus on the time complexity. It is known that any algorithm for black-hole search in a ring requires at least 2(n − 2) time in the worst case. The best algorithm achieves this bound with a team of n − 1 agents with an average time cost 2(n − 2), equal to the worst case. In this paper we first show how the same number of agents using 2 extra time units from optimal in the worst case, can solve the problem in only ]> time on the average. We then prove that the optimal average case complexity ]> can be achieved without increasing the worst case using 2(n − 1) agents Finally we design an algorithm that achieves asymptotically optimal both worst case and average case time complexity employing an optimal team of k = 2 agents, thus improving on the earlier results that required O(n) agents. [ABSTRACT FROM AUTHOR]

In this paper we investigate the basic problem of Exploration of a graph by a group of identical mobile computational entities, called robots, operating autonomously and asynchronously. In particular we are concerned with what graphs can be explored, and how, if the robots do not remember the past and have no explicit means of communication. This model of robots is used when the spatial universe in which the robots operate is continuous (e.g., a curve, a polygonal region, a plane, etc.). The case when the spatial universe is discrete (i.e., a graph) has been also studied but only for the classes of acyclic graphs and of simple cycles. In this paper we consider networks of arbitrary topology modeled as connected graphs with local orientation (locally distinct edge labels). We concentrate on class ]> of asymmetric configurations with k robots. Our results indicate that the explorability of graphs in this class depends on the number k of robots participating in the exploration. In particular, exploration is impossible for k<3 robots. When there are only k=3 robots, only a subset of ]> can be explored; we provide a complete characterization of the networks that can be explored. When there are k=4 robots, we prove that all networks in ]> can be explored. Finally, we prove that for any odd k>4 all networks in ]> can be explored by presenting a general algorithm. The determination of which networks can be explored when k>4 is even, is still open but can be reduced to the existence of a gathering algorithm for ]> . [ABSTRACT FROM AUTHOR]

Network decontamination (or disinfection) is a widely studied problem in distributed computing. Network sites are assumed to be contaminated (e.g., by a virus) and a team of agents is deployed to decontaminate the whole network. In the vast literature a variety of assumptions are made on the power of the agents, which can typically communicate, exchange information, remember the past, etc. In this paper we consider the problem in a much weaker setting; in fact we wish to describe the global disinfection process by a set of cellular automata local rules without the use of active agents. We consider the grid, which is naturally described by a 2-dimensional cellular automata, and we devise disinfection rules both in the common situation where after being disinfected a cell is prone to re-contamination by contact, and in a new setting where disinfection leaves the cells immune to recontamination for a certain amount of time (temporal immunity). We also distinguish between Von Neuman and Moore neighborhood, showing that, not surprisingly, a bigger neighborhood allows for a more efficient disinfection. [ABSTRACT FROM AUTHOR]

We study the computability and complexity of the exploration problem in a class of highly dynamic graphs: periodically varying (PV) graphs, where the edges exist only at some (unknown) times defined by the periodic movements of carriers. These graphs naturally model highly dynamic infrastructure-less networks such as public transports with fixed timetables, low earth orbiting (LEO) satellite systems, security guards΄ tours, etc. We establish necessary conditions for the problem to be solved. We also derive lower bounds on the amount of time required in general, as well as for the PV graphs defined by restricted classes of carriers movements: simple routes, and circular routes. We then prove that the limitations on computability and complexity we have established are indeed tight. We do so constructively presenting two worst case optimal solution algorithms, one for anonymous systems, and one for those with distinct nodes ids. [ABSTRACT FROM AUTHOR]

Consider a tree network that has been contaminated by a persistent and active virus: when infected, a network site will continuously attempt to spread the virus to all its neighbours. The decontamination problem is that of disinfecting the entire network using a team of mobile antiviral system agents, called cleaners, avoiding any recontamination of decontaminated areas. A cleaner is able to decontaminate any infected node it visits; once the cleaner departs, the decontaminated node is immune for t ≥ 0 time units to viral attacks from infected neighbours. After the immunity timet is elapsed, re-contamination can occur. The primary research objective is to determine the minimum team size, as well as the solution strategy, that is the protocol that would enable such a minimal team of cleaners to perform the task. The network decontamination problem has been extensively investigated in the literature, and a very large number of studies exist on the subject. However, all the existing work is limited to the special case t = 0. In this paper we examine the tree decontamination problem for any value t ≥ 0. We determine the minimum team size necessary to disinfect any given tree with immunity time t. Further we show how to compute for all nodes of the tree the minimum team size and implicitly the solution strategy starting from each starting node; these computations use a total of Θ(n) time (serially) or Θ(n) messages (distributively). We then provide a complete structural characterization of the class of trees that can be decontaminated with k agents and immunity time t; we do so by identifying the forbidden subgraphs and analyzing their properties. Finally, we consider generic decontamination algorithms, i.e. protocols that work unchanged in a large class of trees, with little knowledge of their topological structure. We prove that, for each immunity time t ≥ 0, all trees of height at most h can be decontaminated by a team of ]> agents whose only knowledge of the tree is the bound h. The proof is constructive. [ABSTRACT FROM AUTHOR]

We prove that, for the black hole search problem, the pure token model is computationally as powerful as the whiteboard model; furthermore the complexity is exactly the same. More precisely, we prove that a team of two asynchronous agents, each endowed with a single identical pebble (that can be placed only on nodes, and with no more than one pebble per node) can locate the black hole in an arbitrary network of known topology; this can be done with Θ(n logn) moves, where n is the number of nodes, even when the links are not FIFO. [ABSTRACT FROM AUTHOR]

In the effort to understand the algorithmic limitations of computing by a swarm of robots, the research has focused on the minimal capabilities that allow a problem to be solved. The weakest of the commonly used models is Asynch where the autonomous mobile robots, endowed with visibility sensors (but otherwise unable to communicate), operate in Look-Compute-Move cycles performed asynchronously for each robot. The robots are often assumed (or required to be) oblivious: they keep no memory of observations and computations made in previous cycles. We consider the setting when the robots are dispersed in an anonymous and unlabeled graph, and they must perform the very basic task of exploration: within finite time every node must be visited by at least one robot and the robots must enter a quiescent state. The complexity measure of a solution is the number of robots used to perform the task. We study the case when the graph is an arbitrary tree and establish some unexpected results. We first prove that there are n-node trees where Ώ(n) robots are necessary; this holds even if the maximum degree is 4. On the other hand, we show that if the maximum degree is 3, it is possible to explore with only ]> robots. The proof of the result is constructive. Finally, we prove that the size of the team is asymptotically optimal: we show that there are trees of degree 3 whose exploration requires ]> robots. [ABSTRACT FROM AUTHOR]

The recently established computational equivalence between the traditional message-passing model and the mobile-agents model is based on the existence of a mobile-agents algorithm that simulates the execution of message-passing algorithms. Like most existing protocols for mobile agents, this simulation protocol works correctly only if the agents are fault-free. We consider the problem of performing the simulation of message-passing algorithms when the simulating agents may crash unexpectedly. We show how to simulate any distributed algorithm for the message-passing model in a mobile-agents system with k agents, tolerating up to f ≤ k − 1 crashes during the simulation. Two fault-tolerant simulation algorithms are presented, one for non-anonymous settings (i.e., where either the networks nodes or the agents or both have distinct identities), and one for anonymous systems (where both the network nodes and the agents are anonymous). In both cases, the simulation overhead is polynomial. Unlike the existing fault-free simulation algorithm, both our protocols are able to detect termination even if the simulated algorithm has no explicit termination detection. [ABSTRACT FROM AUTHOR]

We consider the problem of exploring an anonymous unoriented ring by a team of k identical, oblivious, asynchronous mobile robots that can view the environment but cannot communicate. This weak scenario is standard when the spatial universe in which the robots operate is the two-dimentional plane, but (with one exception) has not been investigated before. We indeed show that, although the lack of these capabilities renders the problems considerably more difficult, ring exploration is still possible. We show that the minimum number ρ(n) of robots that can explore a ring of size n is O(logn) and that ρ(n) = Ώ(logn) for arbitrarily large n. On one hand we give an algorithm that explores the ring starting from any initial configuration, provided that n and k are co-prime, and we show that there always exist such k in O(logn). On the other hand we show that Ώ(logn) agents are necessary for arbitrarily large n. Notice that, when k and n are not co-prime, the problem is sometimes unsolvable (i.e., there are initial configurations for which the exploration cannot be done). This is the case, e.g., when k divides n. [ABSTRACT FROM AUTHOR]

In this paper we propose a new strategy for tracking mobile agents in a network. Our proposal is based on a semi-cooperative approach: while performing its own prescribed task, a mobile agent moves keeping in mind that a searching agent might be looking for it. In doing so we want a fully distributed solution that does not rely on a central server, and we also want to avoid the use of long forwarding pointers. Our proposal is based on appropriate delays that the mobile agents must perform while moving on the network so to facilitate its tracking, should it be needed. The searching agent computes a particular searching path that will guarantee the tracking within one traversal of the network. The delays to be computed depend on structural properties of the network. We perform several experiments following different strategies for computing the searching path and we compare our results. [ABSTRACT FROM AUTHOR]

In many network applications the computation takes place on the minimum-cost spanning tree (MST) of the network; unfortunately, a single link or node failure disconnects the tree. In this paper we consider for the first time the problem of computing all the replacement minimum-cost spanning trees distributively, and we efficiently solve the problem. We design a solution protocol and we prove that the total amount of data items communicated during the computation is O(n2). This communication can be achieved either transmitting O(n) long messages, if the system so allows, or O(n2) standard messages. Even in systems that do not allow long messages, the proposed protocol constitutes a significant improvement over the individual computation of the replacement trees. [ABSTRACT FROM AUTHOR]

Mobile Agents have been extensively studied for several years by researchers in Artificial Intelligence and in Software Engineering. They offer a simple and natural way to describe distributed settings where mobility is inherent, and an explicit and direct way to describe the entities of those settings, such as mobile code, software agents, viruses, robots, web crawlers, etc. Further, they allow to express immediately notions such as selfish behaviour, negotiation, cooperation, etc arising in the new computing environments. As a programming paradigm, they allow a new philosophy of protocol and software design, bound to have an impact as strong as that caused by that of object-oriented programming. As a computational paradigm, mobile agents systems are an immediate and natural extension of the traditional message-passing settings studied in distributed computing. In spite of all this, mobile agents systems have been largely ignored by the mainstream distributed computing community. It is only in the last few years that several researchers, some motivated by long investigated and well established problems in automata theory, computational complexity, and graph theory, have started to systematically explore this new and exciting distributed computational universe. In this paper we describe some interesting problems and solution techniques developed in this investigations. [ABSTRACT FROM AUTHOR]

We consider the self-deployment problem in a ring for a network of identical sensors: starting from some initial random placement in the ring, the sensors in the network must move, in a purely decentralized and distributed fashion, so to reach in finite time a state of static equilibrium in which they evenly cover the ring. A self-deployment algorithm is exact if within finite time the sensors reach a static uniform configuration: the distance between any two consecutive sensors along the ring is the same, d; the self-deployment algorithm is ε-approximate if the distance between two consecutive sensors is between d − ε and d + ε. We prove that exact self-deployment is impossible if the sensors do not share a common orientation of the ring. We then consider the problem in an oriented ring. We prove that if the sensors know the desired final distance d, then exact self-deployment is possible. Otherwise, we present another protocol based on a very simple strategy and prove that it is ε -approximate for any chosen ε> 0. Our results show that a shared orientation of the ring is an important computational and complexity factor for a network of mobile sensors operating in a ring. [ABSTRACT FROM AUTHOR]

A group of mobile autonomous robots, each with very limited capabilities, can form (complex) patterns in the space it occupies. These patterns can be used to program the robots to accomplish high-level tasks (e.g., surrounding and removal of a mine). The basic research questions address which patterns can be formed, and how they can be formed. These questions have been studied mostly from an empirical point of view. Most solutions do not have any guarantee of correctness; actually many solutions never terminate and never form the desired pattern. On the contrary, we are interested in (provably correct) solutions which always form the pattern within finite time. With this goal, we have been studying what patterns can be formed and how; in this paper we describe the results of our investigations. [ABSTRACT FROM AUTHOR]

We present a space and (asymptotically) time optimal self-stabilizing algorithm for simultaneously activating non-adjacent processes in a rooted tree (Algorithm $\mathcal{SSDST}$). We then give two applications of the proposed algorithm: a time and space optimal solution to the local mutual exclusion problem (Algorithm $\mathcal{LMET}$) and a space and (asymptotically) time optimal distributed algorithm to place the values in min-heap order (Algorithm ${\mathcal{HEAP}}$). All algorithms are self-stabilizing and uniform, and they work under any unfair distributed daemon. In proving the time complexity of the heap construction, we use the notion of pseudo-time. Pseudo-time is similar to logical time introduced by Lamport [12]. Keywords: heap, local mutual exclusion, self-stabilization. [ABSTRACT FROM AUTHOR]

Many graph problems seem to require knowledge that extends beyond the immediate neighbors of a node. The usual self-stabilizing model only allows for nodes to make decisions based on the states of their immediate neighbors. We provide a general polynomial transformation for constructing self-stabilizing algorithms which utilize distance-k knowledge, with a slowdown of nO(logk ) . Our main application is a polynomial-time self-stabilizing algorithm for finding maximal irredundant sets, a problem which seems to require distance-4 information. We also show how to find maximal k-packings in polynomial-time. Our techniques extend results in a recent paper by Gairing et al. for achieving distance-two information. [ABSTRACT FROM AUTHOR]

We consider a distributed system where each node has a local count for each item (similar to elections where nodes are ballot boxes and items are candidates). A top-k query in such a system asks which are the k items whose sum of counts, across all nodes in the system, is the largest. In this paper we present a Monte-Carlo algorithm that outputs, with high probability, a set of k candidates which approximates the top-k items. The algorithm is motivated by sensor networks in that it focuses on reducing the individual communication complexity. In contrast to previous algorithms, the communication complexity depends only on the global scores and not on the partition of scores among nodes. If the number of nodes is large, our algorithm dramatically reduces the communication complexity when compared with deterministic algorithms. We show that the complexity of our algorithm is close to a lower bound on the cell-probe complexity of any non-interactive top-k approximation algorithm. We show that for some natural global distributions (such as the Geometric or Zipf distributions), our algorithm needs only polylogarithmic number of communication bits per node. [ABSTRACT FROM AUTHOR]

We present four new asymmetric communication protocols, with which a server with high bandwidth can help clients with low bandwidth send it messages. Three of our protocols are the first to use only a single round of communication for each message. Unlike previous authors, we do not assume the server knows the messages' distribution. [ABSTRACT FROM AUTHOR]

Let a communication network be modeled by a graph G = (V,E) of n nodes and m edges, where with each edge is associated a pair of values, namely its cost and its length. Assume now that each edge is controlled by a selfish agent, which privately holds the cost of the edge. In this paper we analyze the problem of designing in this non-cooperative scenario a truthful mechanism for building a broadcasting tree aiming to balance costs and lengths. More precisely, given a root node r ∈V and a real value λ≥1, we want to find a minimum cost (as computed w.r.t. the edge costs) spanning tree of G rooted at r such that the maximum stretching factor on the distances from the root (as computed w.r.t. the edge lengths) is λ. We call such a tree the Minimum-cost λ-Approximate Shortest-paths Tree (λ-MAST). First, we prove that, already for the unit length case, the λ-MAST problem is hard to approximate within better than a logarithmic factor, unless NP admits slightly superpolynomial time algorithms. After, assuming that the graph G is directed, we provide a (1 + ε)(n - 1)-approximate truthful mechanism for solving the problem, for any ε> 0. Finally, we analyze a variant of the problem in which the edge lengths coincide with the private costs, and we provide: (i) a constant lower bound (depending on λ) to the approximation ratio that can be achieved by any truthful mechanism; (ii) a $(1+ {{n-1}\over{\lambda}})$-approximate truthful mechanism. Keywords: Algorithmic Mechanism Design, Bicriteria Network Design Problems, Broadcasting Tree, Truthful Mechanisms. [ABSTRACT FROM AUTHOR]

In sparse approximation theory, the fundamental problem is to reconstruct a signal ${\mathbf A} \in {\mathbb{R}}^n$ from linear measurements $\langle{{\mathbf A}}{\psi_i}\rangle$ with respect to a dictionary of ψi's. Recently, there is focus on the novel direction of Compressed Sensing [9] where the reconstruction can be done with very few—O(k logn)—linear measurements over a modified dictionary if the signal is compressible, that is, its information is concentrated in k coefficients with the original dictionary. In particular, these results [9, 4, 23] prove that there exists a single O(k logn) ×n measurement matrix such that any such signal can be reconstructed from these measurements, with error at most O(1) times the worst case error for the class of such signals. Compressed sensing has generated tremendous excitement both because of the sophisticated underlying Mathematics and because of its potential applications. In this paper, we address outstanding open problems in Compressed Sensing. Our main result is an explicit construction of a non-adaptive measurement matrix and the corresponding reconstruction algorithm so that with a number of measurements polynomial in k, logn, 1/ε, we can reconstruct compressible signals. This is the first known polynomial time explicit construction of any such measurement matrix. In addition, our result improves the error guarantee from O(1) to 1 + ε and improves the reconstruction time from poly(n) to poly(k logn). Our second result is a randomized construction of O(kpolylog (n)) measurements that work for each signal with high probability and gives per-instance approximation guarantees rather than over the class of all signals. Previous work on Compressed Sensing does not provide such per-instance approximation guarantees; our result improves the best known number of measurements known from prior work in other areas including Learning Theory [20, 21], Streaming algorithms [11, 12, 6] and Complexity Theory [1] for this case. Our approach is combinatorial. In particular, we use two parallel sets of group tests, one to filter and the other to certify and estimate; the resulting algorithms are quite simple to implement. [ABSTRACT FROM AUTHOR]

An L(h,1,1)-labeling of a graph is an assignment of labels from the set of integers {0, ⋯, λ} to the vertices of the graph such that adjacent vertices are assigned integers of at least distance h ≥1 apart and all vertices of distance three or less must be assigned different labels. The aim of the L(h,1,1)-labeling problem is to minimize λ, denoted by λh,1,1 and called span of the L(h,1,1)-labeling. As outerplanar graphs have bounded treewidth, the L(1,1,1)-labeling problem on outerplanar graphs can be exactly solved in O(n3), but the multiplicative factor depends on the maximum degree Δ and is too big to be of practical use. In this paper we give a linear time approximation algorithm for computing the more general L(h,1,1)-labeling for outerplanar graphs that is within additive constants of the optimum values. [ABSTRACT FROM AUTHOR]

Radio networks model wireless synchronous communication with only one wave frequency used for transmissions. In the problem of many-to-all (M2A) communication, some nodes hold input rumors, and the goal is to have all nodes learn all the rumors. We study the average time complexity of distributed many-to-all communication by deterministic protocols in directed networks under two scenarios: of combined messages, in which all input rumors can be sent in one packet, and of separate messages, in which every rumor requires a separate packet to be transmitted. Let n denote the size of a network and k be the number of nodes activated with rumors; the case when k = n is called gossiping. We give a gossiping protocol for combined messages that works in the average time ${\mathcal O}(n/\log n)$, which is shown to be optimal. For the general M2A communication problem, we show that it can be performed in the average time ${\mathcal O}(\min\{k\log(n/k),n/\log n\})$ with combined messages, and that Ω(k/logn + logn) is a lower bound. We give a gossiping protocol for separate messages that works in the average time ${\mathcal O}(n\log n)$, which is shown to be optimal. For the general M2A communication problem, we develop a protocol for separate messages with the average time ${\mathcal O}(k\log(n/k)\log n)$, and show that Ω(k logn) is a lower bound. [ABSTRACT FROM AUTHOR]

The Minimum Energy Broadcast Routing problem was extensively studied during the last years. Given a sample space where wireless devices are distributed, the aim is to perform the broadcast pattern of communication from a given source while minimizing the total energy consumption. While many papers deal with the 2-dimensional case where the sample space is given by a flat area, few results are known about the more interesting and practical 3-dimensional case. In this paper we study such a case and we present a tighter analysis of the minimum spanning tree heuristic in order to considerably decrease its approximation factor from the known 26 to roughly 18.8. This decreases the gap with the known lower bound of 12 given by the so called kissing number. [ABSTRACT FROM AUTHOR]

The Minimum Energy Broadcast problem consists in finding the minimum-energy range assignment for a given set S of n stations of an ad hoc wireless network that allows a source station to perform broadcast operations over S. We prove a nearly tight asymptotical bound on the optimal cost for the Minimum Energy Broadcast problem on square grids. We emphasize that finding tight bounds for this problem restriction is far to be easy: it involves the Gauss's Circle problem and the Apollonian Circle Packing. We also derive near-tight bounds for the Bounded-Hop version of this problem. Our results imply that the best-known heuristic, the MST-based one, for the Minimum Energy Broadcast problem is far to achieve optimal solutions (even) on very regular, well-spread instances: its worst-case approximation ratio is about π and it yields $\Omega(\sqrt{n})$ hops. As a by product, we get nearly tight bounds for the Minimum Disk Cover problem and for its restriction in which the allowed disks must have non-constant radius. Finally, we emphasize that our upper bounds are obtained via polynomial time constructions. [ABSTRACT FROM AUTHOR]

We study the problem of Byzantine-robust topology discovery in an arbitrary asynchronous network. We formally state the weak and strong versions of the problem. The weak version requires that either each node discovers the topology of the network or at least one node detects the presence of a faulty node. The strong version requires that each node discovers the topology regardless of faults. We focus on non-cryptographic solutions to these problems. We explore their bounds. We prove that the weak topology discovery problem is solvable only if the connectivity of the network exceeds the number of faults in the system. Similarly, we show that the strong version of the problem is solvable only if the network connectivity is more than twice the number of faults. We present solutions to both versions of the problem. Our solutions match the established graph connectivity bounds. The programs are terminating, they do not require the individual nodes to know either the diameter or the size of the network. The message complexity of both programs is low polynomial with respect to the network size. [ABSTRACT FROM AUTHOR]

Unlike localized communication failures that occur on a fixed (although a priori unknown) set of links, dynamic faults can occur on any link. Known also as mobile or ubiquitous faults, their presence makes many tasks difficult if not impossible to solve even in synchronous systems. Their analysis and the development of fault-tolerant protocols have been carried out under two main models. In this paper, we introduce a new model for dynamic faults in synchronous distributed systems. This model includes as special cases the existing settings studied in the literature. We focus on the hardest setting of this model, called simple threshold, where to be guaranteed that at least one message is delivered in a time step, the total number of transmitted messages in that time step must reach a threshold T ≤c(G), where c(G) is the edge connectivity of the network. We investigate the problem of broadcasting under this model for the worst threshold T = c(G) in several classes of graphs as well as in arbitrary networks. We design solution protocols, proving that broadcast is possible even in this harsh environment. We analyze the time costs showing that broadcast can be completed in (low) polynomial time for several networks including rings (with or without knowledge of n), complete graphs (with or without chordal sense of direction), hypercubes (with or without orientation), and constant-degree networks (with or without full topological knowledge). [ABSTRACT FROM AUTHOR]

The k-set agreement problem is a generalization of the consensus problem: considering a system made up of n processes where each process proposes a value, each non-faulty process has to decide a value such that a decided value is a proposed value, and no more than k different values are decided. It has recently be shown that, in the crash failure model, min$(\lfloor \frac{f}{k} \rfloor+2,\lfloor \frac{t}{k} \rfloor +1)$ is a lower bound on the number of rounds for the non-faulty processes to decide (where t is an upper bound on the number of process crashes, and f, 0 ≤f ≤t, the actual number of crashes). This paper considers the k-set agreement problem in synchronous systems where up to t < n /2 processes can experience general omission failures (i.e., a process can crash or omit sending or receiving messages). It first introduces a new property, called strong termination. This property is on the processes that decide. It is satisfied if, not only every non-faulty process, but any process that neither crashes nor commits receive omission failures decides. The paper then presents a k-set agreement protocol that enjoys the following features. First, it is strongly terminating (to our knowledge, it is the first agreement protocol to satisfy this property, whatever the failure model considered). Then, it is early deciding and stopping in the sense that a process that either is non-faulty or commits only send omission failures decides and halts by round min$(\lfloor \frac{f}{k} \rfloor+2,\lfloor \frac{t}{k} \rfloor +1)$. To our knowledge, this is the first early deciding k-set agreement protocol for the general omission failure model. Moreover, the protocol provides also the following additional early stopping property: a process that commits receive omission failures (and does not crash) executes at most min$(\lceil \frac{f}{k} \rceil +2,\lfloor \frac{t}{k} \rfloor +1)$ rounds. It is worth noticing that the protocol allows each property (strong termination vs early deciding/stopping vs early stopping) not to be obtained at the detriment of the two others. The combination of the fact that min$(\lfloor \frac{f}{k} \rfloor+2,\lfloor \frac{t}{k} \rfloor +1)$ is lower bound on the number of rounds in the crash failure model, and the very existence of the proposed protocol has two very interesting consequences. First, it shows that, although general omission failure model is more severe than the crash failure model, both models have the same lower bound for the non-faulty processes to decide. Second, it shows that, in the general omission failure model, that bound applies also the processes that commit only send omission failures. Keywords: Agreement problem, Crash failure, Strong Termination, Early decision, Early stopping, Efficiency, k-set agreement, Message-passing system, Receive omission failure, Round-based computation, Send omission failure, Synchronous system. [ABSTRACT FROM AUTHOR]