In this paper we investigate distributed computation in dynamic networks in which the network topology changes from round to round. We consider a worst-case model in which the communication links for each round are chosen by an adversary, and nodes do not know who their neighbors for the current round are before they broadcast their messages. The model captures mobile networks and wireless networks, in which mobility and interference render communication unpredictable. In contrast to much of the existing work on dynamic networks, we do not assume that the network eventually stops changing; we require correctness and termination even in networks that change continually. We introduce a stability property called T -interval connectivity (for T ≥ 1), which stipulates that for every T consecutive rounds there exists a stable connected spanning subgraph. For T = 1 this means that the graph is connected in every round, but changes arbitrarily between rounds.We show that in 1-interval connected graphs it is possible for nodes to determine the size of the network and compute any computable function of their initial inputs in O(n 2 ) rounds using messages of size O(log n + d), where d is the size of the input to a single node. Further, if the graph is T -interval connected for T > 1, the computation can be sped up by a factor of T , and any function can be computed in O(n + n 2 /T ) rounds using messages of size O(log n + d). We also give two lower bounds on the token dissemination problem, which requires the nodes to disseminate k pieces of information to all the nodes in the network.The T-interval connected dynamic graph model is a novel model, which we believe opens new avenues for research in the theory of distributed computing in wireless, mobile and dynamic networks.

In this paper we present GOAFR, a new geometric ad-hoc routing algorithm combining greedy and face routing. We evaluate this algorithm by both rigorous analysis and comprehensive simulation. GOAFR is the first ad-hoc algorithm to be both asymptotically optimal and average-case efficient. For our simulations we identify a network density range critical for any routing algorithm. We study a dozen of routing algorithms and show that GOAFR outperforms other prominent algorithms, such as GPSR or AFR.

We study the computation power of the congested clique, a model of distributed computation where n players communicate with each other over a complete network in order to compute some function of their inputs. The number of bits that can be sent on any edge in a round is bounded by a parameter b. We consider two versions of the model: in the first, the players communicate by unicast, allowing them to send a different message on each of their links in one round; in the second, the players communicate by broadcast, sending one message to all their neighbors.It is known that the unicast version of the model is quite powerful; to date, no lower bounds for this model are known. In this paper we provide a partial explanation by showing that the unicast congested clique can simulate powerful classes of bounded-depth circuits, implying that even slightly super-constant lower bounds for the congested clique would give new lower bounds in circuit complexity. Moreover, under a widely-believed conjecture on matrix multiplication, the triangle detection problem, studied in [8], can be solved in O(n ) time for any > 0.The broadcast version of the congested clique is the wellknown multi-party shared-blackboard model of communication complexity (with number-in-hand input). This version is more amenable to lower bounds, and in this paper we show that the subgraph detection problem studied in [8] requires polynomially many rounds for several classes of subgraphs. We also give upper bounds for the subgraph detection problem, and relate the hardness of triangle detection in the broadcast congested clique to the communication complexity of set disjointness in the 3-party number-on-forehead model.

The question of what can be computed, and how efficiently, are at the core of computer science. Not surprisingly, in distributed systems and networking research, an equally fundamental question is what can be computed in a distributed fashion. More precisely, if nodes of a network must base their decision on information in their local neighborhood only, how well can they compute or approximate a global (optimization) problem? In this paper we give the first poly-logarithmic lower bound on such local computation for (optimization) problems including minimum vertex cover, minimum (connected) dominating set, maximum matching, maximal independent set, and maximal matching. In addition we present a new distributed algorithm for solving general covering and packing linear programs. For some problems this algorithm is tight with the lower bounds, for others it is a distributed approximation scheme. Together, our lower and upper bounds establish the local computability and approximability of a large class of problems, characterizing how much local information is required to solve these tasks. ]. We are grateful to and Schwartzman [7] for pointing out an error in an earlier draft [30] of this paper. |V | = n, and a parameter k (k might depend on n or some other property of G). At each node v ∈ V there is an independent agent (for simplicity, we identify the agent at node v with v as well). Every node v ∈ V has a unique identifier id(v) 1 and possibly some additional input. We assume that each node v ∈ V can learn the complete neighborhood Γ k (v) up to distance k in G (see below for a formal definition of Γ k (v)). Based on this information, all nodes need to make independent computations and need to individually decide on their outputs without communicating with each other. Hence, the output of each node v ∈ V can be computed as a function of it's k-neighborhood Γ k (v).Synchronous Message Passing Model: The described graph-theoretic local computation model is equivalent to the classic message passing model of distributed computing. In this model, the distributed system is modeled as a point-to-point communication network, described by an undirected graph G = (V, E), in which each vertex v ∈ V represents a node (host, device, processor, . . . ) of the network, and an edge (u, v) ∈ E is a bidirectional communication channel that connects the two nodes. Initially, nodes have no knowledge about the network graph; they only know their own identifier and potential additional inputs. All nodes wake up simultaneously and computation proceeds in synchronous rounds. In each round, every node can send one, arbitrarily long message to each of its neighbors. Since we consider point-to-point networks, a node may send different messages to different neighbors in the same round. Additionally, every node is allowed to perform local computations based on information obtained in messages of previous rounds. Communication is reliable, i.e., every message that is sent during a communication round is correctly received by the end of the ro...

We give time lower bounds for the distributed approximation of minimum vertex cover (MVC) and related problems such as minimum dominating set (MDS). In k communication rounds, MVC and MDS can only be approximated by factors Ω(n c/k 2 /k) and Ω(∆ 1/k /k) for some constant c, where n and ∆ denote the number of nodes and the largest degree in the graph. The number of rounds required in order to achieve a constant or even only a polylogarithmic approximation ratio is at least Ω(log n/ log log n) and Ω(log ∆/ log log ∆). By a simple reduction, the latter lower bounds also hold for the construction of maximal matchings and maximal independent sets.

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This paper is centered on the complexity of graph problems in the well-studied LOCAL model of distributed computing, introduced by Linial [FOCS '87]. It is widely known that for many of the classic distributed graph problems (including maximal independent set (MIS) and (∆ + 1)-vertex coloring), the randomized complexity is at most polylogarithmic in the size n of the network, while the best deterministic complexity is typically 2 O( √ log n) . Understanding and potentially narrowing down this exponential gap is considered to be one of the central long-standing open questions in the area of distributed graph algorithms.We investigate the problem by introducing a complexity-theoretic framework that allows us to shed some light on the role of randomness in the LOCAL model. We define the SLOCAL model as a sequential version of the LOCAL model. Our framework allows us to prove completeness results with respect to the class of problems which can be solved efficiently in the SLOCAL model, implying that if any of the complete problems can be solved deterministically in poly log n rounds in the LOCAL model, we can deterministically solve all efficient SLOCAL-problems (including MIS and (∆ + 1)-coloring) in poly log n rounds in the LOCAL model.Perhaps most surprisingly, we show that a rather rudimentary looking graph coloring problem is complete in the above sense: Color the nodes of a graph with colors red and blue such that each node of sufficiently large polylogarithmic degree has at least one neighbor of each color. The problem admits a trivial zero-round randomized solution. The result can be viewed as showing that the only obstacle to getting efficient determinstic algorithms in the LOCAL model is an efficient algorithm to approximately round fractional values into integer values.In addition, our formal framework also allows us to develop polylogarithmic-time randomized distributed algorithms in a simpler way. As a result, we provide a polylog-time distributed approximation scheme for arbitrary distributed covering and packing integer linear programs.The question of whether a given distributed problem can be solved locally has been at the center of the theory of distributed graph algorithms since the 1980s, especially starting with the seminal work of Awerbuch, Goldberg, Luby, and Plotkin [AGLP89], Linial [Lin92], and Naor and Stockmeyer [NS95]. The locality of distributed computations is captured by the LOCAL model [Lin92, Pel00], defined as follows: a network is modeled as an undirected graph G = (V, E), the nodes V are the network devices, and the edges E are bidirectional communication links. Time is divided into synchronous communication rounds. In each round, each node can perform some arbitrary internal computation, send a message of possibly arbitrary size to each of its neighbors, and receive the messages sent to it by its neighbors. A typical objective in this setting is to solve some given graph problem on the network G by a distributed algorithm. For example, classic problems include computing a vertex or...

The distributed (∆ + 1)-coloring problem is one of most fundamental and well-studied problems in Distributed Algorithms. Starting with the work of Cole and Vishkin in 86, there was a long line of gradually improving algorithms published. The current state-of-the-art running time is O(∆ log ∆+log * n), due to Kuhn and Wattenhofer, PODC'06. Linial (FOCS'87) has proved a lower bound of 1 2 log * n for the problem, and Szegedy and Vishwanathan (STOC'93) provided a heuristic argument that shows that algorithms from a wide family of locally iterative algorithms are unlikely to achieve running time smaller than Θ(∆ log ∆). We present a deterministic (∆+1)-coloring distributed algorithm with running time O(∆)+ 1 2 log * n. We also present a tradeoff between the running time and the number of colors , and devise an O(∆ · t)-coloring algorithm with running time O(∆/t+log * n), for any parameter t, 1 < t ≤ ∆ 1− , for an arbitrarily small constant , 0 < < 1. Our algorithm breaks the heuristic barrier of Szegedy and Vishwanathan, and achieves running time which is linear in the maximum degree ∆. On the other hand, the conjecture of Szegedy and Vishwanathan may still be true, as our algorithm is not from the family of locally iterative algorithms. On the way to this result we study a generalization of the notion of graph coloring, which is called defective coloring. In an m-defective p-coloring the vertices are colored with p colors so that each vertex has up to m neighbors with the same color. We show that an m-defective p-coloring with reasonably small m and p can be computed very efficiently. We also develop a technique to employ multiple defective colorings of various subgraphs of the original graph G for computing a (∆ + 1)-coloring of G. We believe that these techniques are of independent interest. *