Rotem Oshman

Distributed computation in dynamic networks

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 com- putable function of their initial inputs in O(n2) 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 + n2/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.

Coordinated consensus in dynamic networks

We study several variants of coordinated consensus in dynamic networks. We assume a synchronous model, where the communication graph for each round is chosen by a worst-case adversary. The network topology is always connected, but can change completely from one round to the next. The model captures mobile and wireless networks, where communication can be unpredictable. In this setting we study the fundamental problems of eventual, simultaneous, and Δ-coordinated consensus, as well as their relationship to other distributed problems, such as determining the size of the network. We show that in the absence of a good initial upper bound on the size of the network, eventual consensus is as hard as computing deterministic functions of the input, e.g., the minimum or maximum of inputs to the nodes. We also give an algorithm for computing such functions that is optimal in every execution. Next, we show that simultaneous consensus can never be achieved in less than n - 1 rounds in any execution, where n is the size of the network; consequently, simultaneous consensus is as hard as computing an upper bound on the number of nodes in the network. For Δ-coordinated consensus, we show that if the ratio between nodes with input 0 and input 1 is bounded away from 1, it is possible to decide in time n-Θ(√ nΔ), where Δ bounds the time from the first decision until all nodes decide. If the dynamic graph has diameter D, the time to decide is min{O(nD/Δ),n-Ω(nΔ/D)}, even if D is not known in advance. Finally, we show that (a) there is a dynamic graph such that for every input, no node can decide before time n-O(Δ0.28n0.72); and (b) for any diameter D = O(Δ), there is an execution with diameter D where no node can decide before time Ω(nD / Δ). To our knowledge, our work constitutes the first study of Δ-coordinated consensus in general graphs.

On the power of the congested clique model

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(nε) time for any ε > 0. The broadcast version of the congested clique is the well-known 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.

A Tight Bound for Set Disjointness in the Message-Passing Model

In a multiparty message-passing model of communication, there are k players. Each player has a private input, and they communicate by sending messages to one another over private channels. While this model has been used extensively in distributed computing and in secure multiparty computation, lower bounds on communication complexity in this model and related models have been somewhat scarce. In recent work [25], [29], [30], strong lower bounds of the form Ω(n·k) were obtained for several functions in the message-passing model; however, a lower bound on the classical set disjointness problem remained elusive. In this paper, we prove a tight lower bound of Ω(n · k) for the set disjointness problem in the message passing model. Our bound is obtained by developing information complexity tools for the message-passing model and proving an information complexity lower bound for set disjointness.

Broadcasting in unreliable radio networks

Practitioners agree that unreliable links, which sometimes deliver messages and sometime do not, are an important characteristic of wireless networks. In contrast, most theoretical models of radio networks fix a static set of links and assume that these links are reliable. This gap between theory and practice motivates us to investigate how unreliable links affect theoretical bounds on broadcast in radio networks. To that end we consider a model that includes two types of links: reliable links, which always deliver messages, and unreliable links, which sometimes fail to deliver messages. We assume that the reliable links induce a connected graph, and that unreliable links are controlled by a worst-case adversary. In the new model we show an Ω(n log n) lower bound on deterministic broadcast in undirected graphs, even when all processes are initially awake and have collision detection, and an Ω(n) lower bound on randomized broadcast in undirected networks of constant diameter. This separates the new model from the classical, reliable model. On the positive side, we give two algorithms that tolerate unreliability: an O(n3/2 √log n)-time deterministic algorithm and a randomized algorithm which terminates in O(n log2 n) rounds with high probability.

Gradient clock synchronization in dynamic networks

Over the last years, large-scale decentralized computer networks such as peer-to-peer and mobile ad hoc networks have become increasingly prevalent. The topologies of many of these networks are often highly dynamic. This is especially true for ad hoc networks formed by mobile wireless devices. In this paper, we study the fundamental problem of clock synchronization in dynamic networks. We show that there is an inherent trade-off between the skew <i>S</i> guaranteed along sufficiently old links and the time needed to guarantee a small skew along new links. For any sufficiently large initial skew on a new link, there are executions in which the time required to reduce the skew on the link to <i>O</i>(<i>S</i>) is at least Ω(<i>n</i>/<i>S</i>). We show that this bound is tight for moderately small values of <i>S</i>. Assuming a fixed set of

Optimal gradient clock synchronization in dynamic networks

n

The communication complexity of distributed task allocation

nodes and an arbitrary pattern of edge insertions and removals, a weak dynamic connectivity requirement suffices to prove the following results. We present an algorithm that always maintains a skew of <i>O</i>(<i>n</i>) between any two nodes in the network. For a parameter <i>S</i>=Ω(√Á<i>n</i>), where Á is the maximum hardware clock drift, it is further guaranteed that if a communication link between two nodes <i>u</i>, <i>v</i> persists in the network for Θ(<i>n</i>/<i>S</i>) time, the clock skew between <i>u</i> and <i>v</i> is reduced to no more than <i>O</i>(<i>S</i>).

The complexity of data aggregation in directed networks

We study the problem of clock synchronization in highly dynamic networks, where communication links can appear or disappear at any time. The nodes in the network are equipped with hardware clocks, but the rate of the hardware clocks can vary arbitrarily within specific bounds, and the estimates that nodes can obtain about the clock values of other nodes are inherently inaccurate. Our goal in this setting is to output a logical clock at each node, such that the logical clocks of any two nodes are not too far apart, and nodes that remain close to each other in the network for a long time are better synchronized than distant nodes. This property is called gradient clock synchronization. Gradient clock synchronization has been widely studied in the static setting. We show that the bounds for the static case also apply to our highly dynamic setting: if two nodes remain at distance d from each other for sufficiently long, it is possible to synchronize their clocks to within O(d log(D/d)), where D is the diameter of the network. This is known to be optimal for static networks, and since a static network is a special case of a dynamic network, it is optimal for dynamic networks as well. Furthermore, we show that our algorithm has optimal stabilization time: when a path of length d appears between two nodes, the time required until the skew between the two nodes is reduced to O(d log(D/d)) is O(D), which we prove is optimal.

The SkipTrie: low-depth concurrent search without rebalancing

We consider a distributed task allocation problem in which <i>m</i> players must divide a set of <i>n</i> tasks between them. Each player <i>i</i> receives as input a set <i>X_i</i> of tasks such that the union of all input sets covers the task set. The goal is for each player to output a subset <i>Y_i</i> ⊆ <i>X_i</i>, such that the outputs (<i>Y₁</i>,...,<i>Y_m</i>) form a partition of the set of tasks. The problem can be viewed as a distributed one-shot variant of the well-known <i>k</i>-server problem, and we also show that it is closely related to the problem of finding a rooted spanning tree in directed broadcast networks. We study the communication complexity and round complexity of the task allocation problem. We begin with the classical two-player communication model, and show that the randomized communication complexity of task allocation is Ω(<i>n</i>), even when the set of tasks is known to the players in advance. For the multi-player setting with <i>m</i> = <i>O</i>(<i>n</i>) we give two upper bounds in the shared-blackboard model of communication. We show that the problem can be solved in <i>O</i>(log <i>n</i>) rounds and <i>O</i>(<i>n</i> log <i>n</i>) total bits for arbitrary inputs; moreover, if for any set <i>X</i> of tasks, there are at least α|<i>X</i>| players that have at least one task from <i>X</i> in their inputs, then <i>O</i>((1/α + log <i>m</i>)log <i>n</i>) rounds suffice even if each player can only write <i>O</i>(log <i>n</i>) bits on the blackboard in each round. Finally, we extend our results to the case where the players communicate over an arbitrary directed communication graph instead of a shared blackboard. As an application of these results, we also consider the related problem of constructing a directed spanning tree in strongly-connected directed networks and we show lower and upper bounds for that problem.