We use network coding to improve the speed of distributed computation in the dynamic network model of Kuhn, Lynch and Oshman [STOC '10]. In this model an adversary adaptively chooses a new network topology in every round, making even basic distributed computations challenging. show that n nodes, each starting with a d-bit token, can broadcast them to all nodes in time O(n^2) using b-bit messages, where b > d + log n. Their algorithms take the natural approach of token forwarding : in every round each node broadcasts some particular token it knows. They prove matching Omega(n^2) lower bounds for a natural class of token forwarding algorithms and an Omega(n log n) lower bound that applies to all token-forwarding algorithms. We use network coding, transmitting random linear combinations of tokens, to break both lower bounds. Our algorithm's performance is quadratic in the message size b, broadcasting the n tokens in roughly d b^2 * n^2 rounds. For b = d = O(log n) our algorithms use O(n^2 log n) rounds, breaking the first lower bound, while for larger message sizes we obtain linear-time algorithms. We also consider networks that change only every T rounds, and achieve an additional factor T^2 speedup. This contrasts with related lower and upper bounds of implying that for natural token-forwarding algorithms a speedup of T, but not more, can be obtained. Lastly, we give a general way to derandomize random linear network coding, that also leads to new deterministic information dissemination algorithms.
First, @cite_1 , as well as all prior work on algebraic gossip, assumes that the additive overhead of the network coding header, which is linear in the number of coded packets, is negligible compared to the size of a packet. This assumption is backed up by many practical implementations in which this overhead is indeed less than one percent. But a rigorous theoretical treatment, like that of @cite_7 , must account for this overhead which may be significant if message-sizes are small.
@cite_1 @cite_7
2010682628 2146466672 2132612558 2952030388
In this paper we study gossip based information spreading with bounded message sizes. We use algebraic gossip to disseminate k distinct messages to all n nodes in a network. For arbitrary networks we provide a new upper bound for uniform algebraic gossip of O((k + log n + D)Δ) rounds with high probability, where D and Δ are the diameter and the maximum degree in the network, respectively. For many topologies and selections of k this bound improves previous results, in particular, for graphs with a constant maximum degree it implies that uniform gossip is order optimal and the stopping time is Θ(k + D). To eliminate the factor of Δ from the upper bound we propose a non-uniform gossip protocol, TAG, which is based on algebraic gossip and an arbitrary spanning tree protocol S. The stopping time of TAG is O(k+ log n + d(S) + t(S)), where t(S) is the stopping time of the spanning tree protocol, and d(S) is the diameter of the spanning tree. We provide two general cases in which this bound leads to an order optimal protocol. The first is for k=Ω(n), where, using a simple gossip broadcast protocol that creates a spanning tree in at most linear time, we show that TAG finishes after Θ(n) rounds for any graph. The second uses a sophisticated, recent gossip protocol to build a fast spanning tree on graphs with large weak conductance. In turn, this leads to the optimally of TAG on these graphs for k=Ω(polylog(n)). The technique used in our proofs relies on queuing theory, which is an interesting approach that can be useful in future gossip analysis. We study the stopping times of gossip algorithms for network coding. We analyze algebraic gossip (i.e., random linear coding) and consider three gossip algorithms for information spreading Pull, Push, and Exchange. The stopping time of algebraic gossip is known to be linear for the complete graph, but the question of determining a tight upper bound or lower bounds for general graphs is still open. We take a major step in solving this question, and prove that algebraic gossip on any graph of size n is O(Δn) where Δ is the maximum degree of the graph. This leads to a tight bound of Θ(n) for bounded degree graphs and an upper bound of O(n2) for general graphs. We show that the latter bound is tight by providing an example of a graph with a stopping time of Ω(n2). Our proofs use a novel method that relies on Jackson's queuing theorem to analyze the stopping time of network coding; this technique is likely to become useful for future research. In this paper, we study two related iterative randomized algorithms for distributed computation of averages. The first algorithm is the Broadcast Gossip Algorithm, in which at each iteration one randomly selected node broadcasts its own state to its neighbors. The second algorithm is a novel variation of the former, in which at each iteration every node is allowed to broadcast: hence, this algorithm, which we call Collision Broadcast Gossip Algorithm (CBGA), is affected by interference among messages. The performance of both algorithms is evaluated in terms of rate of convergence and asymptotical error: focusing on large Abelian Cayley networks, we highlight the role of topology and of design parameters. We show that on fully connected graphs the rate of convergence is bounded away from one, whereas the asymptotical error is bounded away from zero. On the contrary, on sparse graphs the rate of convergence goes to one and the asymptotical error goes to zero, as the size of the network grows larger. Our results also show that the performance of the CBGA is close to the performance of the BGA: this indicates the robustness of broadcast gossip algorithms to interferences. We give a new technique to analyze the stopping time of gossip protocols that are based on random linear network coding (RLNC). Our analysis drastically simplifies, extends and strengthens previous results. We analyze RLNC gossip in a general framework for network and communication models that encompasses and unifies the models used previously in this context. We show, in most settings for the first time, that it converges with high probability in the information-theoretically optimal time. Most stopping times are of the form O(k + T) where k is the number of messages to be distributed and T is the time it takes to disseminate one message. This means RLNC gossip achieves "perfect pipelining". Our analysis directly extends to highly dynamic networks in which the topology can change completely at any time. This remains true even if the network dynamics are controlled by a fully adaptive adversary that knows the complete network state. Virtually nothing besides simple O(kT) sequential flooding protocols was previously known for such a setting. While RLNC gossip works in this wide variety of networks its analysis remains the same and extremely simple. This contrasts with more complex proofs that were put forward to give less strong results for various special cases.