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.
@cite_1: 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.
@cite_7: 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.
Secondly, in all prior literature including @cite_1 , it is also assumed that tokens are uniquely numbered indexed and that this index is known to any node that starts with a token. This is needed to allow nodes to specify in the coding header which packets are coded together in a message. In this paper such an assumption would be unacceptable. For example, for the task of counting the number of nodes in a dynamic network @cite_7 having the IDs consecutively indexed would essentially amount to assuming that a solution to the counting problem is already part of the input.