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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(∆n) where ∆ 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(n 2 ) for general graphs. We show that the latter bound is tight by providing an example of a graph with a stopping time of Ω(n 2 ). 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.

This paper studies the resilient routing and (in-band) fast failover mechanisms supported in Software-Defined Networks (SDN). We analyze the potential benefits and limitations of such failover mechanisms, and focus on two main metrics: (1) correctness (in terms of connectivity and loop-freeness) and (2) load-balancing. We make the following contributions. First, we show that in the worst-case (i.e., under adversarial link failures), the usefulness of local failover is rather limited: already a small number of failures will violate connectivity properties under any fast failover policy, even though the underlying substrate network remains highly connected. We then present randomized and deterministic algorithms to compute resilient forwarding sets; these algorithms achieve an almost optimal tradeoff. Our worst-case analysis is complemented with a simulation study.

Abstract-This paper initiates the study of locally selfadjusting networks: networks whose topology adapts dynamically and in a decentralized manner, to the communication pattern σ. Our vision can be seen as a distributed generalization of the selfadjusting datastructures introduced by Sleator and Tarjan [22]: In contrast to their splay trees which dynamically optimize the lookup costs from a single node (namely the tree root), we seek to minimize the routing cost between arbitrary communication pairs in the network.As a first step, we study distributed binary search trees (BSTs), which are attractive for their support of greedy routing. We introduce a simple model which captures the fundamental tradeoff between the benefits and costs of self-adjusting networks. We present the SplayNet algorithm and formally analyze its performance, and prove its optimality in specific case studies. We also introduce lower bound techniques based on interval cuts and edge expansion, to study the limitations of any demand-optimized network. Finally, we extend our study to multi-tree networks, and highlight an intriguing difference between classic and distributed splay trees.

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)∆) rounds with high probability, where D and ∆ 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 ∆ 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.One of the most basic information spreading applications is that of disseminating information stored at a subset of source nodes to a set of sink nodes. Here we consider the k-dissemination case: k initial messages (k ≤ n) located at some nodes (a node can hold more than one initial message) need to reach all n nodes. The all-to-all communication -each of n nodes has an initial value that is needed to be disseminated to all nodes -is a special case of k-dissemination. The goal is to perform this task in the lowest possible number of time steps when messages have limited size (i.e., a node may not be able to send all its data in one message).Gossiping, or rumor-spreading, is a simple stochastic process for dissemination of information across a network. In a synchronous round of gossip, each node chooses a single neighbor as the communication partner and takes an action. In an asynchronous time model a single node wake-ups and chooses the communication partner and n consecutive steps are considered as one round. The gossip communication model defines how to select this neighbor, e.g., uniform gossip is when the communication partner is selected uniformly at random from the set of all neighbors. We then consider three possible actions: either the node pushes information to the partner (PUSH), pulls information from the partner (PULL), or does both (EXCHANGE), but here we mostly present results about EXCHANGE.A gossip protocol uses a gossip communication model in conjunction with ...