On the Complexity of Information Spreading in Dynamic Networks [chapter]

Chinmoy Dutta, Gopal Pandurangan, Rajmohan Rajaraman, Zhifeng Sun, Emanuele Viola
2013 Proceedings of the Twenty-Fourth Annual ACM-SIAM Symposium on Discrete Algorithms  
We study how to spread k tokens of information to every node on an n-node dynamic network, the edges of which are changing at each round. This basic gossip problem can be completed in O(n+k) rounds in any static network, and determining its complexity in dynamic networks is central to understanding the algorithmic limits and capabilities of various dynamic network models. Our focus is on token-forwarding algorithms, which do not manipulate tokens in any way other than storing, copying and
more » ... ding them. We first consider the strongly adaptive adversary model where in each round, each node first chooses a token to broadcast to all its neighbors (without knowing who they are), and then an adversary chooses an arbitrary connected communication network for that round with the knowledge of the tokens chosen by each node. We show that Ω(nk/ log n + n) rounds are needed for any randomized (centralized or distributed) token-forwarding algorithm to disseminate the k tokens, thus resolving an open problem raised in [KLO10] . The bound applies to a wide class of initial token distributions, including those in which each token is held by exactly one node and well-mixed ones in which each node has each token independently with a constant probability. Our result for the strongly adaptive adversary model motivates us to study the weakly adaptive adversary model where in each round, the adversary is required to lay down the network first, and then each node sends a possibly distinct token to each of its neighbors. We propose a simple randomized distributed algorithm where in each round, along every edge (u, v), a token sampled uniformly at random from the symmetric difference of the sets of tokens held by node u and node v is exchanged. We prove that starting from any well-mixed distribution of tokens where each node has each token independently with a constant probability, this algorithm solves the k-gossip problem in O((n + k) log n log k) rounds with high probability over the initial token distribution and the randomness of the protocol. We then show how the above uniform sampling problem can be solved usingÕ(log k) bits of communication, making the overall algorithm communication-efficient. We next present a centralized algorithm that solves the gossip problem for every initial distribution in O((n + k) log 2 n) rounds in the offline setting where the entire sequence of communication networks is known to the algorithm in advance. Finally, we present an O(n min{k, √ k log n})-round centralized offline algorithm in which each node can only broadcast a single token to all of its neighbors in each round. Theorem 1. (a) Any randomized token-forwarding algorithm (centralized or distributed) for kgossip needs Ω(nk/ log n + n) rounds in the strongly adaptive adversary model starting from any initial token distribution in which each of k ≤ n tokens is held by exactly one node. (b) In addition, the same bound holds with high probability over an initial token distribution where each of the n nodes receives each of k ≤ n tokens independently with probability 3/4. This result resolves an open problem raised in [KLO10], improving their lower bound of Ω(n log n) for k = ω(log n log log n), and matching their upper bound to within a logarithmic factor. Our lower bound also enables a better comparison of token-forwarding with an alternative approach based on network coding due to [Hae11, HK11] . Assuming the size of each message is bounded by the size of a token, network coding completes k-gossip in O(nk/ log n + n) rounds for O(log n)-bit tokens, and O(n + k) rounds for Ω(n log n) bit tokens. Thus, for large token sizes, our result establishes a factor Ω(min{n, k}/ log n) gap between token-forwarding and network coding, a significant new bound on the network coding advantage for information dissemination. 1 Furthermore, for small token and message sizes (e.g., O(polylog(n)) bits), we do not know of any algorithm (network coding, or otherwise) that completes k-gossip against a strongly adaptive adversary in o(nk/polylog(n)) rounds. Our lower bound for the strongly adaptive adversary model motivates us to study models which restrict the power of the adversary and/or strengthen the capabilities of the algorithm. We would like to restrict the adversary power as little as possible and yet design fast algorithms.
doi:10.1137/1.9781611973105.52 dblp:conf/soda/DuttaPRSV13 fatcat:dtqtpi2yzjaztbamql35ggmonm