On the Complexity of Information Spreading in Dynamic Networks

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 forwarding 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 using Õ(log k) bits of communication, making the overall algorithm communication-efficient. ∗College of Computer and Information Science, Northeastern University, Boston, 02115, USA. Email: {chinmoy,rraj,austin,viola}@ccs.neu.edu. Chinmoy Dutta is supported in part by NSF grant CCF0845003 and a Microsoft grant to Ravi Sundaram; Rajmohan Rajaraman and Zhifeng Sun are supported in part by NSF grant CNS-0915985; Emanuele Viola is supported by NSF grant CCF-0845003. †Division of Mathematical Sciences, Nanyang Technological University, Singapore 637371 and Department of Computer Science, Brown University, Providence, RI 02912, USA. Email: [email protected]. Supported in part by the following research grants: Nanyang Technological University grant M58110000, Singapore Ministry of Education (MOE) Academic Research Fund (AcRF) Tier 2 grant MOE2010-T2-2-082, and a grant from the US-Israel Binational Science Foundation (BSF). We next present a centralized algorithm that solves the gossip problem for every initial distribution in O((n + k) log 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.