Simple, Fast and Deterministic Gossip and Rumor Spreading
  We study gossip algorithms for the rumor spreading problem which asks each node to deliver a rumor to all nodes in an unknown network. Gossip algorithms allow nodes only to call one neighbor per round and have recently attracted attention as message efficient, simple and robust solutions to the rumor spreading problem.   Recently, non-uniform random gossip schemes were devised to allow efficient rumor spreading in networks with bottlenecks. In particular, [Censor-Hillel et al., STOC'12] gave an O(log^3 n) algorithm to solve the 1-local broadcast problem in which each node wants to exchange rumors locally with its 1-neighborhood. By repeatedly applying this protocol one can solve the global rumor spreading quickly for all networks with small diameter, independently of the conductance.   This and all prior gossip algorithms for the rumor spreading problem have been inherently randomized in their design and analysis. This resulted in a parallel research direction trying to reduce and determine the amount of randomness needed for efficient rumor spreading. This has been done via lower bounds for restricted models and by designing gossip algorithms with a reduced need for randomness. The general intuition and consensus of these results has been that randomization plays a important role in effectively spreading rumors.   In this paper we improves over this state of the art in several ways by presenting a deterministic gossip algorithm that solves the the k-local broadcast problem in 2(k+log n)log n rounds. Besides being the first efficient deterministic solution to the rumor spreading problem this algorithm is interesting in many aspects: It is simpler, more natural, more robust and faster than its randomized pendant and guarantees success with certainty instead of with high probability. Its analysis is furthermore simple, self-contained and fundamentally different from prior works. 
Bernhard Haeupler
cs.DS cs.DC math.CO