SIAM Journal on Applied Mathematics
Proceedings of the nineteenth annual ACM-SIAM symposium on Discrete algorithms
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In this paper, we provide a detailed comparison between a fully randomized protocol for rumor spreading on a complete graph and a quasirandom protocol introduced by Doerr, Friedrich, and Sauerwald [Quasirandom rumor spreading, in Proceedings of the 19th Annual ACM-SIAM Symposium on Discrete Algorithms, ACM, New York, SIAM, Philadelphia, 2008, pp. 773-781]. In the former, initially there is one vertex which holds a piece of information, and during each round every one of the informed vertices chooses uniformly at random and independently one of its neighbors and informs it. In the quasirandom version of this method (cf. Doerr, Friedrich, and Sauerwald) each vertex has a cyclic list of its neighbors. Once a vertex has been informed, it chooses uniformly at random only one neighbor. In the following round, it informs this neighbor, and at each subsequent round it picks the next neighbor from its list and informs it. We give a precise analysis of the evolution of the quasirandom protocol on the complete graph with $n$ vertices and show that it evolves essentially in the same way as the randomized protocol. In particular, if $S(n)$ denotes the number of rounds that are needed until all vertices are informed, we show that for any slowly growing function $\omega(n)$, we have $\log_2 n + \ln n - 4 \ln \ln n \leq S(n) \leq \log_2 n + \ln n + \omega(n)$, with probability $1-o(1)$.