Efficient gossip and robust distributed computation

This paper presents an efficient deterministic gossip algorithm for p synchronous, crash-prone, message-passing processors. The algorithm has time complexity T = O(log(2)p) and message complexity M = O(p(1+epsilon)), for any epsilon > 0. This substantially improves the message complexity of the previous best algorithm that has M = O(p(1.77)), while maintaining the same time complexity. The strength of the new algorithm is demonstrated by constructing a deterministic algorithm for performing n tasks in this distributed setting. Previous solutions used coordinator or check-pointing approaches, immediately incurring a work penalty Omega(n + f (.) p) for f crashes, or relied on strong communication primitives, such as reliable broadcast, or had work too close to the trivial Theta(p (.) n) bound of oblivious algorithms. The new algorithm uses p crash-prone processors to perform n similar and idempotent tasks so long as one processor remains active. The work of the algorithm is W = O(n + p (.) min {f + 1, log(3)p}) and its message complexity is M = O(fp(epsilon) + pmin {f + 1, log p}), for any epsilon > 0. This substantially improves the work complexity of previous solutions using simple point-to-point messaging, while "meeting or beating" the corresponding message complexity bounds. The new algorithms use communication graphs and permutations with certain combinatorial properties that are shown to exist. The algorithms are correct for any permutations, and in particular, the same expected bounds can be achieved using random permutations.