This paper studies the problem of computing the most frequent element (the mode) by means of a distributed algorithm where the elements are located at the nodes of a network. Let k denote the number of distinct elements and further let mi be the number of occurrences of the element ei in the ordered list of occurrences m1 > m2 ≥ ... ≥ mk. We give a deterministic distributed algorithm with time complexity O(D+k) where D denotes the diameter of the graph, which is essentially tight. As our main contribution, a Monte Carlo algorithm is presented which computes the mode in O(D + F2/m2 1 · log k) time with high probability, where the frequency moment F` is de�ned as F` = Pk i=1 m` i . This algorithm is substantially faster than the deterministic algorithm for various relevant frequency distributions. Moreover, we provide a lower bound of Ω(D +F5/(m5 1B)), where B is the maximum message size, that captures the e�ect of the frequency distribution on the time complexity to compute the mode.

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