| Abstract |
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 m_i be the number of occurrences of the element e_i in the ordered list of occurrences m_1>m_2>=...>=m_k. 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 + F_2/m_1^2 log k) time with high probability, where the frequency moment F_l is defined as F_l = Sum_(i=1)^k m_i^l. This algorithm is substantially faster than the deterministic algorithm for various relevant frequency distributions. Moreover, we provide a lower bound of Ω(D + F_5/(m_1^5 B)), where B is the maximum message size, that captures the eect of the frequency distribution on the time complexity to compute the mode. |
