Tight bounds for distributed selection

We revisit the problem of distributed <i>k</i>-selection where, given a general connected graph of diameter <i>D</i> consisting of <i>n</i> nodes in which each node holds a numeric element, the goal is to determine the <i>k^th</i> smallest of these elements. In our model, there is no imposed relation between the magnitude of the stored elements and the number of nodes in the graph. We propose a randomized algorithm whose time complexity is <i>O</i>(<i>D</i>log_<i>D</i> <i>n</i>) with high probability. Additionally, a deterministic algorithm with a worst-case time complexity of <i>O</i>(<i>D</i>log2<over><i>D</i> <i>n</i>) is presented which considerably improves the best known bound for deterministic algorithms. Moreover, we prove a lower bound of Ω(<i>D</i> log_<i>D</i><i>n</i>) for any randomized or deterministic algorithm, implying that the randomized algorithm is asymptotically optimal.