A nearly optimal distributed algorithm for computing the weighted girth

Computing the weighted girth, which is the sum of weights of edges in the minimum weight cycle, is an important problem in network analysis. The problem for distributively computing girth in unweighted graphs has garnered lots of attention, but there are few studies in weighted graphs. In this paper, we propose a distributed randomized algorithm for computing the weighted girth in weighted graphs with integral edge weights in the range [1, nc], where n is the number of vertices and c is a constant. The algorithm is devised under the standard synchronous $${\cal C}{\cal O}{\cal N}{\cal G}{\cal E}{\cal S}{\cal T}$$ model, which limits each vertex can only transfer O(log n) bits information along each incident edge in a round. The upper bound of the algorithm is O(n log2n) rounds. We also prove the lower bound for computing the weighted girth is Ω(D + n/ log n) where D is the hop diameter of the weighted graph. This means our distributed algorithm is optimal within a factor of O(log3n).