A simple randomized scheme for constructing low-weight k-connected spanning subgraphs with applications to distributed algorithms

The main focus of this paper is the analysis of a simple randomized scheme for constructing low-weight k-connected spanning subgraphs. We first show that our scheme gives a simple approximation algorithm to construct a minimum-weight k-connected spanning subgraph in a weighted complete graph, a NP-hard problem even if the weights satisfy the triangle inequality. We show that our algorithm gives an approximation ratio of O(k log n) for a metric graph, O(k) for a random graph with nodes uniformly randomly distributed in [0, 1]2 and O(log nk ) for a complete graph with random edge weights U(0, 1). We show that our scheme is optimal with respect to the amount of “local information” needed to compute any connected spanning subgraph. We then show that our scheme can be applied to design an efficient distributed algorithm for constructing such an approximate k-connected spanning subgraph (for any k ≥ 1) in a point-to-point distributed model, where the processors form a complete network. Our algorithm takes O(log nk ) time and expected O(nk log n k ) messages. Our result in conjunction with a result of Korach et al. ([21]) implies that the expected message complexity of our algorithm is significantly better than the best distributed algorithm that finds an optimal k-connected subgraph. We also show that for the geometric instances, our randomized scheme constructs low-degree k-connected spanning subgraphs which have O(k log n) maximum degree, with high probability.