Message Reduction in the LOCAL Model is a Free Lunch

Abstract

A new spanner construction algorithm is presented, working under the LOCAL model assuming unique edge IDs. Given an n-node communication graph, a spanner with a constant stretch and Õ(n1 + c) edges (for any small constant c > 0) is constructed efficiently --- i.e., in a constant number of rounds and a message complexity of Õ (n1 + 2c) whp. One of the many known applications of spanners is for reducing the number of messages of various algorithms. However, usually, one still needs to pay the cost of constructing the spanner. Due to the efficiency of the spanner construction here, we show that every t-round LOCAL algorithm can be transformed into a randomized one with the same asymptotic time complexity and Õ(t2n1 + O(1/log t)) message complexity. All previous message-reduction schemes for LOCAL algorithms incur either an O(log n)-multiplicative or an O(polylog (n))-additive blow-up of the round complexity.