Improved Bounds for Max Consensus in Wireless Networks
Aida Nowzari, Michael Rabbat · IEEE Transactions on Signal and Information Processing over Networks · 2018
In consensus problems, the goal is for the nodes of a network to converge to a certain quantity or a function of their values using local communications. In the maximum value consensus problem, the objective of these communications is for all the nodes to converge to the maximum of their initial values. There are two existing algorithms for the maximum value consensus problem in asynchronous networks: RANDOM-PAIRWISE-MAX and RANDOM-BROADCAST-MAX for which the bounds on the mean convergence time have been derived in the literature. In this paper, we derive tighter bounds on the expected convergence time of these two algorithms when run on grid networks and random geometric graphs, respectively-two models commonly used to capture salient properties of wireless networks. We show that RANDOM-PAIRWISE-MAX run on a 2-D grid graph with n nodes converges in expectation after O(n3/2) iterations, and RANDOM-BROADCAST-MAX run on a random geometric graph with n nodes converges in expectation after O((n/ log n)3/2) iterations. These bounds improve over the previous best-known upper bounds by factors of √n log n and log n + log2n, respectively. Experiments illustrate that the proposed bounds can be up to 95% tighter than the previous state-of-the-art bounds. Furthermore, we enhance the proposed bounds by introducing probabilistic network link failures, e.g., to model packet drops in wireless networks.