Tight bounds for asynchronous randomized consensus

Abstract

A distributed consensus algorithm allows n processes to reach a common decision value starting from individual inputs. Wait-free consensus, in which a process always terminates within a finite number of its own steps, is impossible in an asynchronous shared-memory system. However, consensus becomes solvable using randomization when a process only has to terminate with probability 1. Randomized consensus algorithms are typically evaluated by their total step complexity , which is the expected total number of steps taken by all processes. This article proves that the total step complexity of randomized consensus is Θ( n 2 ) in an asynchronous shared memory system using multi-writer multi-reader registers. This result is achieved by improving both the lower and the upper bounds for this problem. In addition to improving upon the best previously known result by a factor of log 2 n , the lower bound features a greatly streamlined proof. Both goals are achieved through restricting attention to a set of layered executions and using an isoperimetric inequality for analyzing their behavior. The matching algorithm decreases the expected total step complexity by a log n factor, by leveraging the multi-writing capability of the shared registers. Its correctness proof is facilitated by viewing each execution of the algorithm as a stochastic process and applying Kolmogorov's inequality.