An Inductive-style Procedure for Counting Monochromatic Simplexes of Symmetric Subdivisions with Applications to Distributed Computing

Abstract

In the weak symmetry breaking (WSB) task, each of n+1 processes has to decide either 0 or 1 such that not all processes decide the same value. A WSB algorithm should be wait-free, namely, processes are asynchronous (there is no bound on their relative speeds), and potentially faulty (any proper subset may halt without warning). Also, it should be comparison based, namely, processes can only use comparison operations (<,>,=) on the values they read from the memory.Symmetric chromatic subdivisions of an n-simplex have been used to represent the executions of a distributed algorithm solving the WSB task. Informally, each simplex of such a subdivision corresponds to an execution of the algorithm. Each vertex is labeled with the local state of a process in the execution; it is colored with the process ID, and also with the binary value the process decides in the execution. The symmetry properties of such a complex come from the comparison based requirement of a WSB algorithm.Let C denote the number of monochromatic n-simplexes of such a subdivision, counted by orientation. Previous work has shown that C=1+∑i=0n−1(n+1i+1)ki for some coefficients ki∈Z. This characterization of C implies that the WSB task on n+1 processes is solvable if and only if n is such that the binomial coefficients are relatively prime, or equivalently, if and only if n is not a primer power.This paper presents an inductive style procedure that yields an alternative proof of the characterization of C. Roughly speaking, the proof consists in a procedure for modifying gradually the binary coloring of a symmetric chromatic subdivision, and computing the degree of the maps produced during the procedure.