Combinatorial topology of the standard chromatic subdivision and Weak Symmetry Breaking for 6 processes

Abstract

In this paper we study a family of discrete configuration spaces, the so-called protocol complexes, which are of utmost importance in theoretical distributed computing. Specifically, we consider questions of the existance of compliant binary labelings on the vertices of iterated standard chromatic subdivisions of an n-simplex. The existance of such labelings is equivalent to the existance of distributed protocols solving Weak Symmetry Breaking task in the standard computational model. As a part of our formal model, we introduce function sb(n), defined for natural numbers n, called the symmetry breaking function. From the geometric point of view sb(n) denotes the minimal number of iterations of the standard chromatic subdivision of an (n-1)-simplex, which is needed for the compliant binary labeling to exist. From the point of distributed computing, the function sb(n) measures the minimal number of rounds in a protocol solving the Weak Symmetry Breaking task. In addition to the development of combinatorial topology, which is applicable in a broader context, our main contribution is the proof of new bounds for the function sb(n). Accordingly, the bulk of the paper is taken up by in-depth analysis of the structure of adjacency graph on the set of n-simplices in iterated standard chromatic subdivision of an n-simplex. On the algorithmic side, we provide the first distributed protocol solving Weak Symmetry Breaking task in the layered immediate snapshot computational model for some number of processes. It is well known, that the smallest number of processes for which Weak Symmetry Breaking task is solvable is 6. Based on our analysis, we are able to find a very fast explicit protocol, solving the Weak Symmetry Breaking for 6 processes using only 3 rounds. Furthermore, we show that no protocol can solve Weak Symmetry Breaking in fewer than 2 rounds.