Uniform Partition in Population Protocol Model under Weak Fairness

TL;DR

This paper investigates the space complexity of uniform bipartition in population protocols under weak fairness, establishing necessary and sufficient states for symmetric and asymmetric protocols, and extends results to non-initialized base stations and arbitrary partitions.

Contribution

It determines the exact number of states needed for uniform bipartition protocols under weak fairness with an initialized base station, and explores cases with non-initialized base stations and general k-partitions.

Findings

P states are necessary and sufficient for asymmetric protocols.

P+1 states are necessary and sufficient for symmetric protocols.

Results extend to arbitrary k-partitions.

Abstract

We focus on a uniform partition problem in a population protocol model. The uniform partition problem aims to divide a population into k groups of the same size, where k is a given positive integer. In the case of k=2 (called uniform bipartition), a previous work clarified space complexity under various assumptions: 1) an initialized base station (BS) or no BS, 2) weak or global fairness, 3) designated or arbitrary initial states of agents, and 4) symmetric or asymmetric protocols, except for the setting that agents execute a protocol from arbitrary initial states under weak fairness in the model with an initialized base station. In this paper, we clarify the space complexity for this remaining setting. In this setting, we prove that P states are necessary and sufficient to realize asymmetric protocols, and that P+1 states are necessary and sufficient to realize symmetric protocols, where P is the known upper bound of the number of agents. From these results and the previous work, we have clarified the solvability of the uniform bipartition for each combination of assumptions. Additionally, we newly consider an assumption on a model of a non-initialized BS and clarify solvability and space complexity in the assumption. Moreover, the results in this paper can be applied to the case that k is an arbitrary integer (called uniform k-partition).