# A Self-Stabilizing Minimal k-Grouping Algorithm

## TL;DR

This paper presents a self-stabilizing distributed algorithm for minimal k-grouping in graphs, ensuring efficient partitioning with bounded diameter and a novel composition technique for algorithm concatenation.

## Contribution

It introduces a silent, self-stabilizing asynchronous algorithm for minimal k-grouping with proven time and space complexity, and a new loop composition method for algorithms.

## Key findings

- Algorithm converges in O(nD/k) rounds.
- Number of groups is at most 2n/k+1 after convergence.
- Space complexity per process is O((n + n_false) log n).

## Abstract

We consider the minimal k-grouping problem: given a graph G=(V,E) and a constant k, partition G into subgraphs of diameter no greater than k, such that the union of any two subgraphs has diameter greater than k. We give a silent self-stabilizing asynchronous distributed algorithm for this problem in the composite atomicity model of computation, assuming the network has unique process identifiers. Our algorithm works under the weakly-fair daemon. The time complexity (i.e., the number of rounds to reach a legitimate configuration) of our algorithm is O(nD/k) where n is the number of processes in the network and \diam is the diameter of the network. The space complexity of each process is O((n +n_{false})log n) where n_{false} is the number of false identifiers, i.e., identifiers that do not match the identifier of any process, but which are stored in the local memory of at least one process at the initial configuration. Our algorithm guarantees that the number of groups is at most $2n/k+1$ after convergence. We also give a novel composition technique to concatenate a silent algorithm repeatedly, which we call loop composition.