# Analysis of a Memory-Efficient Self-Stabilizing BFS Spanning Tree

## TL;DR

This paper analyzes a memory-efficient self-stabilizing BFS spanning tree algorithm, proving its stabilization under unfair conditions and establishing an upper bound on its stabilization time.

## Contribution

It demonstrates the self-stabilization of a modified version of Johnen's algorithm under unfair conditions and provides a bound on its stabilization time.

## Key findings

- Proves self-stabilization under unfair daemon.
- Establishes a stabilization time of O(D·n^2) rounds.
- Maintains memory efficiency of Θ(1) bits per edge.

## Abstract

We present results on the last topic we collaborate with our late friend, Professor Ajoy Kumar Datta (1958-2019). In this work, we shed new light on a self-stabilizing wave algorithm proposed by Colette Johnen in 1997. This algorithm constructs a BFS spanning tree in any connected rooted network. Nowadays, it is still the best existing self-stabilizing BFS spanning tree construction in terms of memory requirement, {\em i.e.}, it only requires $\Theta(1)$ bits per edge. However, it has been proven assuming a weakly fair daemon. Moreover, its stabilization time was unknown. Here, we study the slightly modified version of this algorithm, still keeping the same memory requirement. We prove the self-stabilization of this variant under the distributed unfair daemon and show a stabilization time in $O(D.n^2)$ rounds, where $D$ is the network diameter and $n$ the number of processes.