# Self-Stabilizing Disconnected Components Detection and Rooted Shortest-Path Tree Maintenance in Polynomial Steps

## TL;DR

This paper presents a self-stabilizing algorithm for maintaining a rooted shortest-path tree in a network that may become disconnected, ensuring correctness and efficiency under various assumptions and network conditions.

## Contribution

It introduces a silent, self-stabilizing algorithm for shortest-path tree maintenance in semi-anonymous networks without global knowledge, with proven correctness and polynomial stabilization time.

## Key findings

- Stabilization time is at most 3nmax+D rounds.
- Algorithm stabilizes in polynomial steps when edge weights are positive integers.
- Works under the most general distributed unfair daemon.

## Abstract

We deal with the problem of maintaining a shortest-path tree rooted at some process r in a network that may be disconnected after topological changes. The goal is then to maintain a shortest-path tree rooted at r in its connected component, V\_r, and make all processes of other components detecting that r is not part of their connected component. We propose, in the composite atomicity model, a silent self-stabilizing algorithm for this problem working in semi-anonymous networks, where edges have strictly positive weights. This algorithm does not require any a priori knowledge about global parameters of the network. We prove its correctness assuming the distributed unfair daemon, the most general daemon. Its stabilization time in rounds is at most 3nmax+D, where nmax is the maximum number of non-root processes in a connected component and D is the hop-diameter of V\_r. Furthermore, if we additionally assume that edge weights are positive integers, then it stabilizes in a polynomial number of steps: namely, we exhibit a bound in O(maxi nmax^3 n), where maxi is the maximum weight of an edge and n is the number of processes.