Distributed Reconfiguration of Spanning Trees

TL;DR

This paper studies distributed algorithms for reconfiguring spanning trees using new multi-edge update steps, showing efficient transformations for unrooted trees in the CONGEST model and limitations in the LOCAL model.

Contribution

It introduces the $k$-simultaneous add and delete reconfiguration step and analyzes its complexity for rooted and unrooted spanning trees.

Findings

Single-step reconfiguration of rooted trees is possible in one round.

Unrooted trees require linear rounds in the LOCAL model for single-step reconfiguration.

Two-step reconfiguration of unrooted trees can be done in logarithmic rounds in the CONGEST model.

Abstract

In a reconfiguration problem, given a problem and two feasible solutions of the problem, the task is to find a sequence of transformations to reach from one solution to the other such that every intermediate state is also a feasible solution to the problem. In this paper, we study the distributed spanning tree reconfiguration problem and we define a new reconfiguration step, called $k$-simultaneous add and delete, in which every node is allowed to add at most $k$ edges and delete at most $k$ edges such that multiple nodes do not add or delete the same edge. We first observe that, if the two input spanning trees are rooted, then we can do the reconfiguration using a single $1$-simultaneous add and delete step in one round in the CONGEST model. Therefore, we focus our attention towards unrooted spanning trees and show that transforming an unrooted spanning tree into another using a single $1$-simultaneous add and delete step requires $\Omega(n)$ rounds in the LOCAL model. We additionally show that transforming an unrooted spanning tree into another using a single $2$-simultaneous add and delete step can be done in $O(\log n)$ rounds in the CONGEST model.