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.
翻译:在重新配置问题上,考虑到一个问题和问题的两个可行的解决办法,我们的任务是找到从一个解决办法到另一个解决办法的转变顺序,这样每个中间国家也是解决问题的可行办法。在本文中,我们研究分布式横贯树木重新配置问题,我们定义了新的重新配置步骤,称为美元同时添加和删除,其中允许每个节点最多增加1美元边缘,并删除最多不超过1美元边缘,以便多个节点不会增加或删除同一边缘。我们首先看到,如果两个贯穿树木的输入是根植的,那么我们就可以使用单一的1美元同时添加,并删除CONEST模型中一轮中的一步。因此,我们把注意力集中在不扎根的树木上,并表明,用单一的1美元双倍加加加加和删除步骤,将不扎根的树变成另一步,在LOCAL模型中,需要1美元加(n)美元加或删除。我们还要指出,将一个不扎根的模型转换成另一步,在CONEST模型中用1美元加一回合。