书本、走廊与社会蝴蝶：关于滑块拼图的一点注记
组合数学
2024-12-19 v3
摘要
回顾经典的 15-拼图，由 网格中的 15 个滑动方块组成。众所周知，该拼图的构型空间由两个连通分支组成，对应于对称群 的奇置换与偶置换。1974 年，Wilson 将滑块拼图从 网格推广到任意图（考虑具有 个顶点的图上的 个滑动方块），并刻画了对应构型空间连通的图。在本工作中，我们将 Wilson 的刻画推广到具有任意数量方块（可能留下多于一个空顶点）的滑块拼图。对于任意图，我们确定了连通对应构型空间所需的空顶点数量，更一般地，我们提供了一种判定任意两个构型是否连通的算法。我们的工作也可在“朋友与陌生人图”框架下解释，其中空顶点对应“社会蝴蝶”，滑动方块对应“不合群”的人。
引用
@article{arxiv.2303.09459,
title = {Books, Hallways and Social Butterflies: A Note on Sliding Block Puzzles},
author = {Florestan Brunck and Matthew Kwan},
journal= {arXiv preprint arXiv:2303.09459},
year = {2024}
}
备注
Since posting our preprint, we were made aware that our main theorem has previously appeared in the literature. Specifically, this result (in different language) is claimed without proof in a paper of Kornhauser, Miller and Spirakis, appearing in the conference proceedings of FOCS'84 (there are connections to memory management in distributed systems). The proof can be found in Kornhauser's thesis