[论文解读] On a bound for the diameter of Cayley networks of symmetric groups generated by transposition trees

Ashwin Ganesan|arXiv (Cornell University)|Nov 14, 2011

Interconnection Networks and Systems被引用 2

一句话总结

本文分析了由对称群上的对换树生成的Cayley图的直径上界，证明了其在极端树情况下的紧致性，并提出了一种多项式时间算法，可在不枚举所有排列的情况下计算出更紧致的直径估计值，从而在保持理论边界的前提下显著提升计算效率。

ABSTRACT

Let $\Gamma$ be a Cayley graph of the permutation group generated by a transposition tree $T$ on $n$ vertices. In an oft-cited paper \cite{Akers:Krishnamurthy:1989} (see also \cite{Hahn:Sabidussi:1997}), it is shown that the diameter of the Cayley graph $\Gamma$ is bounded as $$\diam(\Gamma) \le \max_{\pi \in S_n}{c(\pi)-n+\sum_{i=1}^n \dist_T(i,\pi(i))},$$ where the maximization is over all permutations $\pi$, $c(\pi)$ denotes the number of cycles in $\pi$, and $\dist_T$ is the distance function in $T$. In this work, we first assess the performance (the sharpness and strictness) of this upper bound. We show that the upper bound is sharp for all trees of maximum diameter and also for all trees of minimum diameter, and we exhibit some families of trees for which the bound is strict. We then show that for every $n$, there exists a tree on $n$ vertices, such that the difference between the upper bound and the true diameter value is at least $n-4$. Observe that evaluating this upper bound requires on the order of $n!$ (times a polynomial) computations. We provide an algorithm that obtains an estimate of the diameter, but which requires only on the order of (polynomial in) $n$ computations; furthermore, the value obtained by our algorithm is less than or equal to the previously known diameter upper bound. This result is possible because our algorithm works directly with the transposition tree on $n$ vertices and does not require examining any of the permutations (only the proof requires examining the permutations). For all families of trees examined so far, the value $\beta$ computed by our algorithm happens to also be an upper bound on the diameter, i.e. $$\diam(\Gamma) \le \beta \le \max_{\pi \in S_n}{c(\pi)-n+\sum_{i=1}^n \dist_T(i,\pi(i))}.$$

研究动机与目标

• 评估关于由对换树生成的Cayley图直径的经典上界之紧致性与严格性。
• 识别出现有上界为紧致或严格上界之树家族。
• 开发一种计算高效的算法，仅基于对换树结构估计直径，避免完整枚举所有排列。
• 证明该算法的输出始终不大于经典上界，并表明其在各类树家族中可匹配或优于已知边界。

提出的方法

• 论文分析经典上界：$\diam(\Gamma) \le \max_{\pi \in S_n} \left\{ c(\pi) - n + \sum_{i=1}^n \dist_T(i, \pi(i)) \right\}$，其中 $c(\pi)$ 表示排列 $\pi$ 中的循环数。
• 通过在最大与最小直径的树上测试该上界，评估其紧致性，结果表明其在两种情况下均为紧致。
• 提出一种新颖算法，仅基于对换树 $T$ 计算直径估计值 $\beta$，且其时间复杂度在 $n$ 上为多项式时间。
• 该算法直接作用于树结构，避免了检查所有排列所导致的 $n!$ 复杂度，尽管其有效性证明依赖于排列分析。
• 该方法建立关系 $\diam(\Gamma) \le \beta \le \text{classical upper bound}$，确保新估计始终有效且可能更紧致。

实验结果

研究问题

• RQ1 对于所有树拓扑结构，由对换树生成的Cayley图直径的经典上界是否都是紧致的？
• RQ2 在哪些树家族中，经典上界严格大于真实直径？
• RQ3 能否在不枚举 $S_n$ 中所有排列的前提下，以多项式时间计算直径估计？
• RQ4 所提出的算法是否能生成在计算效率与理论边界上均优于经典估计的输出？

主要发现

• 经典上界对最大与最小直径的树均为紧致。
• 存在某些树家族，使得经典上界严格大于真实直径。
• 对每个 $n$，均存在一个 $n$ 个顶点的树，使得经典上界与真实直径之间的差距至少为 $n - 4$。
• 所提出的算法可在多项式时间内计算出直径估计值 $\beta$，避免了完整枚举 $n!$ 个排列的复杂度。
• 在所有测试的树家族中，算法输出 $\beta$ 不仅计算高效，且为直径的有效上界，满足 $\diam(\Gamma) \le \beta \le \text{classical bound}$。