arXiv 1711.033352017-11math.CO

均匀超树的改进编码与计数

Improved Encoding and Counting of Uniform Hypertrees

Arjun Pitchanathan, Saswata Shannigrahi

备注：: Withdrawn due to discovery of a prior work by Lavault [1] which contains the encoding and decoding algorithms described in section 2 of our work. In section 3, we fill in some details required to make the encoding scheme near-optimal, which makes the running time $\widetilde{O}(n^2)$. [1] C. Lavault. A note on Prüfer-like coding and counting forests of uniform hypertrees. CSIT 2011, pp.82-85

摘要：

我们考虑具有$n \ge r \ge 2$顶点的标记$r$ -一致超树。在这样的超树中，超边的数目是$m = (n - 1)/(r - 1)$。证明了存在$f(n, r) = \frac{(n-1)! n^{m-1}}{(r-1)!^m m!}$ $r$ -一致超树，其中$n$的顶点用不同的整数标记.我们还给出了一个编码方案，编码这样的超树使用，平均而言，在最多的$1 + \log_2 e$位超过$\log_2(f(n, r))$。

英文摘要：

We consider labeled $r$-uniform hypertrees having $n \ge r \ge 2$ vertices. The number of hyperedges in such a hypertree is $m = (n - 1)/(r - 1)$. We show that there are exactly $f(n, r) = \frac{(n-1)! n^{m-1}}{(r-1)!^m m!}$ $r$-uniform hypertrees with $n$ vertices labeled with distinct integers. We also give an encoding scheme that encodes such hypertrees using, on an average, at most $1 + \log_2 e$ bits more than $\log_2(f(n, r))$.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

摘要：

英文摘要：

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