Improved Encoding and Counting of Uniform Hypertrees

Arjun Pitchanathan; Saswata Shannigrahi

Improved Encoding and Counting of Uniform Hypertrees

Combinatorics 2018-12-18 v4

Authors: Arjun Pitchanathan , Saswata Shannigrahi

Abstract

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 $\log_2(f(n, r))$ .

Keywords

trees phylogenetic trees

Cite

@article{arxiv.1711.03335,
  title  = {Improved Encoding and Counting of Uniform Hypertrees},
  author = {Arjun Pitchanathan and Saswata Shannigrahi},
  journal= {arXiv preprint arXiv:1711.03335},
  year   = {2018}
}

Comments

Withdrawn due to discovery of a prior work by Lavault [1] which contains the encoding and decoding algorithms described in section 2 of our work.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 Pr\"ufer-like coding and counting forests of uniform hypertrees. CSIT 2011, pp.82-85

Improved Encoding and Counting of Uniform Hypertrees

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