Improved Encoding and Counting of Uniform Hypertrees
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 $\log_2(f(n, r))$.
- Publication:
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arXiv e-prints
- Pub Date:
- November 2017
- DOI:
- arXiv:
- arXiv:1711.03335
- Bibcode:
- 2017arXiv171103335P
- Keywords:
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- Mathematics - Combinatorics
- E-Print:
- Withdrawn due to discovery of a prior work by Lavault [1] which contains the encoding and decoding algorithms described in section 2 of our work.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