Algorithmic counting of nonequivalent compact Huffman codes
Abstract
It is known that the following five counting problems lead to the same integer sequence~$f_t(n)$: the number of nonequivalent compact Huffman codes of length~$n$ over an alphabet of $t$ letters, the number of `nonequivalent' canonical rooted $t$-ary trees (level-greedy trees) with $n$~leaves, the number of `proper' words, the number of bounded degree sequences, and the number of ways of writing $1= \frac{1}{t^{x_1}}+ \dots + \frac{1}{t^{x_n}}$ with integers $0 \leq x_1 \leq x_2 \leq \dots \leq x_n$. In this work, we show that one can compute this sequence for \textbf{all} $n<N$ with essentially one power series division. In total we need at most $N^{1+\varepsilon}$ additions and multiplications of integers of $cN$ bits, $c<1$, or $N^{2+\varepsilon}$ bit operations, respectively. This improves an earlier bound by Even and Lempel who needed $O(N^3)$ operations in the integer ring or $O(N^4)$ bit operations, respectively.
- Publication:
-
arXiv e-prints
- Pub Date:
- January 2019
- DOI:
- 10.48550/arXiv.1901.11343
- arXiv:
- arXiv:1901.11343
- Bibcode:
- 2019arXiv190111343E
- Keywords:
-
- Mathematics - Combinatorics;
- Computer Science - Computational Complexity;
- Computer Science - Discrete Mathematics;
- Computer Science - Symbolic Computation;
- 05A15;
- 05C05;
- 05C30;
- 11D68;
- 68P30
- E-Print:
- doi:10.1007/s00200-022-00593-0