The supertree of the compact codes

Narimani, Hamed; Khosravifard, Mohammadali

doi:10.1109/istel.2008.4651381

Cited by 4 publications

(4 citation statements)

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“…The literature describes a number of algorithms constructing the complete list of t-ary Huffman codes of length r = 1 + n(t − 1); see [19,23,26,27]. There is no performance analysis given.…”

Section: Taskmentioning

confidence: 99%

Algorithmic counting of nonequivalent compact Huffman codes

Elsholtz,

Heuberger,

Krenn

2019

Preprint

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It is known that the following five counting problems lead to the same integer sequence ft(n):(1) the number of nonequivalent compact Huffman codes of length n over an alphabet of t letters, (2) the number of "nonequivalent" canonical rooted t-ary trees (level-greedy trees) with n leaves, (3) the number of "proper" words, (4) the number of bounded degree sequences, and (5) the number of ways of writingIn this work, we show that one can compute this sequence for all n < N with essentially one power series division. In total we need at most N 1+ε additions and multiplications of integers of cN bits, c < 1, or N 2+ε 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.

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Section: Taskmentioning

confidence: 99%

Algorithmic counting of nonequivalent compact Huffman codes

Elsholtz,

Heuberger,

Krenn

2019

Preprint

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“…Using the above observation of branching at two positions only, Narimani and Khosravifard [23] describe a recursive algorithm to create all codes counted by f t (r).…”

Section: 3mentioning

confidence: 99%

“…r . However, these approximations have been referred to in the more recent coding literature, see for example [27], [29], [1], [23], [18] and [19].…”

Section: 3mentioning

confidence: 99%

See 1 more Smart Citation

The Number of Huffman Codes, Compact Trees, and Sums of Unit Fractions

Elsholtz

Heuberger

Prodinger

2013

IEEE Trans. Inform. Theory

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The number of "nonequivalent" Huffman codes of length r over an alphabet of size t has been studied frequently. Equivalently, the number of "nonequivalent" complete t-ary trees has been examined. We first survey the literature, unifying several independent approaches to the problem. Then, improving on earlier work we prove a very precise asymptotic result on the counting function, consisting of two main terms and an error term.

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Algorithmic counting of nonequivalent compact Huffman codes

Elsholtz

Heuberger

Krenn

2023

AAECC

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It is known that the following five counting problems lead to the same integer sequence $${f_t}({n})$$ f t ( n ) : the number of nonequivalent compact Huffman codes of length n over an alphabet of t letters, the number of “nonequivalent” complete rooted t-ary trees (level-greedy trees) with n leaves, the number of “proper” words (in the sense of Even and Lempel), the number of bounded degree sequences (in the sense of Komlós, Moser, and Nemetz), and the number of ways of writing $$\begin{aligned} 1= \frac{1}{t^{x_1}}+ \dots + \frac{1}{t^{x_n}} \end{aligned}$$ 1 = 1 t x 1 + ⋯ + 1 t x n with integers $$0 \le x_1 \le x_2 \le \dots \le x_n$$ 0 ≤ x 1 ≤ x 2 ≤ ⋯ ≤ x n . In this work, we show that one can compute this sequence for all$$n<N$$ n < N with essentially one power series division. In total we need at most $$N^{1+\varepsilon }$$ N 1 + ε additions and multiplications of integers of cN bits (for a positive constant $$c<1$$ c < 1 depending on t only) or $$N^{2+\varepsilon }$$ N 2 + ε bit operations, respectively, for any $$\varepsilon >0$$ ε > 0 . This improves an earlier bound by Even and Lempel who needed $${O}({{N^3}})$$ O ( N 3 ) operations in the integer ring or $$O({N^4})$$ O ( N 4 ) bit operations, respectively.

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The supertree of the compact codes

Cited by 4 publications

References 4 publications

Algorithmic counting of nonequivalent compact Huffman codes

Algorithmic counting of nonequivalent compact Huffman codes

The Number of Huffman Codes, Compact Trees, and Sums of Unit Fractions

Algorithmic counting of nonequivalent compact Huffman codes

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