Algorithmic counting of nonequivalent compact Huffman codes

• Published in: Applicable algebra in engineering, communication and computing Vol. 35; no. 6; pp. 887 - 903
• Main Authors: Elsholtz, Christian, Heuberger, Clemens, Krenn, Daniel
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer Berlin Heidelberg 01-11-2024; Springer Nature B.V
• Subjects: Artificial Intelligence; Codes; Computer Hardware; Computer Science; Huffman codes; Integers; Original Paper; Power series; Sequences; Symbolic and Algebraic Manipulation; Theory of Computation; Trees (mathematics); Huffman codes; ary trees; Generating function; 05C30; 68P30; Unit fractions; 05A15; Counting; 05C05; 11D68
• ISSN: 0938-1279; 1432-0622
• DOI: 10.1007/s00200-022-00593-0

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” 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
1
=
1
t
x
1
+
⋯
+
1
t
x
n
with integers
0
≤
x
1
≤
x
2
≤
⋯
≤
x
n
.
In 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 (for a positive constant
c
<
1
depending on
t
only) or
N
2
+
ε
bit operations, respectively, for any 
ε
>
0
. 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.