Papers › Algorithmic counting of nonequivalent compact Huffman codes
Algorithmic counting of nonequivalent compact Huffman codes
Christian Elsholtz, Clemens Heuberger, Daniel Krenn
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It is known that the following five counting problems lead to the same integer sequence~fₜ(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= 1/(t^(x₁))+ …+ 1/(t^(xₙ)) with integers 0 ≤x₁ ≤x₂ ≤…≤xₙ. 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+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(N³) operations in the integer ring or O(N⁴) bit operations, respectively.
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