Algorithmic counting of nonequivalent compact Huffman codes
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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= 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, 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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