Fibonacci-Like Polynomials Produced by m-ary Huffman Codes for Absolutely Ordered Sequences

A non-decreasing sequence of positive integer weights P ={p1, p2,..., pn} (n = N*(m-1) + 1, N is number of non-leaves of m-ary tree) is called absolutely ordered if the intermediate sequences of weights produced by m-ary Huffman algorithm for initial sequence P on i-th step satisfy the following conditions 2 , , ) ( 1 ) ( − = < + N 0 i p p i m i m . Let T be an m-ary tree of size n and M=M(T) be a set of such sequences of positive integer weights that M P ∈ ∀ the tree T is the m-ary Huffman tree of P (|P|=n). A sequence Pmin of n positive integer weights is called a minimizing sequence of the m-ary tree T in the class M ( M P ∈ min ) if Pmin produces the minimal Huffman cost of the tree T over all sequences from M, i.e., M. P P) E(T, ) P E(T, ∈ ∀ ≤ min Theorem 1. A minimizing absolutely ordered sequence of size n = N*(m-1) + 1 for the maximum height m-ary Huffman tree (m > 1) is Pminabs(N, m) = { times ) 1 ( times ) 1 ( 2 2 times ) 1 ( 1 1 0 ) 1 ( ),..., 1 ( ,..., ) 1 ( ),..., 1 ( , ) 1 ( ),..., 1 ( ), 1 ( − − − − − − − − − − m N N m m m G m G m G m G m G m G m G }, where G0(m) = 1, G1(m) = 1, G2(m) = 2, Gi(m) = Gi-1(m) + m*Gi-2(m) when N i , 2 = ■ Polynomials Gi(x) are called Fibonacci-like polynomials. Theorem 2. The cost of maximum height m-ary Huffman tree T of size n = N*(m-1) + 1 for the minimizing absolutely ordered sequence Pminabs(N, m) is E(T, Pminabs(N, m)) = 1 2 ) 1 ( 4 − − − + m m GN – (N+3). ■ Samples of Fibonacci-like polynomials and costs of maximum height m-ary Huffman trees are shown. 0. Preface Absolutely ordered and k-ordered sequences for binary Huffman trees those have maximum height have been investigated in [1] and [2]. In this article the generalization of absolutely ordered sequences for m-ary Huffman trees those have maximum height is considered. Fibonacci-Like Polynomials Produced by m-ary Huffman Codes for Absolutely Ordered Sequences Alex Vinokur Page 2 of 2 1. Main Conceptions and Terminology 1.1. m-ary trees A (strictly) m-ary tree is an oriented ordered tree where each nonleaf node has exactly m children (siblings). Size of an m-ary tree is the total number of leaves of this tree. Let N be number of nonleaves (internal nodes), n be number of leaves of m-ary tree. Number of leaves in m-ary tree satisfies the following condition n = N*(m-1) + 1. (1) An m-tree (m ≥ 2) is called elongated if at least (m-1) of any m sibling nodes are leaves. An elongated binary tree of size n has maximum height among all binary trees of size n. An elongated m-ary tree is called left-sided if only the left node in each m-tuple of sibling nodes can be nonleaf. A m-ary tree is called labeled if a certain positive integer (weight) is set in correspondence with each leaf. Definition. Let T be an m-ary tree with positive weights P={p1,.., pn} at its leaf nodes. The weighted external path length of T is