Varn Codes and Generalized Fibonacci Trees

نویسنده

  • Julia Abrahams
چکیده

Yarn's [6] algorithm solves the problem of finding an optimal code tree, optimal in the sense of minimum average cost, when the code symbols are of unequal cost and the source symbols are equiprobable. He addresses both exhaustive codes, for which the code tree is a full tree, as well as nonexhaustive codes, but only the exhaustive case will be of concern here. In particular, for code symbol costs c(l) < c(2) < • • • < c(r) and a uniform source of size TV, where (N-l)/(r-1) is an integer, the Yarn code tree is generated as follows. Start with an r-ary tree consisting of a root node from which descend r leaf nodes labeled from left to right by c(l), c(2),..., c(r), the costs associated with the corresponding code symbols. Select the lowest cost node, let c be its cost, and let descend from it r leaf nodes labeled from left to right by c + c(l),c + c(2),...,c + c(r). Continue, by selecting the lowest cost node from the new tree, until N leaf nodes have been created. Horibe [3] studied a sequence of binary trees and showed that each tree in the sequence is a Yarn code tree for c(l) = 1, c(2) = 2. In particular, the k tree has the k 1 tree as its left subtree and the k 2 n d tree as its right subtree; for k 1 and k = 2, the tree is only the root; c(l) is associated with the left descendant of a node and c(2) with the right descendant. These trees are called Fibonacci trees, and the number of leaves in the k tree is the k Fibonacci number. Note that some integers N are not equal to the k Fibonacci number for any k so that not every Yarn code tree for c(l) = 1, c(2) = 2 is a Fibonacci tree. Chang [1] studied a sequence of r-ary trees that reduces to Horibe's sequence of Fibonacci trees for r = 2. In particular, the k tree has the k-i tree as its / leftmost subtree, / = 1,..., r; for k 1,..., r, the tree is only the root; and c(i) = i,i = l,...,r is associated with the descendants of a node in left to right order. For these particular costs, c(i) = /, i = 1,..., r, Chang's trees are Yarn code trees, and the number of leaves in the k^ tree is determined according to an integer sequence that general-izes the Fibonacci sequence. It is the purpose of this note to examine sequences of trees that are recursively constructed and are Varn code trees for integer costs c(l),..., c(r) whose greatest common divisor is 1. Since common factors shared by all costs do not affect Yarn's algorithm, the costs considered here are essentially all rational costs or all sets of rational costs with a common irrational multiplier. Thus, previous work on recursive characterizations of Varn code trees for particular integer code symbol costs is extended to the case of arbitrary integer code symbol costs.

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تاریخ انتشار 1995