Alphabetic coding with exponential costs

Abstract

This note considers an alphabetic binary tree formulation in a family of nonlinear problems. An application of this family occurs when a random outcome needs to be determined via alphabetically ordered search within a stochastic time window. Rather than finding a decision tree minimizing n i=1 w(i)l(i), this variant involves minimizing log a n i=1 w(i)a l(i) for a given a ∈ (0, 1). Herein a dynamic programming algorithm finds the optimal solution in O (n 3 ) time and O (n 2 ) space; methods traditionally used to improve the speed of optimizations in related problems, such as the Hu-Tucker procedure, fail for this problem. This note thus also introduces two algorithms which can find a suboptimal solution in linear time (for one) or O (n log n) time (for the other), with associated redundancy bounds guaranteeing their coding efficiency.

On the Huffman and Alphabetic Tree Problem with General Cost Functions

“…The authors identified a class of cost functions where the Hu-Tucker Algorithm is applicable, including power summations of the type cost(T ) = ∑ i w i t d i for any constant t ≥ 1. Baer [Bae10] recently showed that for t < 1 neither the Hu-Tucker algorithm nor the approach of Knuth leads to optimal solutions, and he proposes to use the O(n 3 ) algorithm by Gilbert and Moore instead.…”