Alphabetic Coding with Exponential Costs

TL;DR

This paper introduces a new alphabetic coding problem with exponential costs, providing a polynomial-time dynamic programming solution and two linear or near-linear approximation algorithms with redundancy bounds.

Contribution

It formulates a novel alphabetic coding problem with exponential costs, develops a polynomial-time optimal algorithm, and proposes two efficient approximation algorithms with bounds.

Findings

Optimal solution found in polynomial time.

Traditional optimization methods fail for this problem.

Two approximation algorithms achieve near-optimal solutions efficiently.

Abstract

An alphabetic binary tree formulation applies to problems in which an outcome needs to be determined via alphabetically ordered search prior to the termination of some window of opportunity. Rather than finding a decision tree minimizing $\sum_{i=1}^n w(i) l(i)$, this variant involves minimizing $\log_a \sum_{i=1}^n w(i) a^{l(i)}$ for a given $a \in (0,1)$. This note introduces a dynamic programming algorithm that finds the optimal solution in polynomial time and space, and shows that 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 approximation algorithms which can find a suboptimal solution in linear time (for one) or $\order(n \log n)$ time (for the other), with associated coding redundancy bounds.