Lower Bounds on the Redundancy of Huffman Codes with Known and Unknown Probabilities

TL;DR

This paper introduces a method to compute tight lower bounds on Huffman code redundancy when the underlying symbol probabilities are partially unknown, applicable to various alphabet sizes, and provides insights into the structure of minimal redundancy.

Contribution

It presents a novel computational approach to determine lower bounds on Huffman code redundancy with incomplete probability information, enhancing understanding of optimal coding under uncertainty.

Findings

Method yields closed-form bounds for various alphabet sizes.

Applicable to both known and unknown probability scenarios.

Provides structural insights into minimal redundancy configurations.

Abstract

In this paper we provide a method to obtain tight lower bounds on the minimum redundancy achievable by a Huffman code when the probability distribution underlying an alphabet is only partially known. In particular, we address the case where the occurrence probabilities are unknown for some of the symbols in an alphabet. Bounds can be obtained for alphabets of a given size, for alphabets of up to a given size, and for alphabets of arbitrary size. The method operates on a Computer Algebra System, yielding closed-form numbers for all results. Finally, we show the potential of the proposed method to shed some light on the structure of the minimum redundancy achievable by the Huffman code.