Tight Bounds on the Average Length, Entropy, and Redundancy of Anti-Uniform Huffman Codes

TL;DR

This paper establishes tight bounds on the average length, entropy, and redundancy of anti-uniform Huffman codes, highlighting the role of Fibonacci distributions in maximizing these metrics for a given alphabet size.

Contribution

It provides the first tight bounds on key properties of anti-uniform Huffman codes and introduces Fibonacci distributions as extremal cases.

Findings

Fibonacci distributions maximize average length and entropy for AUH codes.

Derived tight bounds relate alphabet size to code length, entropy, and redundancy.

Connected bounds on entropy and average length, extending previous results.

Abstract

In this paper we consider the class of anti-uniform Huffman codes and derive tight lower and upper bounds on the average length, entropy, and redundancy of such codes in terms of the alphabet size of the source. The Fibonacci distributions are introduced which play a fundamental role in AUH codes. It is shown that such distributions maximize the average length and the entropy of the code for a given alphabet size. Another previously known bound on the entropy for given average length follows immediately from our results.