Tight Bounds on the Redundancy of Huffman Codes

TL;DR

This paper establishes tight upper and lower bounds on the redundancy of Huffman codes for sources with known symbol probabilities, confirming a conjecture and extending to related problems.

Contribution

It proves a conjecture on the redundancy bounds of Huffman codes and derives tight bounds for sources with known symbol probabilities, also applying the method to related problems.

Findings

Proved a conjecture on Huffman code redundancy bounds.

Derived tight upper and lower bounds for sources with known symbol probabilities.

Extended the method to derive other known bounds in related problems.

Abstract

In this paper we study the redundancy of Huffman codes. In particular, we consider sources for which the probability of one of the source symbols is known. We prove a conjecture of Ye and Yeung regarding the upper bound on the redundancy of such Huffman codes, which yields in a tight upper bound. We also derive a tight lower bound for the redundancy under the same assumption. We further apply the method introduced in this paper to other related problems. It is shown that several other previously known bounds with different constraints follow immediately from our results.