Optimal Prefix Free Code in Linear Time

TL;DR

This paper presents a linear-time algorithm for computing optimal prefix free codes from unsorted weights, significantly improving over previous methods and leveraging specific hardware instructions and theoretical insights.

Contribution

It introduces a novel linear-time algorithm for Huffman coding that outperforms existing complexity bounds in standard computational models.

Findings

Achieves linear time complexity in the number of machine words.

Improves over previous $O(N ext{log}N)$ and $O(N ext{log} ext{log}N)$ algorithms.

Utilizes hardware-specific instructions and theoretical results for optimal prefix codes.

Abstract

We describe an algorithm computing an optimal prefix free code from $N$ unsorted positive integer weights in time linear in the number of machine words holding those weights. This algorithm takes advantage of common non-algebraic instructions, and of specific results on optimal prefix free codes. This result improves over the state of the art complexities of $O(N\lg N)$ in the algebraic decision tree model and $O(N\lg\lg N)$ in the RAM model for the computation of Huffman's codes, a landmark in compression and coding since 1952.