Optimality of Huffman Code in the Class of 1-bit Delay Decodable Codes

TL;DR

This paper proves that Huffman codes are optimal among 1-bit delay decodable codes with finite code tables for i.i.d. sources, extending the understanding of optimality beyond instantaneous codes.

Contribution

It establishes that no 1-bit delay decodable code with multiple code tables can outperform Huffman codes in average length for i.i.d. sources.

Findings

Huffman code is optimal among 1-bit delay decodable codes.

No shorter average length code exists within the 1-bit delay decodable class.

The result extends Huffman's optimality to a broader class of codes.

Abstract

For a given independent and identically distributed (i.i.d.) source, Huffman code achieves the optimal average codeword length in the class of instantaneous code with a single code table. However, it is known that there exist time-variant encoders, which achieve a shorter average codeword length than the Huffman code, using multiple code tables and allowing at most k-bit decoding delay for k = 2, 3, 4, . . .. On the other hand, it is not known whether there exists a 1-bit delay decodable code, which achieves a shorter average length than the Huffman code. This paper proves that for a given i.i.d. source, a Huffman code achieves the optimal average codeword length in the class of 1-bit delay decodable codes with a finite number of code tables.