Analysis and Practice of Uniquely Decodable One-to-One Code

TL;DR

This paper studies uniquely decodable one-to-one codes (UDOOCs), analyzing their combinatorial properties, bounds on average codeword length, and providing efficient algorithms, demonstrating their potential for effective data compression.

Contribution

It introduces a comprehensive analysis of UDOOCs, including enumeration formulas, bounds on average length, and practical encoding/decoding algorithms, highlighting their compression efficiency.

Findings

UDOOCs can achieve compression rates comparable to Huffman codes.

They can yield smaller average codeword lengths than Lempel-Ziv codes.

Theoretical bounds are established for sources with infinite alphabets.

Abstract

In this paper, we consider the so-called uniquely decodable one-to-one code (UDOOC) that is formed by inserting a "comma" indicator, termed the unique word (UW), between consecutive one-to-one codewords for separation. Along this research direction, we first investigate several general combinatorial properties of UDOOCs, in particular the enumeration of the number of UDOOC codewords for any (finite) codeword length. Based on the obtained formula on the number of length-n codewords for a given UW, the per-letter average codeword length of UDOOC for the optimal compression of a given source statistics can be computed. Several upper bounds on the average codeword length of such UDOOCs are next established. The analysis on the bounds of average codeword length then leads to two asymptotic bounds for sources having infinitely many alphabets, one of which is achievable and hence tight for a certain source statistics and UW, and the other of which proves the achievability of source entropy rate of UDOOCs when both the block size of source letters for UDOOC compression and UW length go to infinity. Efficient encoding and decoding algorithms for UDOOCs are also given in this paper. Numerical results show that the proposed UDOOCs can potentially result in comparable compression rate to the Huffman code under similar decoding complexity and yield a smaller average codeword length than that of the Lempel-Ziv code, thereby confirming the practicability of UDOOCs.