Subset sum phase transitions and data compression

TL;DR

This paper analyzes the phase transition in the subset sum problem within lossless data compression, linking it to the transition from ambiguous to unambiguous decompression, and extends the analysis to various complex scenarios.

Contribution

It introduces a rigorous analysis method for subset sum in data compression, applicable to non-binary alphabets, side information, and multiple subset sums, with insights into critical behavior.

Findings

Identifies phase transition points related to decompression ambiguity.

Extends analysis to non-binary alphabets and side information.

Demonstrates the technique's applicability to complex sequence compositions.

Abstract

We propose a rigorous analysis approach for the subset sum problem in the context of lossless data compression, where the phase transition of the subset sum problem is directly related to the passage between ambiguous and non-ambiguous decompression, for a compression scheme that is based on specifying the sequence composition. The proposed analysis lends itself to straightforward extensions in several directions of interest, including non-binary alphabets, incorporation of side information at the decoder (Slepian-Wolf coding), and coding schemes based on multiple subset sums. It is also demonstrated that the proposed technique can be used to analyze the critical behavior in a more involved situation where the sequence composition is not specified by the encoder.