Universal almost optimal compression and Slepian-Wolf coding in probabilistic polynomial time

TL;DR

This paper introduces a universal, nearly optimal lossless compression method that operates in probabilistic polynomial time, achieving near-minimal code lengths with minimal overhead, and extends to distributed compression scenarios like Slepian-Wolf coding.

Contribution

It presents the first polynomial-time universal compressor with polylogarithmic overhead that approximates any compression scheme, including non-computable ones, and applies to distributed compression.

Findings

Existence of a universal compressor with polylogarithmic overhead

Achieves near-optimal compression rates in distributed settings

Applicable even when the number of sources exceeds string length

Abstract

In a lossless compression system with target lengths, a compressor ${\cal C}$ maps an integer $m$ and a binary string $x$ to an $m$-bit code $p$, and if $m$ is sufficiently large, a decompressor ${\cal D}$ reconstructs $x$ from $p$. We call a pair $(m,x)$ $\textit{achievable}$ for $({\cal C},{\cal D})$ if this reconstruction is successful. We introduce the notion of an optimal compressor ${\cal C}_\text{opt}$, by the following universality property: For any compressor-decompressor pair $({\cal C}, {\cal D})$, there exists a decompressor ${\cal D}'$ such that if $(m,x)$ is achievable for $({\cal C},{\cal D})$, then $(m+\Delta, x)$ is achievable for $({\cal C}_\text{opt}, {\cal D}')$, where $\Delta$ is some small value called the overhead. We show that there exists an optimal compressor that has only polylogarithmic overhead and works in probabilistic polynomial time. Differently said, for any pair $({\cal C}, {\cal D})$, no matter how slow ${\cal C}$ is, or even if ${\cal C}$ is non-computable, ${\cal C}_{\text{opt}}$ is a fixed compressor that in polynomial time produces codes almost as short as those of ${\cal C}$. The cost is that the corresponding decompressor is slower. We also show that each such optimal compressor can be used for distributed compression, in which case it can achieve optimal compression rates, as given in the Slepian-Wolf theorem, and even for the Kolmogorov complexity variant of this theorem. Moreover, the overhead is logarithmic in the number of sources, and unlike previous implementations of Slepian-Wolf coding, meaningful compression can still be achieved if the number of sources is much larger than the length of the compressed strings.