On the space complexity of one-pass compression

TL;DR

This paper investigates the memory requirements of one-pass compression algorithms, establishing bounds on their space complexity relative to multi-pass algorithms and empirical entropy.

Contribution

It introduces the concept of (f(n, ll))-footprint compressors and proves bounds on their existence based on memory usage and entropy.

Findings

Existence of (f(n, ll))-footprint compressors for certain space bounds.

Non-existence of such compressors below specific space thresholds.

Quantitative bounds relating memory and compression performance.

Abstract

We study how much memory one-pass compression algorithms need to compete with the best multi-pass algorithms. We call a one-pass algorithm an (f (n, \ell))-footprint compressor if, given $n$, $\ell$ and an $n$-ary string $S$, it stores $S$ in ((\rule{0ex}{2ex} O (H_\ell (S)) + o (\log n)) |S| + O (n^{\ell + 1} \log n)) bits -- where (H_\ell (S)) is the $\ell$th-order empirical entropy of $S$ -- while using at most (f (n, \ell)) bits of memory. We prove that, for any (\epsilon > 0) and some (f (n, \ell) \in O (n^{\ell + \epsilon} \log n)), there is an (f (n, \ell))-footprint compressor; on the other hand, there is no (f (n, \ell))-footprint compressor for (f (n, \ell) \in o (n^\ell \log n)).