A nearly tight memory-redundancy trade-off for one-pass compression

TL;DR

This paper establishes a nearly tight trade-off between memory usage and redundancy in one-pass string compression algorithms, showing optimal bounds for encoding efficiency with limited memory.

Contribution

It provides a fundamental characterization of the memory-redundancy trade-off in one-pass compression, achieving nearly optimal bounds for all k simultaneously.

Findings

Achieves encoding within n H_k(s) + O(σ^k n^{1 - c + ε}) bits using O(n) time and O(n^c) memory.

Proves that surpassing certain redundancy bounds is impossible with limited memory, even with unlimited time.

Defines tight bounds for the redundancy-memory trade-off in one-pass compression algorithms.

Abstract

Let $s$ be a string of length $n$ over an alphabet of constant size $\sigma$ and let $c$ and $\epsilon$ be constants with (1 \geq c \geq 0) and (\epsilon > 0). Using (O (n)) time, (O (n^c)) bits of memory and one pass we can always encode $s$ in (n H_k (s) + O (\sigma^k n^{1 - c + \epsilon})) bits for all integers (k \geq 0) simultaneously. On the other hand, even with unlimited time, using (O (n^c)) bits of memory and one pass we cannot always encode $s$ in (O (n H_k (s) + \sigma^k n^{1 - c - \epsilon})) bits for, e.g., (k = \lceil (c + \epsilon / 2) \log_\sigma n \rceil).