Substring Complexities on Run-length Compressed Strings

TL;DR

This paper introduces an efficient method to compute the substring complexity measure elta in run-length compressed strings, enabling better analysis of string repetitiveness with optimal time and space complexity.

Contribution

It presents a novel algorithm to compute elta directly from run-length compressed strings in near-optimal time and linear space, improving analysis of repetitive string structures.

Findings

elta can be computed in _{sort}(r, n) time

The algorithm operates in O(r) space complexity

Efficient analysis of highly-repetitive strings is enabled

Abstract

Let $S_{T}(k)$ denote the set of distinct substrings of length $k$ in a string $T$, then the $k$-th substring complexity is defined by its cardinality $|S_{T}(k)|$. Recently, $\delta = \max \{ |S_{T}(k)| / k : k \ge 1 \}$ is shown to be a good compressibility measure of highly-repetitive strings. In this paper, given $T$ of length $n$ in the run-length compressed form of size $r$, we show that $\delta$ can be computed in $\mathit{C}_{\mathsf{sort}}(r, n)$ time and $O(r)$ space, where $\mathit{C}_{\mathsf{sort}}(r, n) = O(\min (r \lg\lg r, r \lg_{r} n))$ is the time complexity for sorting $r$ $O(\lg n)$-bit integers in $O(r)$ space in the Word-RAM model with word size $\Omega(\lg n)$.