Grammar-Based Construction of Indexes for Binary Jumbled Pattern Matching

TL;DR

This paper introduces a grammar-based indexing method for binary strings that enables efficient pattern matching queries related to substring composition, significantly improving query times with a linear-space index.

Contribution

It presents a novel construction of indexes from straight-line programs for binary strings, achieving fast query times for substring composition problems.

Findings

Index construction in $O(g^{2/3} n^{4/3})$ time

Constant-time queries for substring presence with specific composition

Linear-space index allows listing substrings efficiently

Abstract

We show how, given a straight-line program with $g$ rules for a binary string $B$ of length $n$, in $O(g^{2 / 3} n^{4 / 3})$ time we can build a linear-space index such that, given $m$ and $c$, in O(1) time we can determine whether there is a substring of $B$ with length $m$ containing exactly $c$ copies of 1. If we use $O(n \log n)$ space for the index, then we can list all such substrings using $O(m)$ time per substring.