Detecting One-variable Patterns

TL;DR

This paper presents an efficient algorithm for detecting all instances of a specific one-variable pattern, including reversed variables, within a string, with a focus on compact representation and linear time complexity.

Contribution

It introduces a novel $O(rn)$ time algorithm for constructing a compact representation of pattern instances involving reversed variables in strings.

Findings

Constructs a compact representation in $O(rn)$ time.

Reports pattern instances in $O(P)$ time.

Applicable to strings over polynomially bounded integer alphabets.

Abstract

Given a pattern $p = s_1x_1s_2x_2\cdots s_{r-1}x_{r-1}s_r$ such that $x_1,x_2,\ldots,x_{r-1}\in\{x,\overset{{}_{\leftarrow}}{x}\}$, where $x$ is a variable and $\overset{{}_{\leftarrow}}{x}$ its reversal, and $s_1,s_2,\ldots,s_r$ are strings that contain no variables, we describe an algorithm that constructs in $O(rn)$ time a compact representation of all $P$ instances of $p$ in an input string of length $n$ over a polynomially bounded integer alphabet, so that one can report those instances in $O(P)$ time.