# New Variants of Pattern Matching with Constants and Variables

## TL;DR

This paper explores new variants of pattern matching problems involving constants and variables, proposing convolution and KMP-based algorithms with analyzed complexities for these problems.

## Contribution

It introduces novel variants of pattern matching with constants and variables and provides two algorithmic solutions with complexity analysis.

## Key findings

- Convolution-based algorithms for pattern matching variants
- Extended KMP-based algorithms for pattern matching variants
- Complexity analysis of proposed algorithms

## Abstract

Given a text and a pattern over two types of symbols called constants and variables, the parameterized pattern matching problem is to find all occurrences of substrings of the text that the pattern matches by substituting a variable in the text for each variable in the pattern, where the substitution should be injective. The function matching problem is a variant of it that lifts the injection constraint. In this paper, we discuss variants of those problems, where one can substitute a constant or a variable for each variable of the pattern. We give two kinds of algorithms for both problems, a convolution-based method and an extended KMP-based method, and analyze their complexity.