Cadences in Grammar-Compressed Strings

TL;DR

This paper introduces efficient algorithms for detecting 3-cadences in binary strings, both compressed and uncompressed, and proves NP-completeness for certain variants of the problem in grammar-compressed strings.

Contribution

It provides the first polynomial-time detection algorithm for 3-cadences in grammar-compressed binary strings and establishes NP-completeness results for related problems.

Findings

Polynomial-time algorithm for 3-cadence detection in grammar-compressed binary strings

Linear-time detection algorithm for uncompressed binary strings

NP-completeness of several cadence detection variants in grammar-compressed strings

Abstract

Cadences are structurally maximal arithmetic progressions of indices corresponding to equal characters in an underlying string. This paper provides a polynomial time detection algorithm for 3-cadences in grammar-compressed binary strings. This algorithm also translates to a linear time detection algorithm for 3-cadences in uncompressed binary strings. Furthermore, this paper proves that several variants of the cadence detection problem are NP-complete for grammar-compressed strings. As a consequence, the equidistant subsequence matching problem with patterns of length three is NP-complete for grammar-compressed ternary strings.