Minimal Suffix and Rotation of a Substring in Optimal Time

TL;DR

This paper introduces an optimal data structure for minimal suffix and rotation queries on substrings, enabling constant-time answers after linear preprocessing, leveraging Lyndon words and advanced algorithmic tools.

Contribution

It presents a novel, optimal data structure for minimal suffix and rotation queries, extending to concatenations, with linear preprocessing and constant query time.

Findings

Achieves constant-time minimal suffix and rotation queries after linear preprocessing.

Extends query capabilities to concatenations of O(1) substrings.

Utilizes Lyndon words, fusion trees, and order isomorphism in data structure design.

Abstract

For a text given in advance, the substring minimal suffix queries ask to determine the lexicographically minimal non-empty suffix of a substring specified by the location of its occurrence in the text. We develop a data structure answering such queries optimally: in constant time after linear-time preprocessing. This improves upon the results of Babenko et al. (CPM 2014), whose trade-off solution is characterized by $\Theta(n\log n)$ product of these time complexities. Next, we extend our queries to support concatenations of $O(1)$ substrings, for which the construction and query time is preserved. We apply these generalized queries to compute lexicographically minimal and maximal rotations of a given substring in constant time after linear-time preprocessing. Our data structures mainly rely on properties of Lyndon words and Lyndon factorizations. We combine them with further algorithmic and combinatorial tools, such as fusion trees and the notion of order isomorphism of strings.