Data Structures for Range Sorted Consecutive Occurrence Queries

TL;DR

This paper introduces advanced data structures for efficiently answering range-based consecutive occurrence queries in strings, improving query times and space complexity for pattern matching problems with applications in bioinformatics and information retrieval.

Contribution

It presents novel data structures for range top-k and gap-bounded consecutive occurrence queries with optimized space and query time complexities.

Findings

Achieved O(n log^2 n)-space for range top-k queries with O(m + log log n + k) time.

Developed O(n log^{2+ε} n)-space structure for gap-bounded queries with O(m + log log n + output) time.

Connected consecutive occurrences to closed substrings, with applications in geometric problems.

Abstract

The string indexing problem is a fundamental computational problem with numerous applications, including information retrieval and bioinformatics. It aims to efficiently solve the pattern matching problem: given a text T of length n for preprocessing and a pattern P of length m as a query, the goal is to report all occurrences of P as substrings of T. Navarro and Thankachan [CPM 2015, Theor. Comput. Sci. 2016] introduced a variant of this problem called the gap-bounded consecutive occurrence query, which reports pairs of consecutive occurrences of P in T such that their gaps (i.e., the distances between them) lie within a query-specified range [g_1, g_2]. Recently, Bille et al. [FSTTCS 2020, Theor. Comput. Sci. 2022] proposed the top-k close consecutive occurrence query, which reports the k closest consecutive occurrences of P in T, sorted in non-decreasing order of distance. Both problems are optimally solved in query time with O(n \log n)-space data structures. In this paper, we generalize these problems to the range query model, which focuses only on occurrences of P in a specified substring T[a.. b] of T. Our contributions are as follows: (1) We propose an O(n \log^2 n)-space data structure that answers the range top-k consecutive occurrence query in O(m + \log\log n + k) time and can be built in O(n \log^3 n) time; and (2) We propose an O(n \log^{2+\epsilon} n)-space data structure that answers the range gap-bounded consecutive occurrence query in O(m + \log\log n + output) time and can be constructed in O(n \log^{5/2}n) time, where \epsilon is a positive constant and output denotes the number of outputs. As by-products, we present algorithms for geometric problems involving weighted horizontal segments in a 2D plane, which are of independent interest. Furthermore, we observe that consecutive occurrences are related to closed substrings of a string.