Sorted Consecutive Occurrence Queries in Substrings

2024-11-18

Annual Symposium on Combinatorial Pattern Matching

AI Summary

This paper studies the **range consecutive occurrences query** problem in string indexing: given a text $T$, a pattern $P$, and a substring interval $[a,b]$, efficiently locate all *consecutive* occurrences of $P$ within $T[a..b]$. We introduce two novel query types: (1) **range top-$k$ nearest occurrences**, returning the $k$ occurrences closest to $[a,b]$ in ascending order of distance; and (2) **range gap-bounded occurrences**, reporting only those occurrences whose inter-occurrence distances lie within $[g_1,g_2]$. This work is the first to generalize consecutive occurrence queries from the entire text to arbitrary substrings, breaking the traditional global-text assumption. We design a hierarchical index combining suffix arrays, interval trees, and 2D geometric range reporting techniques. Our structure supports top-$k$ queries in $O(|P| + log log n + k)$ time using $O(n log^2 n)$ space, and gap-bounded queries in $O(|P| + log log n + mathrm{output})$ time using $O(n log^{2+varepsilon} n)$ space—delivering the first efficient solution for range string matching with both ordering and inter-occurrence gap constraints.

Technology Category

• Search and Optimization: Distributed Search
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Data Mining & Knowledge Management: Intelligent Query Processing

Application Category

• Graph Algorithms and Modeling for the Web: Querying, indexing, and retrieval in Web-related graphs
• Search and Retrieval-Augmented AI: Web crawling and indexing
• Web Mining and Content Analysis: Bridging structured and unstructured data

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-descending 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 $O(|P| + loglog n + k)$ time. (2) We propose an $O(n log^{2+epsilon} n)$-space data structure that answers the range gap-bounded consecutive occurrence query in $O(|P| + loglog n + mathit{output})$ time, where $epsilon$ is a positive constant and $mathit{output}$ denotes the number of outputs. Additionally, as by-products, we present algorithms for geometric problems involving weighted horizontal segments in a 2D plane, which are of independent interest.

Problem

• Generalizing consecutive occurrence queries to range query model
• Proposing data structures for range top-k consecutive occurrences
• Addressing range gap-bounded consecutive occurrence queries efficiently

Innovation

• Range top-k consecutive occurrence query data structure
• Range gap-bounded consecutive occurrence query solution
• Geometric algorithms for weighted horizontal segments