String Indexing for Top-$k$ Close Consecutive Occurrences

TL;DR

This paper introduces the string indexing for top-$k$ close consecutive occurrences problem, providing efficient data structures with various space-time trade-offs for reporting closest consecutive pattern occurrences.

Contribution

It formulates a new string indexing problem and offers three different data structures balancing space and query time, including novel techniques like line segment intersection translation and recursive clustering.

Findings

Optimal query time of O(m+k) with O(n log n) space.

Linear space solutions with query times O(m+k^{1+ε}) and O(m+log^{1+ε} n).

Development of new techniques such as line segment intersection translation and recursive clustering.

Abstract

The classic string indexing problem is to preprocess a string $S$ into a compact data structure that supports efficient subsequent pattern matching queries, that is, given a pattern string $P$, report all occurrences of $P$ within $S$. In this paper, we study a basic and natural extension of string indexing called the string indexing for top-$k$ close consecutive occurrences problem (SITCCO). Here, a consecutive occurrence is a pair $(i,j)$, $i < j$, such that $P$ occurs at positions $i$ and $j$ in $S$ and there is no occurrence of $P$ between $i$ and $j$, and their distance is defined as $j-i$. Given a pattern $P$ and a parameter $k$, the goal is to report the top-$k$ consecutive occurrences of $P$ in $S$ of minimal distance. The challenge is to compactly represent $S$ while supporting queries in time close to the length of $P$ and $k$. We give three time-space trade-offs for the problem. Let $n$ be the length of $S$, $m$ the length of $P$, and $\epsilon\in(0,1]$. Our first result achieves $O(n\log n)$ space and optimal query time of $O(m+k)$. Our second and third results achieve linear space and query times either $O(m+k^{1+\epsilon})$ or $O(m + k \log^{1+\epsilon} n)$. Along the way, we develop several techniques of independent interest, including a new translation of the problem into a line segment intersection problem and a new recursive clustering technique for trees.