Gapped String Indexing in Subquadratic Space and Sublinear Query Time

TL;DR

This work breaks the long-standing quadratic-space barrier for Gapped String Indexing by introducing polynomially subquadratic space and polynomially sublinear query-time trade-offs. The key technique is Shifted Set Intersection, shown to be equivalent to the indexing variant of 3SUM Indexing, enabling reductions that transfer efficient data-structure results across problems. The authors develop Gapped Set Intersection and Gapped String Indexing w. Reporting, achieving $\tilde{O}(n^{2-\delta/3})$ or $\tilde{O}(n^{3-2\delta})$ space with $\tilde{O}(|P_1|+|P_2|+n^{\delta}\cdot(occ+1))$ query time for any $0\le \delta \le 1$, and they derive improvements for Jumbled Indexing under constant alphabet sizes. The results establish deep connections between gap-based string indexing and classic sumset problems, with implications for data structures in computational biology and text mining, and open avenues for further improvements and applications in related indexing tasks.

Abstract

In Gapped String Indexing, the goal is to compactly represent a string $S$ of length $n$ such that for any query consisting of two strings $P_1$ and $P_2$, called patterns, and an integer interval $[α, β]$, called gap range, we can quickly find occurrences of $P_1$ and $P_2$ in $S$ with distance in $[α, β]$. Gapped String Indexing is a central problem in computational biology and text mining and has thus received significant research interest, including parameterized and heuristic approaches. Despite this interest, the best-known time-space trade-offs for Gapped String Indexing are the straightforward $O(n)$ space and $O(n+occ)$ query time or $Ω(n^2)$ space and $\tilde{O}(|P_1| + |P_2| + occ)$ query time. We break through this barrier obtaining the first interesting trade-offs with polynomially subquadratic space and polynomially sublinear query time. In particular, we show that, for every $0\leq δ\leq 1$, there is a data structure for Gapped String Indexing with either $\tilde{O}(n^{2-δ/3})$ or $\tilde{O}(n^{3-2δ})$ space and $\tilde{O}(|P_1| + |P_2| + n^δ\cdot (occ+1))$ query time, where $occ$ is the number of reported occurrences. As a new tool towards obtaining our main result, we introduce the Shifted Set Intersection problem. We show that this problem is equivalent to the indexing variant of 3SUM (3SUM Indexing). Via a series of reductions, we obtain a solution to the Gapped String Indexing problem. Furthermore, we enhance our data structure for deciding Shifted Set Intersection, so that we can support the reporting variant of the problem. Via the obtained equivalence to 3SUM Indexing, we thus give new improved data structures for the reporting variant of 3SUM Indexing, and we show how this improves upon the state-of-the-art solution for Jumbled Indexing for any alphabet of constant size $σ>5$.

Figures (2)

• Figure 1: We show that Shifted Set Intersection and 3SUM Indexing are equivalent, and our reduction also transfers to the reporting variants of the two problems. In fact, we solve a problem that is even harder than the Shifted Set Intersection problem; namely, the Gapped Set Intersection problem, which leads to a solution to the Gapped String Indexing problem.
• Figure 2: Illustration of the phases from the proof of Theorem \ref{['thm:intervals']}: At the $l$th phase of the algorithm, we use at most three $2^{l}$-approximate queries to cover at least $2\cdot 2^l$ elements in $[\alpha,\beta]$ which were not covered by previous phases. The centers of the queries are shown by short vertical line segments. The covered elements for each query are shown dashed, and the uncertain elements are shown by the black horizontal arrows. The dots at the bottom show all potential centers for the approximate queries; the ones our algorithm uses are marked in red.

Theorems & Definitions (40)

• Theorem 1
• Theorem 2: Golovnev et al. DBLP:conf/stoc/GolovnevGHPV20
• Theorem 3
• Corollary 1
• Theorem 4
• Theorem 5
• Definition 1
• Corollary 2
• Proposition 1
• proof