Improved data structures for the orthogonal range successor problem

Abstract

Let P be a set of n points that lie on an n × n grid. The well-known orthogonal range reporting problem is to preprocess P so that for any query rectangle R, we can report all points in P ∩ R efficiently. In many applications driven by the information retrieval or the bioinformatics communities, we do not need all the points in P ∩ R, but need only just the point that has the smallest y-coordinate; this motivates the study of a variation called the orthogonal range successor problem. If space is the major concern, the best-known result is by Mäkinen and Navarro, which requires an optimal index space of n + o(n) words and supports each query in O (log n) time. In contrast, if query time is the major concern, the best-known result is by Crochemore et al., which supports each query in O (1) time with O (n 1+ε ) index space. In this paper, we first propose another optimal-space index with a faster O (log n/ loglog n) query time. The improvement stems from the design of an index with O (1) query time when the points are restricted to lie on a narrow grid, and the subsequent application of the wavelet tree technique to support the desired query. Based on the proposed index, we directly obtain improved results for the successive indexing problem and the position-restricted pattern matching problem in the literature. We next propose an O (n 1+ε )-word index that supports each query in O (1) time. When compared with the result by Crochemore et al., our scheme is conceptually simpler and easier for construction. In addition, our scheme can be easily extended to work for high-dimensional cases.

Reporting and counting maximal points in a query orthogonal rectangle

“…However, the following result is known from [17]: [17].) If the coordinates of the points are on an integer grid of [ [c, d].…”

“…As RS is an instance of the data structure of [17], which is a linear space data structure, the total storage space needed by Proof. As Subproblem 1 is an instance of the range successor query with points restricted in the range of [1, log n] × [1, n], and as the data structure of [17] takes O (1) time to answer a range successor query, our query algorithm takes O (1) time. 2 Therefore, we conclude: Theorem 5.…”

“…We tackle this issue by proposing an algorithm that identifies the nodes with useful output and by utilizing fast data structures for "restricted 3-sided range successor queries" and "rank/select queries" (see [17]) to support the query on these nodes.…”

“…At each canonical node we report the maximal points by repeating range maxima queries. Thus, reporting the point p is ensured by the correctness of range maxima data structure of [8] while visit to the node φ is ensured by the correctness of the range successor query of [17] and visiting the group leader node GL(m) is ensured by the step 13 of the query algorithm. Proof.…”

Closed factorization

“…We first preprocess the suffix array SA for range successor queries, building the data structure of Yu, Hon and Wang [8]. A range successor query rsq SA (s, e, k) returns, given a range Figure 2.…”