Efficient Data Structures for the Orthogonal Range Successor Problem

Abstract

This paper considers a type of orthogonal range query, called orthogonal range successor query, which is defined as follows: Let P be a set of n points that lie on an n ×n grid. Then, for any given rectangle R, our target is to report, among all points of P ∩ R, the point which has the smallest y-coordinate. We propose two indexing data structures for P so that online orthogonal range successor queries are supported efficiently. The first one is a succinct index where only O(n) words are allowed for the index space. We show that each query can be answered in O(logn / loglogn) time, thus improving the best-known O(logn) time by Mäkinen and Navarro. 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, which in turn extends the recent wavelet tree technique to support the desired query. Our second result is a general framework for indexing points in the d-dimensional grids. We show an O(n ^1 + ε)-space index that supports each d-dimensional query in optimal O(1) time. Our second index is very simple and when d = 2, it is as efficient as the existing index by Crochemore et al.

This research was supported in part by the National Science Council of the Republic of China under the Contracts NSC-95-2213-E-007-029 and NSC-96-2221-E-007-082.