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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 + *** )-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.