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Under the word RAM model, we design three data structures that can beconstructed in $O(n\sqrt{\lg n})$ time over $n$ points in an $n \times n$ grid.The first data structure is an $O(n\lg^{\epsilon} n)$-word structure supportingorthogonal range reporting in $O(\lg\lg n+k)$ time, where $k$ denotes outputsize and $\epsilon$ is an arbitrarily small constant. The second is an$O(n\lg\lg n)$-word structure supporting orthogonal range successor in$O(\lg\lg n)$ time, while the third is an $O(n\lg^{\epsilon} n)$-word structuresupporting sorted range reporting in $O(\lg\lg n+k)$ time. The query times ofthese data structures are optimal when the space costs must be within $O(n\polylog\ n)$ words. Their exact space bounds match those of the best knownresults achieving the same query times, and the $O(n\sqrt{\lg n})$ constructiontime beats the previous bounds on preprocessing. Previously, among 2d rangesearch structures, only the orthogonal range counting structure of Chan andP\v{a}tra\c{s}cu (SODA 2010) and the linear space, $O(\lg^{\epsilon} n)$ querytime structure for orthogonal range successor by Belazzougui and Puglisi (SODA2016) can be built in the same $O(n\sqrt{\lg n})$ time. Hence our work is thefirst that achieve the same preprocessing time for optimal orthogonal rangereporting and range successor. We also apply our results to improve theconstruction time of text indexes.

Fast Preprocessing for Optimal Orthogonal Range Reporting and Range Successor with Applications to Text Indexing