Substring Range Reporting

We revisit various string indexing problems with range reporting features, namely, position-restricted substring searching, indexing substrings with gaps, and indexing substrings with intervals. We obtain the following main results. {itemize} We give efficient reductions for each of the above problems to a new problem, which we call \emph{substring range reporting}. Hence, we unify the previous work by showing that we may restrict our attention to a single problem rather than studying each of the above problems individually. We show how to solve substring range reporting with optimal query time and little space. Combined with our reductions this leads to significantly improved time-space trade-offs for the above problems. In particular, for each problem we obtain the first solutions with optimal time query and $O(n\log^{O(1)} n)$ space, where $n$ is the length of the indexed string. We show that our techniques for substring range reporting generalize to \emph{substring range counting} and \emph{substring range emptiness} variants. We also obtain non-trivial time-space trade-offs for these problems. {itemize} Our bounds for substring range reporting are based on a novel combination of suffix trees and range reporting data structures. The reductions are simple and general and may apply to other combinations of string indexing with range reporting

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

Under the word RAM model, we design three data structures that can be constructed 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 supporting orthogonal range reporting in $O(\lg\lg n+k)$ time, where $k$ denotes output size 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 structure supporting sorted range reporting in $O(\lg\lg n+k)$ time. The query times of these 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 known results achieving the same query times, and the$O(n\sqrt{\lg n})$construction time beats the previous bounds on preprocessing. Previously, among 2d range search structures, only the orthogonal range counting structure of Chan and P\v{a}tra\c{s}cu (SODA 2010) and the linear space,$O(\lg^{\epsilon} n)$query time structure for orthogonal range successor by Belazzougui and Puglisi (SODA 2016) can be built in the same$O(n\sqrt{\lg n})$ time. Hence our work is the first that achieve the same preprocessing time for optimal orthogonal range reporting and range successor. We also apply our results to improve the construction time of text indexes

Finding Patterns in Given Intervals

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