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Orthogonal Range Searching on the RAM, Revisited
, 2011
"... We present a number of new results on one of the most extensively studied topics in computational geometry, orthogonal range searching. All our results are in the standard word RAM model: 1. We present two data structures for 2d orthogonal range emptiness. The first achieves O(n lg lg n) space and ..."
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Cited by 15 (4 self)
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We present a number of new results on one of the most extensively studied topics in computational geometry, orthogonal range searching. All our results are in the standard word RAM model: 1. We present two data structures for 2d orthogonal range emptiness. The first achieves O(n lg lg n) space and O(lg lg n) query time, assuming that the n given points are in rank space. This improves the previous results by Alstrup, Brodal, and Rauhe (FOCS’00), with O(n lg ε n) space and O(lg lg n) query time, or with O(n lg lg n) space and O(lg 2 lg n) query time. Our second data structure uses O(n) space and answers queries in O(lg ε n) time. The best previous O(n)space data structure, due to Nekrich (WADS’07), answers queries in O(lg n / lg lg n) time. 2. We give a data structure for 3d orthogonal range reporting with O(n lg 1+ε n) space and O(lg lg n+ k) query time for points in rank space, for any constant ε> 0. This improves the previous results by Afshani (ESA’08), Karpinski and Nekrich (COCOON’09), and Chan (SODA’11), with O(n lg 3 n) space and O(lg lg n + k) query time, or with O(n lg 1+ε n) space and O(lg 2 lg n + k) query time. Consequently, we obtain improved upper bounds for orthogonal range reporting in all constant dimensions above 3.
Range Selection and Median: Tight Cell Probe Lower Bounds and Adaptive Data Structures
"... Range selection is the problem of preprocessing an input array A of n unique integers, such that given a query (i, j, k), one can report the k’th smallest integer in the subarray A[i], A[i + 1],..., A[j]. In this paper we consider static data structures in the wordRAM for range selection and severa ..."
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Cited by 12 (5 self)
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Range selection is the problem of preprocessing an input array A of n unique integers, such that given a query (i, j, k), one can report the k’th smallest integer in the subarray A[i], A[i + 1],..., A[j]. In this paper we consider static data structures in the wordRAM for range selection and several natural special cases thereof. The first special case is known as range median, which arises when k is fixed to ⌊(j − i + 1)/2⌋. The second case, denoted prefix selection, arises when i is fixed to 0. Finally, we also consider the bounded rank prefix selection problem and the fixed rank range selection problem. In the former, data structures must support prefix selection queries under the assumption that k ≤ κ for some value κ ≤ n given at construction time, while in the latter, data structures must support range selection queries where k is fixed beforehand for all queries. We prove cell probe lower bounds for range selection, prefix selection and range median, stating that any data structure that uses S words of space needs Ω(log n / log(Sw/n)) time to answer a query. In particular, any data structure that uses n log O(1) n space needs Ω(log n / log log n) time to answer a query, and any data structure that supports queries in constant time, needs n 1+Ω(1) space. For data structures that uses n log O(1) n space this matches the best known upper bound. Additionally, we present a linear space data structure that supports range selection queries in O(log k / log log n + log log n) time. Finally, we prove that any data structure that uses S space, needs Ω(log κ / log(Sw/n)) time to answer a bounded rank prefix selection query and Ω(log k / log(Sw/n)) time to answer a fixed rank range selection query. This shows that our data structure is optimal except for small values of k. 1
Three Problems about Dynamic Convex Hulls
, 2011
"... We present three results related to dynamic convex hulls: • A fully dynamic data structure for maintaining a set of n points in the plane so that we can find the edges of the convex hull intersecting a query line, with expected query and amortized update time O(log 1+ε n) for an arbitrarily small co ..."
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Cited by 1 (0 self)
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We present three results related to dynamic convex hulls: • A fully dynamic data structure for maintaining a set of n points in the plane so that we can find the edges of the convex hull intersecting a query line, with expected query and amortized update time O(log 1+ε n) for an arbitrarily small constant ε> 0. This improves the previous bound of O(log 3/2 n). • A fully dynamic data structure for maintaining a set of n points in the plane to support halfplane range reporting queries in O(log n+k) time with O(polylog n) expected amortized update time. A similar result holds for 3dimensional orthogonal range reporting. For 3dimensional halfspace range reporting, the query time increases to O(log 2 n / log log n+k). • A semionline dynamic data structure for maintaining a set of n line segments in the plane, so that we can decide whether a query line segment lies completely above the lower envelope, with query time O(log n) and amortized update time O(n ε). As a corollary, we can solve the following problem in O(n 1+ε) time: given a triangulated terrain in 3d of size n, identify all faces that are partially visible from a fixed viewpoint. 1
Author manuscript, published in "String Processing and Information Retrieval, Cartagena de Indias: Colombia (2012)" DOI: 10.1007/9783642341090_32 Computing Discriminating and Generic Words
, 2013
"... Abstract. We study the following three problems of computing generic or discriminating words for a given collection of documents. Given a pattern P and a threshold d, we want to report (i) all longest extensions of P which occur in at least d documents, (ii) all shortest extensions of P which occur ..."
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Abstract. We study the following three problems of computing generic or discriminating words for a given collection of documents. Given a pattern P and a threshold d, we want to report (i) all longest extensions of P which occur in at least d documents, (ii) all shortest extensions of P which occur in less than d documents, and (iii) all shortest extensions of P which occur only in d selected documents. For these problems, we propose efficient algorithms based on suffix trees and using advanced data structure techniques. For problem (i), we propose an optimal solution with constant running time per output word. 1
Adaptive and Approximate Orthogonal Range Counting ∗
"... We present three new results on one of the most basic problems in geometric data structures, 2D orthogonal range counting. All the results are in the wbit word RAM model. • It is well known that there are linearspace data structures for 2D orthogonal range counting with worstcase optimal query t ..."
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We present three new results on one of the most basic problems in geometric data structures, 2D orthogonal range counting. All the results are in the wbit word RAM model. • It is well known that there are linearspace data structures for 2D orthogonal range counting with worstcase optimal query time O(logw n). We give an O(n log log n)space adaptive data structure that improves the query time to O(log log n + logw k), where k is the output count. When k = O(1), our bounds match the state of the art for the 2D orthogonal range emptiness problem [Chan, Larsen, and Pătra¸scu, SoCG 2011]. • We give an O(n log log n)space data structure for approximate 2D orthogonal range counting that can compute a (1 + δ)factor approximation to the count in O(log log n) time for any fixed constant δ> 0. Again, our bounds match the state of the art for the 2D orthogonal range emptiness problem. • Lastly, we consider the 1D range selection problem, where a query in an array involves finding the kth least element in a given subarray. This problem is closely related to 2D 3sided orthogonal range counting. Recently, Jørgensen and Larsen [SODA 2011] presented a linearspace adaptive data structure with query time O(log log n + log w k). We give a new linearspace structure that improves the query time to O(1 + log w k), exactly matching the lower bound proved by Jørgensen and Larsen. 1