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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
Transdichotomous Results in Computational Geometry, I: Point Location in Sublogarithmic Time
, 2008
"... Given a planar subdivision whose coordinates are integers bounded by U ≤ 2 w, we present a linearspace data structure that can answer point location queries in O(min{lg n / lg lg n, √ lg U/lg lg U}) time on the unitcost RAM with word size w. Thisisthe first result to beat the standard Θ(lg n) bou ..."
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Cited by 10 (3 self)
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Given a planar subdivision whose coordinates are integers bounded by U ≤ 2 w, we present a linearspace data structure that can answer point location queries in O(min{lg n / lg lg n, √ lg U/lg lg U}) time on the unitcost RAM with word size w. Thisisthe first result to beat the standard Θ(lg n) bound for infinite precision models. As a consequence, we obtain the first o(n lg n) (randomized) algorithms for many fundamental problems in computational geometry for arbitrary integer input on the word RAM, including: constructing the convex hull of a threedimensional point set, computing the Voronoi diagram or the Euclidean minimum spanning tree of a planar point set, triangulating a polygon with holes, and finding intersections among a set of line segments. Higherdimensional extensions and applications are also discussed. Though computational geometry with bounded precision input has been investigated for a long time, improvements have been limited largely to problems of an orthogonal flavor. Our results surpass this longstanding limitation, answering, for example, a question of Willard (SODA’92).
LinearSpace Data Structures for Range Mode Query in Arrays ∗
"... A mode of a multiset S is an element a ∈ S of maximum multiplicity; that is, a occurs at least as frequently as any other element in S. Given an array A[1: n] of n elements, we consider a basic problem: constructing a static data structure that efficiently answers range mode queries on A. Each query ..."
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Cited by 8 (2 self)
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A mode of a multiset S is an element a ∈ S of maximum multiplicity; that is, a occurs at least as frequently as any other element in S. Given an array A[1: n] of n elements, we consider a basic problem: constructing a static data structure that efficiently answers range mode queries on A. Each query consists of an input pair of indices (i, j) for which a mode of A[i: j] must be returned. The best previous data structure with linear space, by Krizanc, Morin, and Smid (ISAAC 2003), requires O ( √ n log log n) query time. We improve their result and present an O(n)space data structure that supports range mode queries in O ( p n / log n) worstcase time. Furthermore, we present strong evidence that a query time significantly below √ n cannot be achieved by purely combinatorial techniques; we show that boolean matrix multiplication of two √ n × √ n matrices reduces to n range mode queries in an array of size O(n). Additionally, we give linearspace data structures for orthogonal range mode in higher dimensions (queries in near O(n 1−1/2d) time) and for halfspace range mode in higher dimensions (queries in O(n 1−1/d2) time).
Persistent Predecessor Search and Orthogonal Point Location on the Word RAM
"... We answer a basic data structuring question (for example, raised by Dietz and Raman back in SODA 1991): can van Emde Boas trees be made persistent, without changing their asymptotic query/update time? We present a (partially) persistent data structure that supports predecessor search in a set of int ..."
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Cited by 6 (3 self)
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We answer a basic data structuring question (for example, raised by Dietz and Raman back in SODA 1991): can van Emde Boas trees be made persistent, without changing their asymptotic query/update time? We present a (partially) persistent data structure that supports predecessor search in a set of integers in {1,..., U} under an arbitrary sequence of n insertions and deletions, with O(log log U) expected query time and expected amortized update time, and O(n) space. The query bound is optimal in U for linearspace structures and improves previous nearO((log log U) 2) methods. The same method solves a fundamental problem from computational geometry: point location in orthogonal planar subdivisions (where edges are vertical or horizontal). We obtain the first static data structure achieving O(log log U) worstcase query time and linear space. This result is again optimal in U for linearspace structures and improves the previous O((log log U) 2) method by de Berg, Snoeyink, and van Kreveld (1992). The same result also holds for higherdimensional subdivisions that are orthogonal binary space partitions, and for certain nonorthogonal planar subdivisions such as triangulations without small angles. Many geometric applications follow, including improved query times for orthogonal range reporting for dimensions ≥ 3 on the RAM. Our key technique is an interesting new vanEmdeBoas–style recursion that alternates between two strategies, both quite simple.
EntropyBounded Representation of Point Grids
"... Abstract. We give the first fully compressed representation of a set of m points on an n×n grid, taking H +o(H) bits of space, where H = lg ( n 2) m is the entropy of the set. This representation supports range counting, range reporting, and point selection queries, with a performance that is compar ..."
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Cited by 3 (1 self)
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Abstract. We give the first fully compressed representation of a set of m points on an n×n grid, taking H +o(H) bits of space, where H = lg ( n 2) m is the entropy of the set. This representation supports range counting, range reporting, and point selection queries, with a performance that is comparable to that of uncompressed structures and that improves upon the only previous compressed structure. Operating within entropybounded space opens a new line of research on an otherwise wellstudied area, and is becoming extremely important for handling large datasets. 1
EntropyBounded Representation of Point Grids ✩
"... We give the first fully compressed representation of a set of m points on an n× n grid, taking H +o(H) bits of space, where H = lg ( n2) is the entropy of the m set. This representation supports range counting, range reporting, and point selection queries, with complexities that go from O(1) to O ( ..."
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We give the first fully compressed representation of a set of m points on an n× n grid, taking H +o(H) bits of space, where H = lg ( n2) is the entropy of the m set. This representation supports range counting, range reporting, and point selection queries, with complexities that go from O(1) to O ( lg 2 n / lg lg n) per answer as the entropy of the grid decreases. Operating within entropybounded space, as well as relating time complexity with entropy, opens a new line of research on an otherwise wellstudied area. Keywords: Compressed data structures, geometric grids, range queries.
Bichromatic Line Segment Intersection Counting in O(n √ log n) Time
, 2011
"... We give an algorithm for bichromatic line segment intersection counting that runs in O(n √ log n) time under the word RAM model via a reduction to dynamic predecessor search, offline point location, and offline dynamic ranking. This algorithm is the first to solve bichromatic line segment intersecti ..."
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We give an algorithm for bichromatic line segment intersection counting that runs in O(n √ log n) time under the word RAM model via a reduction to dynamic predecessor search, offline point location, and offline dynamic ranking. This algorithm is the first to solve bichromatic line segment intersection counting in o(n log n) time.