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22
Towards InPlace Geometric Algorithms and Data Structures
 In Proceedings of the Twentieth ACM Symposium on Computational Geometry
, 2003
"... For many geometric problems, there are ecient algorithms that surprisingly use very little extra space other than the given array holding the input. For many geometric query problems, there are ecient data structures that need no extra space at all other than an array holding a permutation of the ..."
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Cited by 17 (5 self)
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For many geometric problems, there are ecient algorithms that surprisingly use very little extra space other than the given array holding the input. For many geometric query problems, there are ecient data structures that need no extra space at all other than an array holding a permutation of the input. In this paper, we obtain the rst such spaceeconomical solutions for a number of fundamental problems, including threedimensional convex hulls, twodimensional Delaunay triangulations, xeddimensional range queries, and xeddimensional nearest neighbor queries.
Succinct Geometric Indexes Supporting Point Location Queries
"... We propose to design data structures called succinct geometric indexes of negligible space (more precisely, o(n) bits) that support geometric queries in optimal time, by taking advantage of the n points in the data set permuted and stored elsewhere as a sequence. Our first and main result is a succi ..."
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Cited by 11 (5 self)
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We propose to design data structures called succinct geometric indexes of negligible space (more precisely, o(n) bits) that support geometric queries in optimal time, by taking advantage of the n points in the data set permuted and stored elsewhere as a sequence. Our first and main result is a succinct geometric index that can answer point location queries, a fundamental problem in computational geometry, on planar triangulations in O(lg n) time1. We also design three variants of this index. The first supports point location using lg n +2 √ lg n + O(lg 1/4 n) pointline comparisons. The second supports point location in o(lg n) time when the coordinates are integers bounded by U. The last variant can answer point location queries in O(H +1) expected time, where H is the entropy of the query distribution. These results match the query efficiency of previous point location structures that occupy O(n) words or O(n lg n) bits, while saving drastic amounts of space. We generalize our succinct geometric index to planar subdivisions, and design indexes for other types of queries. Finally, we apply our techniques to design the first implicit data structures that support point location in O(lg² n) time.
Inplace algorithms for computing (layers of) maxima
 In: Proceedings of the 10th Scandinavian Workshop on Algorithm Theory (SWAT ’06
, 2006
"... Abstract. We describe spaceefficient algorithms for solving problems related to finding maxima among points in two and three dimensions. Our algorithms run in optimal O(n log n) time and occupy only constant extra space in addition to the space needed for representing the input. 1 ..."
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Cited by 10 (2 self)
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Abstract. We describe spaceefficient algorithms for solving problems related to finding maxima among points in two and three dimensions. Our algorithms run in optimal O(n log n) time and occupy only constant extra space in addition to the space needed for representing the input. 1
Linesegment intersection made inplace
, 2007
"... We present a spaceefficient algorithm for reporting all k intersections induced by a set of n line segments in the plane. Our algorithm is an inplace variant of Balaban’s algorithm and, in the worst case, runs in O(n log2 n+k) time using O(1) extra words of memory in addition to the space used f ..."
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Cited by 9 (3 self)
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We present a spaceefficient algorithm for reporting all k intersections induced by a set of n line segments in the plane. Our algorithm is an inplace variant of Balaban’s algorithm and, in the worst case, runs in O(n log2 n+k) time using O(1) extra words of memory in addition to the space used for the input to the algorithm.
SpaceEfficient Algorithms for Klee’s Measure Problem
, 2005
"... We give spaceefficient geometric algorithms for three related problems. Given a set of n axisaligned rectangles in the plane, we calculate the area covered by the union of these rectangles (Klee’s measure problem) in O(n 3/2 log n) time with O(√n) extra space. If the input can be destroyed and the ..."
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Cited by 6 (0 self)
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We give spaceefficient geometric algorithms for three related problems. Given a set of n axisaligned rectangles in the plane, we calculate the area covered by the union of these rectangles (Klee’s measure problem) in O(n 3/2 log n) time with O(√n) extra space. If the input can be destroyed and there are no degenerate cases and input coordinates are all integers, we can solve Klee’s measure problem in O(n log² n) time with O(log² n) extra space. Given a set of n points in the plane, we find the axisaligned unit square that covers the maximum number of points in O(n log³ n) time with O(log² n) extra space.
InPlace 2d Nearest Neighbor Search
, 2007
"... Abstract We revisit a classic problem in computational geometry: preprocessing a planar npoint set to answer nearest neighbor queries. In SoCG 2004, Br"onnimann, Chan, and Chen showed that it is possible to design an efficient data structure that takes no extra space at all other than the ..."
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Cited by 4 (1 self)
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Abstract We revisit a classic problem in computational geometry: preprocessing a planar npoint set to answer nearest neighbor queries. In SoCG 2004, Br&quot;onnimann, Chan, and Chen showed that it is possible to design an efficient data structure that takes no extra space at all other than the input array holding a permutation of the points. The best query time known for such &quot;inplace data structures &quot; is O(log 2 n). In this paper, we break the O(log 2 n) barrier by providing a method that answers nearest neighbor queries in time O((log n) log3=2 2 log log n) = O(log
Minimum Enclosing Circle with Few Extra Variables
"... Asano et al. [JoCG 2011] proposed an open problem of computing the minimum enclosing circle of a set of n points in R2 given in a readonly array in subquadratic time. We show that Megiddo’s prune and search algorithm for computing the minimum radius circle enclosing the given points can be tailore ..."
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Cited by 3 (1 self)
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Asano et al. [JoCG 2011] proposed an open problem of computing the minimum enclosing circle of a set of n points in R2 given in a readonly array in subquadratic time. We show that Megiddo’s prune and search algorithm for computing the minimum radius circle enclosing the given points can be tailored to work in a readonly environment in O(n1+ɛ) time using O(log n) extra space, where ɛ is a positive constant less than 1. As a warmup, we first solve the same problem in an inplace setup in linear time with O(1) extra space.
Adaptive Algorithms for Planar Convex Hull Problems?
"... Abstract. We study problems in computational geometry from the viewpoint of adaptive algorithms. Adaptive algorithms have been extensively studied for the sorting problem, and in this paper we generalize the framework to geometric problems. To this end, we think of geometric problems as permutation ..."
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Abstract. We study problems in computational geometry from the viewpoint of adaptive algorithms. Adaptive algorithms have been extensively studied for the sorting problem, and in this paper we generalize the framework to geometric problems. To this end, we think of geometric problems as permutation (or rearranging) problems of arrays, and define the “presortedness ” as a distance from the input array to the desired output array. We call an algorithm adaptive if it runs faster when a given input array is closer to the desired output, and furthermore it does not make use of any information of the presortedness. As a case study, we look into the planar convex hull problem for which we discover two natural formulations as permutation problems. An interesting phenomenon that we prove is that for one formulation the problem can be solved adaptively, but for the other formulation no adaptive algorithm can be better than an optimal outputsensitive algorithm for the planar convex hull problem. 1
InPlace Randomized Slope Selection
"... Abstract. Slope selection is a wellknown algorithmic tool used in the context of computing robust estimators for fitting a line to a collection P of n points in the plane. We demonstrate that it is possible to perform slope selection in expected O(n log n) time using only constant extra space in ad ..."
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Abstract. Slope selection is a wellknown algorithmic tool used in the context of computing robust estimators for fitting a line to a collection P of n points in the plane. We demonstrate that it is possible to perform slope selection in expected O(n log n) time using only constant extra space in addition to the space needed for representing the input. Our solution is based upon a spaceefficient variant of Matouˇsek’s randomized interpolation search, and we believe that the techniques developed in this paper will prove helpful in the design of spaceefficient randomized algorithms using samples. To underline this, we also sketch how to compute the repeated median line estimator in an inplace setting. 1