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15
Spaceefficient geometric divideandconquer algorithms
 Comput. Geom
"... We develop a number of spaceefficient tools including an approach to simulate divideandconquer spaceefficiently, stably selecting and unselecting a subset from a sorted set, and computing the kth smallest element in one dimension from a multidimensional set that is sorted in another dimension. ..."
Abstract

Cited by 16 (4 self)
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We develop a number of spaceefficient tools including an approach to simulate divideandconquer spaceefficiently, stably selecting and unselecting a subset from a sorted set, and computing the kth smallest element in one dimension from a multidimensional set that is sorted in another dimension. We then apply these tools to solve several geometric problems that have solutions using some form of divideandconquer. Specifically, we present solutions running in O(n logn) time using O(1) extra memory given inputs of size n for the closest pair problem and the bichromatic closest pair problem. For the orthogonal line segment intersection problem, we solve the problem in O(n logn + k) time using O(1) extra space where n is the number of horizontal and vertical line segments and k is the number of intersections. 1
SpaceEfficient Algorithms for Computing the Convex Hull of a Simple Polygonal Line in Linear Time
"... We present spaceefficient algorithms for computing the convex hull of a simple polygonal line inplace, in linear time. It turns out that the problem is as hard as stable partition, i.e., if there were a truly simple solution then stable partition would also have a truly simple solution, and vice v ..."
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Cited by 16 (2 self)
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We present spaceefficient algorithms for computing the convex hull of a simple polygonal line inplace, in linear time. It turns out that the problem is as hard as stable partition, i.e., if there were a truly simple solution then stable partition would also have a truly simple solution, and vice versa. Nevertheless, we present a simple selfcontained solution that uses O(log n) space, and indicate how to improve it to O(1) space with the same techniques used for stable partition. If the points inside the convex hull can be discarded, then there is a truly simple solution that uses a single call to stable partition, and even that call can be spared if only extreme points are desired (and not their order). If the polygonal line is closed, then the problem admits a very simple solution which does not call for stable partitioning at all.
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 8 (2 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.
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 6 (1 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
InSitu, Stable Merging by way of the Perfect Shuffle.
, 1999
"... We introduce a novel approach to the classical problem of insitu, stable merging, where "insitu" means the use of no more than O(log 2 n) bits of extra memory for lists of size n. Shufflemerge reduces the merging problem to the problem of realising the "perfect shuffle" permu ..."
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Cited by 1 (0 self)
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We introduce a novel approach to the classical problem of insitu, stable merging, where "insitu" means the use of no more than O(log 2 n) bits of extra memory for lists of size n. Shufflemerge reduces the merging problem to the problem of realising the "perfect shuffle" permutation, that is, the exact interleaving of two, equal length lists. The algorithm is recursive, using a logarithmic number of variables, and so does not use absolutely minimum storage, i.e., a fixed number of variables. A simple method of realising the perfect shuffle uses one extra bit per element, and so is not insitu. We show that the perfect shuffle can be attained using absolutely minimum storage and in linear time, at the expense of doubling the number of moves, relative to the simple method. We note that there is a worst case for Shufflemerge requiring time\Omega\Gamma n log n), where n is the sum of the lengths of the input lists. We also present an analysis of a variant of Shufflemerge which uses a ...
SpaceEfficient Geometric DivideandConquer Algorithms ⋆ Prosenjit Bose a Anil Maheshwari a
"... We develop a number of spaceefficient tools including an approach to simulate divideandconquer spaceefficiently, stably selecting and unselecting a subset from a sorted set, and computing the kth smallest element in one dimension from a multidimensional set that is sorted in another dimension. ..."
Abstract
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We develop a number of spaceefficient tools including an approach to simulate divideandconquer spaceefficiently, stably selecting and unselecting a subset from a sorted set, and computing the kth smallest element in one dimension from a multidimensional set that is sorted in another dimension. We then apply these tools to solve several geometric problems that have solutions using some form of divideandconquer. Specifically, we present a deterministic algorithm running in O(n log n) time using O(1) extra memory given inputs of size n for the closest pair problem and a randomized solution running in O(n log n) expected time and using O(1) extra space for the bichromatic closest pair problem. For the orthogonal line segment intersection problem, we solve the problem in O(n log n + k) time using O(1) extra space where n is the number of horizontal and vertical line segments and k is the number of intersections. 1
unknown title
"... Slope selection, i.e. selecting the slope with rank k among all � � n 2 lines induced by a collection P of points, results in a widely used robust estimator for linefitting. In this paper, we demonstrate that it is possible to perform slope selection in expected O(n·log2 n) time using only constant ..."
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Slope selection, i.e. selecting the slope with rank k among all � � n 2 lines induced by a collection P of points, results in a widely used robust estimator for linefitting. In this paper, we demonstrate that it is possible to perform slope selection in expected O(n·log2 n) time using only constant extra space in addition to the space needed for representing the input. 1
Ratio based stable inplace merging
"... Abstract. We investigate the problem of stable inplace merging from a ratio k = n based point of view where m, n are the sizes of the input m sequences with m ≤ n. We introduce a novel algorithm for this problem that is asymptotically optimal regarding the number of assignments as well as compariso ..."
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Abstract. We investigate the problem of stable inplace merging from a ratio k = n based point of view where m, n are the sizes of the input m sequences with m ≤ n. We introduce a novel algorithm for this problem that is asymptotically optimal regarding the number of assignments as well as comparisons. Our algorithm uses knowledge about the ratio of the input sizes to gain optimality and does not stay in the tradition of Mannila and Ukkonen’s work [8] in contrast to all other stable inplace merging algorithms proposed so far. It has a simple modular structure and does not demand the additional extraction of a movement imitation buffer as needed by its competitors. For its core components we give concrete implementations in form of Pseudo Code. Using benchmarking we prove that our algorithm performs almost always better than its direct competitor proposed in [6]. As additional subresult we show that stable inplace merging is a quite simple problem for every ratio k ≥ √ m by proving that there exists a primitive algorithm that is asymptotically optimal for such ratios. 1