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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. ..."
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Cited by 20 (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
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 10 (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.
MemoryConstrained Algorithms for Simple Polygons
, 2011
"... A constantworkspace algorithm has readonly access to an input array and may use only O(1) additional words of O(log n) bits, where n is the size of the input. We show that we can find a triangulation of a plane straightline graph with n vertices in O(n²) time. We also consider preprocessing a sim ..."
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Cited by 6 (3 self)
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A constantworkspace algorithm has readonly access to an input array and may use only O(1) additional words of O(log n) bits, where n is the size of the input. We show that we can find a triangulation of a plane straightline graph with n vertices in O(n²) time. We also consider preprocessing a simple ngon, which is given by the ordered sequence of its vertices, for shortest path queries when the space constraint is relaxed to allow s words of working space. After a preprocessing of O(n²) time, we are able to solve shortest path queries between any two points inside the polygon in O(n²/s) time.
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 5 (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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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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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.
Geometric Streaming Algorithms with a Sorting Primitive (TR
"... Abstract. We solve several fundamental geometric problems under a new streaming model recently proposed by Ruhl et al. [2, 12]. In this model, in one pass the input stream can be scanned to generate an output stream or be sorted based on a userdefined comparator; all intermediate streams must be of ..."
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Abstract. We solve several fundamental geometric problems under a new streaming model recently proposed by Ruhl et al. [2, 12]. In this model, in one pass the input stream can be scanned to generate an output stream or be sorted based on a userdefined comparator; all intermediate streams must be of size O(n). We obtain the following geometric results for any fixed constant ɛ> 0: – We can construct 2D convex hulls in O(1) passes with O(n ɛ) extra space. – We can construct 3D convex hulls in O(1) expected number of passes with O(n ɛ) extra space. – We can construct a triangulation of a simple polygon in O(1) expected number of passes with O(n ɛ) extra space, where n is the number of vertices on the polygon. – We can report all k intersections of a set of 2D line segments in O(1) passes with O(n ɛ) extra space, if an intermediate stream of size O(n + k) is allowed. We also consider a weaker model, where we do not have the sorting primitive but are allowed to choose a scan direction for every scan pass. Here we can construct a 2D convex hull from an xordered point set in O(1) passes with O(n ɛ) extra space. 1
Succinct and Implicit Data Structures for Computational Geometry
"... Abstract. Many classic data structures have been proposed to support geometric queries, such as range search, point location and nearest neighbor search. For a twodimensional geometric data set consisting of n elements, these structures typically require O(n), close to O(n) or O(n lg n) words of ..."
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Abstract. Many classic data structures have been proposed to support geometric queries, such as range search, point location and nearest neighbor search. For a twodimensional geometric data set consisting of n elements, these structures typically require O(n), close to O(n) or O(n lg n) words of space; while they support efficient queries, their storage costs are often much larger than the space required to encode the given data. As modern applications often process very large geometric data sets, it is often not practical to construct and store these data structures. This article surveys research that addresses this issue by designing spaceefficient geometric data structures. In particular, two different but closely related lines of research will be considered: succinct geometric data structures and implicit geometric data structures. The space usage of succinct geometric data structures is equal to the informationtheoretic minimum space required to encode the given geometric data set plus a lower order term, and these structures also answer queries efficiently. Implicit geometric data structures are encoded as permutations of elements in the data sets, and only zero or O(1) words of extra space is required to support queries. The succinct and implicit data structures surveyed in this article support several fundamental geometric queries and their variants. 1
A LinearTime OptimalLength Encoding of Floorplans with ConstantTime Queries HaoYu Hung ∗ HsuehI Lu†
, 2011
"... A floorplan, which is also known as rectangular drawing, is a division of a rectangle into rectangular faces using horizontal and vertical line segments. Yamanaka and Nakano showed how to encode an nnode floorplan G in 2.5n bits. Chuang reduced the number of bits to at most 2.293n. Takahashi, Fujim ..."
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A floorplan, which is also known as rectangular drawing, is a division of a rectangle into rectangular faces using horizontal and vertical line segments. Yamanaka and Nakano showed how to encode an nnode floorplan G in 2.5n bits. Chuang reduced the number of bits to at most 2.293n. Takahashi, Fujimaki, and Inoue further reduced the number of bits to 2n. In this paper, we give an optimal solution to the problem of encoding G. Specifically, the firstorder term of the length of our encoding is informationtheoretically optimal, implying that our encoding has at most 1.878n + o(n) bits. Just like the previous results, our encoding and decoding algorithms run in O(n) time. Moreover, our encoding supports adjacency, degree, and neighborlisting queries in O(1) time per output. 1
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