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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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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.
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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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(lg 2 n) time. 1
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 2 (2 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.
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 inpu ..."
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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"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 "inplace data structures " 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
Spaceefficient Algorithms for Empty Space Recognition among a Point Set in 2D and 3D
"... In this paper, we consider the problem of designing inplace algorithms for computing the maximum area empty rectangle of arbitrary orientation among a set of points in 2D, and the maximum volume empty axisparallel cuboid among a set of points in 3D. If n points are given in an array of size n, the ..."
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In this paper, we consider the problem of designing inplace algorithms for computing the maximum area empty rectangle of arbitrary orientation among a set of points in 2D, and the maximum volume empty axisparallel cuboid among a set of points in 3D. If n points are given in an array of size n, the worst case time complexity of our proposed algorithms for both the problems is O(n 3); both the algorithms use O(1) extra space in addition to the array containing the input points. 1
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.
An InPlace MinMax Priority Search Tree ∗
"... One of the classic data structures for storing point sets in R 2 is the priority search tree, introduced by McCreight in 1985. We show that this data structure can be made inplace, i.e., it can be stored in an array such that each entry stores only one point of the point set and no entry is stored ..."
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One of the classic data structures for storing point sets in R 2 is the priority search tree, introduced by McCreight in 1985. We show that this data structure can be made inplace, i.e., it can be stored in an array such that each entry stores only one point of the point set and no entry is stored in more than one location of that array. It combines a binary search tree with a heap. We show that all the standard query operations can be answered within the same time bounds as for the original priority search tree, while using only O(1) extra space. We introduce the minmax priority search tree which is a combination of a binary search tree and a minmax heap. We show that all the standard queries which can be done in two separate versions of a priority search tree can be done with a single minmax priority search tree. As an application, we present an inplace algorithm to enumerate all maximal empty axisparallel rectangles amongst points in a rectangular region R in R 2 in O(m log n) time with O(1) extraspace, where m is the total number of maximal empty rectangles. 1