New applications of random sampling in computational geometry
TL;DR: This paper gives several new demonstrations of the usefulness of random sampling techniques in computational geometry by creating a search structure for arrangements of hyperplanes by sampling the hyperplanes and using information from the resulting arrangement to divide and conquer.
Abstract: This paper gives several new demonstrations of the usefulness of random sampling techniques in computational geometry. One new algorithm creates a search structure for arrangements of hyperplanes by sampling the hyperplanes and using information from the resulting arrangement to divide and conquer. This algorithm requiresO(sd+?) expected preprocessing time to build a search structure for an arrangement ofs hyperplanes ind dimensions. The expectation, as with all expected times reported here, is with respect to the random behavior of the algorithm, and holds for any input. Given the data structure, and a query pointp, the cell of the arrangement containingp can be found inO(logs) worst-case time. (The bound holds for any fixed ?>0, with the constant factors dependent ond and ?.) Using point-plane duality, the algorithm may be used for answering halfspace range queries. Another algorithm finds random samples of simplices to determine the separation distance of two polytopes. The algorithm uses expectedO(n[d/2]) time, wheren is the total number of vertices of the two polytopes. This matches previous results [10] for the cased = 3 and extends them. Another algorithm samples points in the plane to determine their orderk Voronoi diagram, and requires expectedO(s1+?k) time fors points. (It is assumed that no four of the points are cocircular.) This sharpens the boundO(sk2 logs) for Lee's algorithm [21], andO(s2 logs+k(s?k) log2s) for Chazelle and Edelsbrunner's algorithm [4]. Finally, random sampling is used to show that any set ofs points inE3 hasO(sk2 log8s/(log logs)6) distinctj-sets withj≤k. (ForS ?Ed, a setS? ?S with |S?| =j is aj-set ofS if there is a half-spaceh+ withS? =S ?h+.) This sharpens with respect tok the previous boundO(sk5) [5]. The proof of the bound given here is an instance of a "probabilistic method" [15].
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01 May 1990
TL;DR: The query problem is solvable in quasiquadratic, and ray shooting queries in randomized expected time and logarithmic preprocessing and storage are solved.
Abstract: In this paper we consider the following problems: given a set T of triangles in 3-space, with |T| = n,answer the query “given a line l, does l stab the set of triangles?” (query problem).find whether a stabbing line exists for the set of triangles (existence problem).Given a ray r, which is the first triangle in T hit by r?The following results are shown.There is an O(n3) lower bound on the descriptive complexity of the set of all stabbers for a set of triangles.The existence problem for triangles on a set of planes with g different plane inclinations can be solved in O(g2n2 log n) time (Theorem 2). The query problem is solvable in quasiquadratic O(n2+e) preprocessing and storage and logarithmic O(log n) query time (Theorem 4).If we are given m rays we can answer ray shooting queries in O(m5/6-δ n5/6+5δ log2 n + m log2 n + n log n log m) randomized expected time and O(m + n) space (Theorem 5).In time O((n+m)5/3+4δ) it is possible to decide whether two non convex polyhedra of complexity m and n intersect (Corollary 1).Given m rays and n axis-oriented boxes we can answer ray shooting queries in randomized expected time O(m3/4-δ n3/4+3δ log4 n + m log4 n + n log n log m) and O(m + n) space (Theorem 6).
41 citations
24 Oct 1992
TL;DR: Using dynamic data structures for half-space range reporting and for maintaining the minima of a decomposable function, the authors obtain efficient dynamic algorithms for a number of geometric problems, including closest/farthest neighbor searching, fixed dimension linear programming, bi-chromatic closest pair, diameter, and Euclidean minimum spanning tree.
Abstract: The authors describe dynamic data structures for half-space range reporting and for maintaining the minima of a decomposable function. Using these data structures, they obtain efficient dynamic algorithms for a number of geometric problems, including closest/farthest neighbor searching, fixed dimension linear programming, bi-chromatic closest pair, diameter, and Euclidean minimum spanning tree. >
41 citations
TL;DR: The results improve the deterministic polynomial-time algorithm of Matoušek and Ramos and the optimal but randomized algorithm of Ramos and lead to efficient derandomization of previous algorithms for numerous well-studied problems in computational geometry.
Abstract: We present optimal deterministic algorithms for constructing shallow cuttings in an arrangement of lines in two dimensions or planes in three dimensions. Our results improve the deterministic polynomial-time algorithm of Matousek (Comput Geom 2(3):169---186, 1992) and the optimal but randomized algorithm of Ramos (Proceedings of the Fifteenth Annual Symposium on Computational Geometry, SoCG'99, 1999). This leads to efficient derandomization of previous algorithms for numerous well-studied problems in computational geometry, including halfspace range reporting in 2-d and 3-d, k nearest neighbors search in 2-d, $$({\le }k)$$(≤k)-levels in 3-d, order-k Voronoi diagrams in 2-d, linear programming with k violations in 2-d, dynamic convex hulls in 3-d, dynamic nearest neighbor search in 2-d, convex layers (onion peeling) in 3-d, $$\varepsilon $$?-nets for halfspace ranges in 3-d, and more. As a side product we also describe an optimal deterministic algorithm for constructing standard (non-shallow) cuttings in two dimensions, which is arguably simpler than the known optimal algorithms by Matousek (Discrete Comput Geom 6(1):385---406, 1991) and Chazelle (Discrete Comput Geom 9(1):145---158, 1993).
40 citations
22 Jan 1995
TL;DR: This paper gives an algorithm for output-sensitive construction of an j-face polytope that is defined by n halfspaces in E4 and is the first algorithm within a polylogarithmic factor of optimal O(nlogf + j) time over the whole range of j.
Abstract: In this paper, we give an algorithm for output-sensitive construction of an j-face polytope that is defined by n halfspaces in E4. Our algorithm runs in O((n + f)log2 f) time and uses O(n + j) space. This is the first algorithm within a polylogarithmic factor of optimal O(nlogf + j) time over the whole range of j. By a standard lifting map, we obtain output-sensitive algorithms for the Voronoi diagram or Delaunay triangulation in E3 and for the portion of a Voronoi diagram that is clipped to a convex polytope. Our approach also simplifies the “ultimate convex hull algorithm” of Kirkpatrick and Seidel in E2.
39 citations
TL;DR: In this paper, a convex hull of a set of n planar convex objects of fixed type m is computed in O(nβ(h,m log h) time.
Abstract: A set of planar objects is said to be of type m if the convex hull of any two objects has its size bounded by 2m. In this paper, we present an algorithm based on the marriage-before-conquest paradigm to compute the convex hull of a set of n planar convex objects of fixed type m. The algorithm is output-sensitive, i.e. its time complexity depends on the size h of the computed convex hull. The main ingredient of this algorithm is a linear method to find a bridge, i.e. a facet of the convex hull intersected by a given line. We obtain an O(nβ(h,m log h)-time convex hull algorithm for planar objects. Here β(h,2)=O(1) and β(h,m) is an extremely slowly growing function. As a direct consequence, we can compute in optimal Θ(n log h) time the convex hull of disks, convex homothets, non-overlapping objects. The method described in this paper also applies to compute lower envelopes of functions. In particular, we obtain an optimal Θ(n log h)-time algorithm to compute the upper envelope of line segments.
39 citations
References
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Book•
01 Jan 1968
TL;DR: The arrangement of this invention provides a strong vibration free hold-down mechanism while avoiding a large pressure drop to the flow of coolant fluid.
Abstract: A fuel pin hold-down and spacing apparatus for use in nuclear reactors is disclosed. Fuel pins forming a hexagonal array are spaced apart from each other and held-down at their lower end, securely attached at two places along their length to one of a plurality of vertically disposed parallel plates arranged in horizontally spaced rows. These plates are in turn spaced apart from each other and held together by a combination of spacing and fastening means. The arrangement of this invention provides a strong vibration free hold-down mechanism while avoiding a large pressure drop to the flow of coolant fluid. This apparatus is particularly useful in connection with liquid cooled reactors such as liquid metal cooled fast breeder reactors.
17,939 citations
01 Jan 1985
TL;DR: This book offers a coherent treatment, at the graduate textbook level, of the field that has come to be known in the last decade or so as computational geometry.
Abstract: From the reviews: "This book offers a coherent treatment, at the graduate textbook level, of the field that has come to be known in the last decade or so as computational geometry...The book is well organized and lucidly written; a timely contribution by two founders of the field. It clearly demonstrates that computational geometry in the plane is now a fairly well-understood branch of computer science and mathematics. It also points the way to the solution of the more challenging problems in dimensions higher than two."
6,525 citations
"New applications of random sampling..." refers background in this paper
...(In fact the mapping γ is not unique in this regard: see [13, 23, 2]....
[...]
Book•
01 Jan 1973
4,243 citations
TL;DR: This chapter reproduces the English translation by B. Seckler of the paper by Vapnik and Chervonenkis in which they gave proofs for the innovative results they had obtained in a draft form in July 1966 and announced in 1968 in their note in Soviet Mathematics Doklady.
Abstract: This chapter reproduces the English translation by B. Seckler of the paper by Vapnik and Chervonenkis in which they gave proofs for the innovative results they had obtained in a draft form in July 1966 and announced in 1968 in their note in Soviet Mathematics Doklady. The paper was first published in Russian as Вапник В. Н. and Червоненкис А. Я. О равномерноЙ сходимости частот появления событиЙ к их вероятностям. Теория вероятностеЙ и ее применения 16(2), 264–279 (1971).
3,939 citations
"New applications of random sampling..." refers background in this paper
...Vapnik and Chervonenkis [27] have derived general conditions under which several probabilities may be uniformly estimated using one random sample....
[...]
Book•
01 Jan 1978TL;DR: In this article, the authors present a coherent treatment of computational geometry in the plane, at the graduate textbook level, and point out the way to the solution of the more challenging problems in dimensions higher than two.
Abstract: From the reviews: "This book offers a coherent treatment, at the graduate textbook level, of the field that has come to be known in the last decade or so as computational geometry...The book is well organized and lucidly written; a timely contribution by two founders of the field. It clearly demonstrates that computational geometry in the plane is now a fairly well-understood branch of computer science and mathematics. It also points the way to the solution of the more challenging problems in dimensions higher than two."
3,419 citations