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29
Efficient Binary Space Partitions for HiddenSurface Removal and Solid Modeling
, 1990
"... We consider schemes for recursively dividing a set of geometric objects by hyperplanes until all objects are separated. Such a binary space partition, or BSP, is naturally considered as a binary tree where each internal node corresponds to a division. The goal is to choose the hyperplanes properly ..."
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Cited by 91 (0 self)
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We consider schemes for recursively dividing a set of geometric objects by hyperplanes until all objects are separated. Such a binary space partition, or BSP, is naturally considered as a binary tree where each internal node corresponds to a division. The goal is to choose the hyperplanes properly so that the size of the BSP, i.e., the number of resulting fragments of the objects, is minimized. For the twodimensional case, we construct BSPs of size O(n log n) for n edges, while in three dimensions, we obtain BSPs of size O(n²) for n planar facets and prove a matching lower bound of Ω(n²). Two applications of efficient BSPs are given. The first is an O(n²)sized data structure for implementing a hiddensurface removal scheme of Fuchs et al. [6]. The second application is in solid modeling: given a polyhedron described by its n faces, we show how to generate an O(n²)sized CSG (constructivesolidgeometry) formula whose literals correspond to halfspaces supporting the faces of the polyhedron. The best previous results for both of these problems were O(n³).
On the convex layers of a planar set
 IEEE Transactions on Information Theory
, 1985
"... AbstractLet S be a set of n points in the Euclidean plane. The convex layers of S are the convex polygons obtained by iterating on the following procedure: compute the convex hull of S and remove its vertices from S. This process of peeling a planar point set is central in the study of robust estim ..."
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Cited by 55 (1 self)
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AbstractLet S be a set of n points in the Euclidean plane. The convex layers of S are the convex polygons obtained by iterating on the following procedure: compute the convex hull of S and remove its vertices from S. This process of peeling a planar point set is central in the study of robust estimators in statistics. It also provides valuable information on the morphology of a set of sites and has proven to be an efficient preconditioning for range search problems. An optimal algorithm is described for computing the convex layers of S. The algorithm runs in O ( n log n) time and requires O(n) space. Also addressed is the problem of determining the depth of a query point within the convex layers of S, i.e., the number of layers that enclose the query point. This is essentially a planar point location problem, for which optimal solutions are therefore known. Taking advantage of structural properties of the problem, however, a much simpler optimal solution is derived. L I.
Dynamic Trees and Dynamic Point Location
 In Proc. 23rd Annu. ACM Sympos. Theory Comput
, 1991
"... This paper describes new methods for maintaining a pointlocation data structure for a dynamicallychanging monotone subdivision S. The main approach is based on the maintenance of two interlaced spanning trees, one for S and one for the graphtheoretic planar dual of S. Queries are answered by using ..."
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Cited by 43 (9 self)
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This paper describes new methods for maintaining a pointlocation data structure for a dynamicallychanging monotone subdivision S. The main approach is based on the maintenance of two interlaced spanning trees, one for S and one for the graphtheoretic planar dual of S. Queries are answered by using a centroid decomposition of the dual tree to drive searches in the primal tree. These trees are maintained via the linkcut trees structure of Sleator and Tarjan, leading to a scheme that achieves vertex insertion/deletion in O(log n) time, insertion/deletion of kedge monotone chains in O(log n + k) time, and answers queries in O(log 2 n) time, with O(n) space, where n is the current size of subdivision S. The techniques described also allow for the dual operations expand and contract to be implemented in O(log n) time, leading to an improved method for spatial pointlocation in a 3dimensional convex subdivision. In addition, the interlacedtree approach is applied to online pointlo...
Approximating Shortest Paths on Weighted Polyhedral Surfaces
"... Shortest path problems are among the... In this paper we propose several simple and practical algorithms (schemes) to compute an approximated weighted shortest path Π'(s, t) points s and t on the surface of a polyhedron P. ..."
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Cited by 29 (6 self)
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Shortest path problems are among the... In this paper we propose several simple and practical algorithms (schemes) to compute an approximated weighted shortest path Π'(s, t) points s and t on the surface of a polyhedron P.
Efficient Visibility Queries in Simple Polygons
"... We present a method of decomposing a simple polygon that allows the preprocessing of the polygon to efficiently answer visibility queries of various forms in an output sensitive manner. Using O(n3 log n) preprocessing time and O(n3) space, we can, given a query point q inside or outside an n verte ..."
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Cited by 24 (2 self)
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We present a method of decomposing a simple polygon that allows the preprocessing of the polygon to efficiently answer visibility queries of various forms in an output sensitive manner. Using O(n3 log n) preprocessing time and O(n3) space, we can, given a query point q inside or outside an n vertex polygon, recover the visibility polygon of q in O(log n + k) time, where k is the size of the visibility polygon, and recover the number of vertices visible from q in O(log n) time. The key notion
Shortest Paths on a Polyhedron, Part I: Computing Shortest Paths
 INTERNATIONAL JOURNAL OF COMPUTATIONAL GEOMETRY & APPLICATIONS
, 1990
"... We present an algorithm for determining the shortest path between any two points along the surface of a polyhedron which need not be convex. This algorithm also computes for any source point on the surface of a polyhedron the inward layout and the subdivision of the polyhedron which can be used for ..."
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Cited by 24 (0 self)
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We present an algorithm for determining the shortest path between any two points along the surface of a polyhedron which need not be convex. This algorithm also computes for any source point on the surface of a polyhedron the inward layout and the subdivision of the polyhedron which can be used for processing queries of shortest paths between the source point and any destination point. Our algorithm uses a new approach which deviates from the conventional "continuous Dijkstra" technique. Our algorithm has time complexity O(n²) and space complexity \Theta (n).
Methods for Achieving Fast Query Times in Point Location Data Structures
, 1997
"... Given a collection S of n line segments in the plane, the planar point location problem is to construct a data structure that can efficiently determine for a given query point p the first segment(s) in S intersected by vertical rays emanating out from p. It is well known that linearspace data struc ..."
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Cited by 21 (1 self)
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Given a collection S of n line segments in the plane, the planar point location problem is to construct a data structure that can efficiently determine for a given query point p the first segment(s) in S intersected by vertical rays emanating out from p. It is well known that linearspace data structures can be constructed so as to achieve O(log n) query times. But applications, such as those common in geographic information systems, motivate a reexamination of this problem with the goal of improving query times further while also simplifying the methods needed to achieve such query times. In this paper we perform such a reexamination, focusing on the issues that arise in three different classes of pointlocation query sequences: ffl sequences that are reasonably uniform spatially and temporally (in which case the constant factors in the query times become critical), ffl sequences that are nonuniform spatially or temporally (in which case one desires data structures that adapt to s...
New Results on Binary Space Partitions in the Plane
 COMPUT. GEOM. THEORY APPL
, 1994
"... We prove the existence of linear size binary space partitions for sets of objects in the plane under certain conditions that are often satisfied in practical situations. In particular, we construct linear size binary space partitions for sets of fat objects, for sets of line segments where the ra ..."
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Cited by 19 (6 self)
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We prove the existence of linear size binary space partitions for sets of objects in the plane under certain conditions that are often satisfied in practical situations. In particular, we construct linear size binary space partitions for sets of fat objects, for sets of line segments where the ratio between the lengths of the longest and shortest segment is bounded by a constant, and for homothetic objects. For all cases we also show how to turn the existence proofs into efficient algorithms.
Nearly Optimal ExpectedCase Planar Point Location
"... We consider the planar point location problem from the perspective of expected search time. We are given a planar polygonal subdivision S and for each polygon of the subdivision the probability that a query point lies within this polygon. The goal is to compute a search structure to determine which ..."
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Cited by 17 (5 self)
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We consider the planar point location problem from the perspective of expected search time. We are given a planar polygonal subdivision S and for each polygon of the subdivision the probability that a query point lies within this polygon. The goal is to compute a search structure to determine which cell of the subdivision contains a given query point, so as to minimize the expected search time. This is a generalization of the classical problem of computing an optimal binary search tree for onedimensional keys. In the onedimensional case it has long been known that the entropy H of the distribution is the dominant term in the lower bound on the expectedcase search time, and further there exist search trees achieving expected search times of at most H + 2. Prior to this work, there has been no known structure for planar point location with an expected search time better than 2H, and this result required strong assumptions on the nature of the query point distribution. Here we present a data structure whose expected search time is nearly equal to the entropy lower bound, namely H + o(H). The result holds for any polygonal subdivision in which the number of sides of each of the polygonal cells is bounded, and there are no assumptions on the query distribution within each cell. We extend these results to subdivisions with convex cells, assuming a uniform query distribution within each cell.
Dynamization of the Trapezoid Method for Planar Point Location
, 1991
"... We present a fully dynamic data structure for point location in a monotone subdivision, based on the trapezoid method. The operations supported are insertion and deletion of vertices and edges, and horizontal translation of vertices. Let n be the current number of vertices of the subdivision. Point ..."
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Cited by 14 (4 self)
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We present a fully dynamic data structure for point location in a monotone subdivision, based on the trapezoid method. The operations supported are insertion and deletion of vertices and edges, and horizontal translation of vertices. Let n be the current number of vertices of the subdivision. Point location queries take O(log n) time, while updates take O(log2 n) time. The space requirement is O(n log n). This is the first fully dynamic point location data structure for monotone subdivisions that achieves optimal query time.