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DavenportSchinzel Sequences and Their Geometric Applications
, 1998
"... An (n; s) DavenportSchinzel sequence, for positive integers n and s, is a sequence composed of n distinct symbols with the properties that no two adjacent elements are equal, and that it does not contain, as a (possibly noncontiguous) subsequence, any alternation a \Delta \Delta \Delta b \Delta \ ..."
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Cited by 476 (123 self)
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An (n; s) DavenportSchinzel sequence, for positive integers n and s, is a sequence composed of n distinct symbols with the properties that no two adjacent elements are equal, and that it does not contain, as a (possibly noncontiguous) subsequence, any alternation a \Delta \Delta \Delta b \Delta \Delta \Delta a \Delta \Delta \Delta b \Delta \Delta \Delta of length s + 2 between two distinct symbols a and b. The close relationship between DavenportSchinzel sequences and the combinatorial structure of lower envelopes of collections of functions make the sequences very attractive because a variety of geometric problems can be formulated in terms of lower envelopes. A nearlinear bound on the maximum length of DavenportSchinzel sequences enable us to derive sharp bounds on the combinatorial structure underlying various geometric problems, which in turn yields efficient algorithms for these problems.
Realistic Input Models for Geometric Algorithms
 IN PROC. 13TH ANNU. ACM SYMPOS. COMPUT. GEOM
, 1997
"... Many algorithms developed in computational geometry are needlessly complicated and slow because they have to be prepared for very complicated, hypothetical inputs. To avoid this, realistic models are needed that describe the properties that realistic inputs have, so that algorithms can de designed t ..."
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Cited by 100 (20 self)
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Many algorithms developed in computational geometry are needlessly complicated and slow because they have to be prepared for very complicated, hypothetical inputs. To avoid this, realistic models are needed that describe the properties that realistic inputs have, so that algorithms can de designed that take advantage of these properties. This can lead to algorithms that are provably efficient in realistic situations. We obtain some fundamental results in this research direction. In particular, we have the following results. ffl We show the relations between various models that have been proposed in the literature. ffl For several of these models, we give algorithms to compute the model parameter(s) for a given scene; these algorithms can be used to verify whether a model is appropriate for typical scenes in some application area. ffl As a case study, we give some experimental results on the appropriateness of some of the models for one particular type of scenes often encountered in ...
Arrangements and Their Applications
 Handbook of Computational Geometry
, 1998
"... The arrangement of a finite collection of geometric objects is the decomposition of the space into connected cells induced by them. We survey combinatorial and algorithmic properties of arrangements of arcs in the plane and of surface patches in higher dimensions. We present many applications of arr ..."
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Cited by 89 (20 self)
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The arrangement of a finite collection of geometric objects is the decomposition of the space into connected cells induced by them. We survey combinatorial and algorithmic properties of arrangements of arcs in the plane and of surface patches in higher dimensions. We present many applications of arrangements to problems in motion planning, visualization, range searching, molecular modeling, and geometric optimization. Some results involving planar arrangements of arcs have been presented in a companion chapter in this book, and are extended in this chapter to higher dimensions. Work by P.A. was supported by Army Research Office MURI grant DAAH049610013, by a Sloan fellowship, by an NYI award, and by a grant from the U.S.Israeli Binational Science Foundation. Work by M.S. was supported by NSF Grants CCR9122103 and CCR9311127, by a MaxPlanck Research Award, and by grants from the U.S.Israeli Binational Science Foundation, the Israel Science Fund administered by the Israeli Ac...
On approximating the depth and related problems
 SIAM J. COMPUT
, 2008
"... We study the question of finding a deepest point in an arrangement of regions, and provide a fast algorithm for this problem using random sampling, showing it sufficient to solve this problem when the deepest point is shallow. This implies, among other results, a fast algorithm for solving linear pr ..."
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Cited by 72 (14 self)
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We study the question of finding a deepest point in an arrangement of regions, and provide a fast algorithm for this problem using random sampling, showing it sufficient to solve this problem when the deepest point is shallow. This implies, among other results, a fast algorithm for solving linear programming with violations approximately. We also use this technique to approximate the disk covering the largest number of red points, while avoiding all the blue points, given two such sets in the plane. Using similar techniques imply that approximate range counting queries have roughly the same time and space complexity as emptiness range queries.
Spheres, Molecules, and Hidden Surface Removal
, 1996
"... We devise techniques to manipulate a collection of loosely interpenetrating spheres in threedimensional space. Our study is motivated by the representation and manipulation of molecular con gurations, modeled by a collection of spheres. We analyze the sphere model and point toitsfavorable properties ..."
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Cited by 50 (13 self)
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We devise techniques to manipulate a collection of loosely interpenetrating spheres in threedimensional space. Our study is motivated by the representation and manipulation of molecular con gurations, modeled by a collection of spheres. We analyze the sphere model and point toitsfavorable properties that make it more easy to manipulate than an arbitrary collection of spheres. For this special sphere model we present e cient algorithms for computing its union boundary and for hidden surface removal. The e ciency and practicality of our approach are demonstrated by experiments on actual molecule data.
Analysis of a bounding box heuristic for object intersection
 Journal of the ACM
, 1999
"... Abstract. Bounding boxes are commonly used in computer graphics and other fields to improve the performance of algorithms that should process only the intersecting objects. A boundingboxbased heuristic avoids unnecessary intersection processing by eliminating the pairs whose bounding boxes are dis ..."
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Cited by 33 (4 self)
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Abstract. Bounding boxes are commonly used in computer graphics and other fields to improve the performance of algorithms that should process only the intersecting objects. A boundingboxbased heuristic avoids unnecessary intersection processing by eliminating the pairs whose bounding boxes are disjoint. Empirical evidence suggests that the heuristic works well in many practical applications, although its worstcase performance can be bad for certain pathological inputs. What is a pathological input, however, is not well understood, and consequently there is no guarantee that the heuristic will always work well in a specific application. In this paper, we analyze the performance of bounding box heuristic in terms of two natural shape parameters, aspect ratio and scale factor. These parameters can be used to realistically measure the degree to which the objects are pathologically shaped. We derive tight worstcase bounds on the performance for bounding box heuristic. One of the significant contributions of our paper is that we only require that objects be well shaped on average. Somewhat surprisingly, the bounds are significantly different from the case when all objects are well shaped.
On Fat Partitioning, Fat Covering and the Union Size of Polygons
, 1993
"... The complexity of the contour of the union of simple polygons with n vertices in total can be O(n 2) in general. A notion of fatness for simple polygons is introduced, which extends most of the existing fatness definitions. It is proved that a set of fat polygons with n vertices in total has unio ..."
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Cited by 32 (2 self)
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The complexity of the contour of the union of simple polygons with n vertices in total can be O(n 2) in general. A notion of fatness for simple polygons is introduced, which extends most of the existing fatness definitions. It is proved that a set of fat polygons with n vertices in total has union complexity is O(nloglogn), which is a generalization of a similar result for fat triangles [19]. Applications to several basic problems in computational geometry are given, such as efficient hidden surface removal, motion planning, injection molding, etc. The result is based on a new method to partition a fat simple polygon P with n vertices into O(n) fat convex quadrilaterals, and a method to cover (but not partition) a fat convex quadrilateral with O(1) fat triangles. The maximum overlap of the triangles at any point is two, which is optimal for any coveting of a fat simple polygon by a linear number of fat triangles.
The complexity of the union of (α, β)covered objects
 SIAM J. Comput
"... An (α, β)covered object is a simply connected planar region c with the property that for each point p ∈ ∂c there exists a triangle contained in c and having p as a vertex, such that all its angles are at least α and all its edges are at least β ·diam(c)long. This notion extends that of fat convex ..."
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Cited by 31 (2 self)
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An (α, β)covered object is a simply connected planar region c with the property that for each point p ∈ ∂c there exists a triangle contained in c and having p as a vertex, such that all its angles are at least α and all its edges are at least β ·diam(c)long. This notion extends that of fat convex objects. We show that the combinatorial complexity of the union of n (α, β)covered objects of ‘constant description complexity ’ is O(λs+2(n) log 2 n log logn), where s is the maximum number of intersections between the boundaries of any pair of the given objects. 1
Linear Size Binary Space Partitions for Uncluttered Scenes
 Algorithmica
, 1998
"... We describe a new and simple method for constructing binary space partitions in arbitrary dimensions. We also introduce the concept of uncluttered scenes, which are scenes with a certain property that we suspect many realistic scenes exhibit, and we show that our method constructs a BSP of size O ..."
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Cited by 31 (8 self)
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We describe a new and simple method for constructing binary space partitions in arbitrary dimensions. We also introduce the concept of uncluttered scenes, which are scenes with a certain property that we suspect many realistic scenes exhibit, and we show that our method constructs a BSP of size O(n) for an uncluttered scene consisting of n objects. The construction time is O(n log n). Because any set of disjoint fat objects is uncluttered, our result implies an efficient method to construct a linear size BSP for fat objects. We use our BSP to develop a data structure for point location in uncluttered scenes. The query time of our structure is O(log n), and the amount of storage is O(n). This result can in turn be used to perform range queries with nottoosmall ranges in scenes consisting of disjoint fat objects or, more generally, in socalled lowdensity scenes. 1 Introduction Many geometric problems can be solved more easily if a decomposition of the space of interest in...
Computing Depth Orders and Related Problems
 IN PROC. 4TH SCAND. WORKSHOP ALGORITHM THEORY
, 1994
"... Let K be a set of n nonintersecting objects in 3space. A depth order of K, if exists, is a linear order ! of the objects in K such that if K;L 2 K and K lies vertically below L then K ! L. We present a new technique for computing depth orders, and apply it to several special classes of objects. ..."
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Cited by 26 (11 self)
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Let K be a set of n nonintersecting objects in 3space. A depth order of K, if exists, is a linear order ! of the objects in K such that if K;L 2 K and K lies vertically below L then K ! L. We present a new technique for computing depth orders, and apply it to several special classes of objects. Our results include: (i) If K is a set of n triangles whose xyprojections are all `fat', then a depth order for K can be computed in time O(n log 5 n). (ii) If K is a set of n convex and simplyshaped objects whose xyprojections are all `fat' and their sizes are within a constant ratio from one another, then a depth order for K can be computed in time O(n 1=2 s (n) log 4 n), where s is the maximum number of intersections between the boundaries of the xyprojections of any pair of objects in K, and s (n) is the maximum length of (n; s) DavenportSchinzel sequences.