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17
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 78 (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...
Geometric Range Searching
, 1994
"... In geometric range searching, algorithmic problems of the following type are considered: Given an npoint set P in the plane, build a data structure so that, given a query triangle R, the number of points of P lying in R can be determined quickly. Problems of this type are of crucial importance in c ..."
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Cited by 46 (2 self)
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In geometric range searching, algorithmic problems of the following type are considered: Given an npoint set P in the plane, build a data structure so that, given a query triangle R, the number of points of P lying in R can be determined quickly. Problems of this type are of crucial importance in computational geometry, as they can be used as subroutines in many seemingly unrelated algorithms. We present a survey of results and main techniques in this area.
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 44 (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...
Computing envelopes in four dimensions with applications
 SIAM J. Comput
, 1997
"... Abstract. Let F be a collection of ndvariate, possibly partially defined, functions, all algebraic of some constant maximum degree. We present a randomized algorithm that computes the vertices, edges, and 2faces of the lower envelope (i.e., pointwise minimum) of F in expected time O(n d+ε) for any ..."
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Cited by 41 (19 self)
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Abstract. Let F be a collection of ndvariate, possibly partially defined, functions, all algebraic of some constant maximum degree. We present a randomized algorithm that computes the vertices, edges, and 2faces of the lower envelope (i.e., pointwise minimum) of F in expected time O(n d+ε) for any ε>0. For d = 3, by combining this algorithm with the pointlocation technique of Preparata and Tamassia, we can compute, in randomized expected time O(n 3+ε), for any ε>0, a data structure of size O(n 3+ε) that, for any query point q, can determine in O(log 2 n) time the function(s) of F that attain the lower envelope at q. As a consequence, we obtain improved algorithmic solutions to several problems in computational geometry, including (a) computing the width of a point set in 3space, (b) computing the “biggest stick ” in a simple polygon in the plane, and (c) computing the smallestwidth annulus covering a planar point set. The solutions to these problems run in randomized expected time O(n 17/11+ε), for any ε>0, improving previous solutions that run in time O(n 8/5+ε). We also present data structures for (i) performing nearestneighbor and related queries for fairly general collections of objects in 3space and for collections of moving objects in the plane and (ii) performing rayshooting and related queries among n spheres or more general objects in 3space. Both of these data structures require O(n 3+ε) storage and preprocessing time, for any ε>0, and support polylogarithmictime queries. These structures improve previous solutions to these problems.
Constructing Approximate Shortest Path Maps in Three Dimensions
"... We define two results on approximate shortest path maps in IR 3 . (i) Given a polyhedral surface or a convex polytope P with n edges in IR 3 , a source point s on P , and a real parameter 0 ! " 1, we present an algorithm that computes a subdivision of P of size O((n=") log(1=")) ..."
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Cited by 30 (6 self)
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We define two results on approximate shortest path maps in IR 3 . (i) Given a polyhedral surface or a convex polytope P with n edges in IR 3 , a source point s on P , and a real parameter 0 ! " 1, we present an algorithm that computes a subdivision of P of size O((n=") log(1=")) which can be used to answer efficiently approximate shortest path queries. Namely, given any point t on P , one can compute, in O(log (n=")) time, a distance \Delta P;s (t), such that dP;s (t) \Delta P;s (t) (1 + ")d P;s (t), where dP;s (t) is the length of a shortest path between s and t on P . The map can be computed in O(n 2 log n + (n=") log (1=") log (n=")) time, for the case of a polyhedral surface, and in O((n=" 3 ) log(1=") + (n=" 1:5 ) log (1=") log n) time if P is a convex polytope. (ii) Given a set of polyhedral obstacles O with a total of n edges in IR 3 , a source point s in IR 3 n int S O2O O, and a real parameter 0 ! " 1, we present an algorithm that computes a subdivision o...
TwoPoint Euclidean Shortest Path Queries in the Plane (Extended Abstract)
, 1999
"... ) To appear in Proc. Tenth Annual ACMSIAM Symposium on Discrete Algorithms (SODA '99), January 1719, 1999 YiJen Chiang Joseph S. B. Mitchell y Abstract We consider the twopoint query version of the fundamental geometric shortest path problem: Given a set h of polygonal obstacles in th ..."
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Cited by 18 (2 self)
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) To appear in Proc. Tenth Annual ACMSIAM Symposium on Discrete Algorithms (SODA '99), January 1719, 1999 YiJen Chiang Joseph S. B. Mitchell y Abstract We consider the twopoint query version of the fundamental geometric shortest path problem: Given a set h of polygonal obstacles in the plane, having a total of n vertices, build a data structure such that for any two query points s and t we can efficiently determine the length, d(s; t), of an Euclidean shortest obstacleavoiding path, ß(s; t), from s to t. Additionally, our data structure should allow one to report the path ß(s; t), in time proportional to its (combinatorial) size. We present various methods for solving this twopoint query problem, including algorithms with o(n), O(log n+h), O(h log n), O(log 2 n) or optimal O(log n) query times, using polynomialspace data structures, with various tradeoffs between space and query time. While several results have been known for approximate twopoint Euclidean shortest p...
Drawing Nice Projections of Objects in Space
, 1995
"... Given a polygonal object (simple polygon, geometric graph, wireframe, skeleton or more generally a set of line segments) in three dimensional Euclidean space, we consider the problem of computing a variety of "nice" parallel (orthographic) projections of the object. We show that given a g ..."
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Cited by 18 (8 self)
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Given a polygonal object (simple polygon, geometric graph, wireframe, skeleton or more generally a set of line segments) in three dimensional Euclidean space, we consider the problem of computing a variety of "nice" parallel (orthographic) projections of the object. We show that given a general polygonal object consisting of n line segments in space, deciding whether it admits a crossingfree projection can be done in O(n 2 log n+k) time and O(n 2 +k) space, where k is the number of edge intersections of forbidden quadrilaterals (i.e. set of directions that admits a crossing) and varies from zero to O(n 4 ). This implies for example that given a simple polygon in 3space we can determine if there exists a plane on which the projection is a simple polygon, within the same complexity. Furthermore, if such a projection does not exist, a minimumcrossing projection can be found in O(n 4 ) time and space. We show that an object always admits a regular projection (of interest to k...
Motion Planning for a Convex Polygon in a Polygonal Environment
 Geom
, 1997
"... We study the motionplanning problem for a convex mgon P in a planar polygonal environment Q bounded by n edges. We give the first algorithm that constructs the entire free configuration space (the 3dimensional space of all free placements of P in Q) in time that is nearquadratic in mn, which i ..."
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Cited by 13 (7 self)
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We study the motionplanning problem for a convex mgon P in a planar polygonal environment Q bounded by n edges. We give the first algorithm that constructs the entire free configuration space (the 3dimensional space of all free placements of P in Q) in time that is nearquadratic in mn, which is nearly optimal in the worst case. The algorithm is also conceptually relatively simple. Previous solutions were incomplete, more expensive, or produced only part of the free configuration space. Combining our solution with parametric searching, we obtain an algorithm that finds the largest placement of P in Q in time that is also nearquadratic in mn. In addition, we describe an algorithm that preprocesses the computed free configuration space so that `reachability' queries can be answered in polylogarithmic time. All three authors have been supported by a grant from the U.S.Israeli Binational Science Foundation. Pankaj Agarwal has also been supported by a National Science Foundation Gr...
Robust Proximity Queries in Implicit Voronoi Diagrams
 IN PROC. 8TH CANAD. CONF. COMPUT. GEOM
, 1996
"... In the context of methodologies intended to confer robustness to geometric algorithms, we elaborate on the exact computation paradigm and formalize the notion of degree of a geometric algorithm, as a worstcase quantification of the precision (number of bits) to which arithmetic calculation have ..."
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Cited by 11 (3 self)
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In the context of methodologies intended to confer robustness to geometric algorithms, we elaborate on the exact computation paradigm and formalize the notion of degree of a geometric algorithm, as a worstcase quantification of the precision (number of bits) to which arithmetic calculation have to be executed in order to guarantee topological correctness. We also propose a formalism for the expeditious evaluation of algorithmic degree. As an application of this paradigm and an illustration of our general approach, we consider the important classical problem of proximity queries in 2 and 3 dimensions, and develop a new technique for the efficient and robust execution of such queries based on an implicit representation of Voronoi diagrams. Our new technique gives both low degree and fast query time, and for 2D queries is optimal with respect to both cost measures of the paradigm, asymptotic number of operations and arithmetic degree.