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Vertical ray shooting and computing depth orders for fat objects
 In Proc. 17th Annual Symposium on Discrete Algorithms
, 2006
"... Scientific Research (NWO) under project no. 639.023.301. We present new results for three problems dealing with a set P of n constantcomplexity fat objects in 3space. (i) We describe a data structure for vertical ray shooting in P that has O(log 2 n) query time and uses O(n log 2 n) storage. (ii) ..."
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Scientific Research (NWO) under project no. 639.023.301. We present new results for three problems dealing with a set P of n constantcomplexity fat objects in 3space. (i) We describe a data structure for vertical ray shooting in P that has O(log 2 n) query time and uses O(n log 2 n) storage. (ii) We give an algorithm to compute in O(n log 3 n) time a depth order on P, if it exists. (iii) We give an algorithm to verify in O(n log 4 n) time whether a given order on P is a valid depth order. All three results improve on previous results. 1
Delaunay Triangulation of Imprecise Points Simplified and Extended
"... Abstract. Suppose we want to compute the Delaunay triangulation of a set P whose points are restricted to a collection R of input regions known in advance. Building on recent work by Löffler and Snoeyink [21], we show how to leverage our knowledge of R for faster Delaunay computation. Our approach n ..."
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Abstract. Suppose we want to compute the Delaunay triangulation of a set P whose points are restricted to a collection R of input regions known in advance. Building on recent work by Löffler and Snoeyink [21], we show how to leverage our knowledge of R for faster Delaunay computation. Our approach needs no fancy machinery and optimally handles a wide variety of inputs, eg, overlapping disks of different sizes and fat regions. 1
Binary Plane Partitions for Disjoint Line Segments
"... A binary space partition (BSP) for a set of disjoint objects in Euclidean space is a recursive decomposition, where each step partitions the space (and some of the objects) along a hyperplane and recurses on the objects clipped in each of the two open halfspaces. The size of a BSP is defined as the ..."
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A binary space partition (BSP) for a set of disjoint objects in Euclidean space is a recursive decomposition, where each step partitions the space (and some of the objects) along a hyperplane and recurses on the objects clipped in each of the two open halfspaces. The size of a BSP is defined as the number of resulting fragments of the input objects. It is shown that every set of n disjoint line segments in the plane admits a BSP of size O(n log n / log log n). This bound is best possible apart from the constant factor. 1
Linear Binary Space Partitions and Hierarchy of Object Classes
, 2003
"... We consider the problem of constructing binary space partitions for the set P of ddimensional objects in ddimensional space. There are several classes of objects defined for such settings, which support design of effective algorithms. We extend the existing the de Berg hierarchy of classes [8] by ..."
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We consider the problem of constructing binary space partitions for the set P of ddimensional objects in ddimensional space. There are several classes of objects defined for such settings, which support design of effective algorithms. We extend the existing the de Berg hierarchy of classes [8] by the definition of new classes derived from that one and we show desirability of such an extension. Moreover we propose a new algorithm, which works on generalized λlow density scenes [20] (defined in this paper) and provides BSP tree of linear size. The tree can be constructed in O(n log 2 n) time and space, where n is the number of objects. Moreover, we can tradeoff between size and balance of the BSP tree fairly simply.
Binary Space Partitions  Recent Developments
, 2004
"... A binary space partition tree is a data structure for the representation of a set of objectsin space. It found an increasing number of applications over the last decades. In recent years, intensifying research focused on its combinatorial properties, which affect directly the efficiency of applica ..."
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A binary space partition tree is a data structure for the representation of a set of objectsin space. It found an increasing number of applications over the last decades. In recent years, intensifying research focused on its combinatorial properties, which affect directly the efficiency of applications. Important advances were made on binary space partitions for disjoint line segments in the plane and for axisaligned objects in higher dimensions. New research directions were also initiated on some realistic polygonal scenes and on kinetic binary space partitions. This paper attempts to give an overview of these results and reiterates some of the most pressing open problems.
LowEntropy Computational Geometry
, 2010
"... The worstcase model for algorithm design does not always reflect the real world: inputs may have additional structure to be exploited, and sometimes data can be imprecise or become available only gradually. To better understand these situations, we examine several scenarios where additional informa ..."
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The worstcase model for algorithm design does not always reflect the real world: inputs may have additional structure to be exploited, and sometimes data can be imprecise or become available only gradually. To better understand these situations, we examine several scenarios where additional information can affect the design and analysis of geometric algorithms. First, we consider hereditary convex hulls: given a threedimensional convex polytope and a twocoloring of its vertices, we can find the individual monochromatic polytopes in linear expected time. This can be generalized in many ways, eg, to more than two colors, and to the offlineproblem where we wish to preprocess a polytope so that any large enough subpolytope can be found quickly. Our techniques can also be used to give a simple analysis of the selfimproving algorithm for planar Delaunay triangulations by Clarkson and Seshadhri [58]. Next, we assume that the point coordinates have a bounded number of bits, and that we can do standard bit manipulations in constant time. Then Delaunay triangulations can be found in expected time O(n √ log log n). Our result is based on a new connection between quadtrees and Delaunay triangulations, which also lets us generalize a recent result by Löffler and Snoeyink about Delaunay triangulations for imprecise points [110]. Finally, we consider randomized incremental constructions when the input permutation is generated by a boundeddegree Markov chain, and show that the resulting running time is almost optimal for chains with a constant eigenvalue gap.
Algorithmica DOI 10.1007/s0045300790194 Kinetic Collision Detection for Convex Fat Objects
"... Abstract We design compact and responsive kinetic data structures for detecting collisions between n convex fat objects in 3dimensional space that can have arbitrary sizes. Our main results are: (i) If the objects are 3dimensional balls that roll on a plane, then we can detect collisions with a KD ..."
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Abstract We design compact and responsive kinetic data structures for detecting collisions between n convex fat objects in 3dimensional space that can have arbitrary sizes. Our main results are: (i) If the objects are 3dimensional balls that roll on a plane, then we can detect collisions with a KDS of size O(nlog n) that can handle events in O(log2 n) time. This structure processes O(n2) events in the worst case, assuming that the objects follow constantdegree algebraic trajectories. (ii) If the objects are convex fat 3dimensional objects of constant complexity that are freeflying in R3, then we can detect collisions with a KDS of O(nlog6 n) size that can handle events in O(log7 n) time. This structure processes O(n2) events in the worst case, assuming that the objects follow constantdegree algebraic trajectories. If the objects have similar sizes then the size of the KDS becomes O(n) and events can be handled in O(log n) time.