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17
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 22 (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...
Computing faces in segment and simplex arrangements
 In Proc. 27th Annu. ACM Sympos. Theory Comput
, 1995
"... For a set S of n line segments in the plane, we give the first workoptimal deterministic parallel algorithm for constructing their arrangement. It runs in O(log 2 n) time using O(n log n + k) work in the EREW PRAM model, where k is the number of intersecting line segment pairs, and provides a fairl ..."
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Cited by 16 (10 self)
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For a set S of n line segments in the plane, we give the first workoptimal deterministic parallel algorithm for constructing their arrangement. It runs in O(log 2 n) time using O(n log n + k) work in the EREW PRAM model, where k is the number of intersecting line segment pairs, and provides a fairly simple divideandconquer alternative to the optimal sequential “planesweep ” algorithm of Chazelle and Edelsbrunner. Moreover, our method can be used to output all k intersecting pairs while using only O(n) working space, which solves an open problem posed by Chazelle and Edelsbrunner. We also describe a sequential algorithm for computing a single face in an arrangement of n line segments that runs in O(n 2 (n) log n) time, which improves on a previous O(n log 2 n) time algorithm. For collections of simplices in IR d, we give methods for constructing a set of m = O(n d,1 log c n+k) cells of constant descriptive complexity that covers their arrangement, where c> 1 is a constant and k is the number of faces in the arrangement. The construction is performed sequentially in O(m) time, or in O(log n) time using O(m) work in the EREW PRAM model. The covering can be augmented to answer point location queries in O(log n) time. In addition to supplying the first parallel methods for these problems, we improve on the previous best sequential methods by reducing the query times (from O(log 2 n) in IR 3 and O(log 3 n) in IR d, d>3), and also the size and construction cost of the covering (from O(n d,1+ + k)). 1
LinearTime Triangulation of a Simple Polygon Made Easier Via Randomization
 In Proc. 16th Annu. ACM Sympos. Comput. Geom
, 2000
"... We describe a randomized algorithm for computing the trapezoidal decomposition of a simple polygon. Its expected running time is linear in the size of the polygon. By a wellknown and simple linear time reduction, this implies a linear time algorithm for triangulating a simple polygon. Our algorithm ..."
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Cited by 10 (0 self)
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We describe a randomized algorithm for computing the trapezoidal decomposition of a simple polygon. Its expected running time is linear in the size of the polygon. By a wellknown and simple linear time reduction, this implies a linear time algorithm for triangulating a simple polygon. Our algorithm is considerably simpler than Chazelle's (1991) celebrated optimal deterministic algorithm and, hence, positively answers his question of whether a simpler randomized algorithm for the problem exists. The new algorithm can be viewed as a combination of Chazelle's algorithm and of nonoptimal randomized algorithms due to Clarkson et al. (1991) and to Seidel (1991), with the essential innovation that sampling is performed on subchains of the initial polygonal chain, rather than on its edges. It is also essential, as in Chazelle's algorithm, to include a bottomup preprocessing phase previous to the topdown construction phase. 1 Introduction Polygon triangulation is a classic problem in comp...
Guarding Scenes against Invasive Hypercubes
, 1998
"... A set of points G is a guarding set for a set of objects O, if any hypercube not containing a point from G in its interior intersects at most objects of O. This definition underlies a new input model, that is both more general than de Berg's unclutteredness, and retains its main property: ..."
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Cited by 9 (3 self)
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A set of points G is a guarding set for a set of objects O, if any hypercube not containing a point from G in its interior intersects at most objects of O. This definition underlies a new input model, that is both more general than de Berg's unclutteredness, and retains its main property: a ddimensional scene satisfying the new model's requirements is known to have a linearsize binary space partition. We propose several algorithms for computing guarding sets, and evaluate them experimentally. One of them appears to be quite practical. 1. Introduction Recently de Berg et al. [4] brought together several of the realistic input models that have been proposed in the literature, namely fatness, low density, unclutteredness, and small simplecover 1 Supported by the ESPRIT IV LTR Project No. 21957 (CGAL). 2 Supported by the Israel Science Foundation founded by the Israel Academy of Sciences and Humanities. 3 Supported by the Netherlands' Organization for Scientific Research...
Approximation of MultiColor Discrepancy
 Randomization, Approximation and Combinatorial Optimization (Proceedings of APPROXRANDOM 1999), volume 1671 of Lecture Notes in Computer Science
, 1999
"... . In this article we introduce (combinatorial) multicolor discrepancy and generalize some classical results from 2color discrepancy theory to c colors. We give a recursive method that constructs ccolorings from approximations to the 2color discrepancy. This method works for a large class of ..."
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Cited by 8 (7 self)
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. In this article we introduce (combinatorial) multicolor discrepancy and generalize some classical results from 2color discrepancy theory to c colors. We give a recursive method that constructs ccolorings from approximations to the 2color discrepancy. This method works for a large class of theorems like the sixstandarddeviation theorem of Spencer, the BeckFiala theorem and the results of Matousek, Welzl and Wernisch for bounded VCdimension. On the other hand there are examples showing that discrepancy in c colors can not be bounded in terms of twocolor discrepancy even if c is a power of 2. For the linear discrepancy version of the BeckFiala theorem the recursive approach also fails. Here we extend the method of floating colors to multicolorings and prove multicolor versions of the the BeckFiala theorem and the BaranyGrunberg theorem. 1 Introduction Combinatorial discrepancy theory deals with the problem of partitioning the vertices of a hypergraph (set...
Models and Motion Planning
, 1998
"... We study the complexity of the motion planning problem for a boundedreach robot in the situation where the n obstacles in its workspace satisfy two of the realistic models proposed in the literature, namely unclutteredness and small simplecover complexity. We show that the maximum complexity of ..."
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Cited by 8 (2 self)
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We study the complexity of the motion planning problem for a boundedreach robot in the situation where the n obstacles in its workspace satisfy two of the realistic models proposed in the literature, namely unclutteredness and small simplecover complexity. We show that the maximum complexity of the free space of a robot with f degrees of freedom in the plane is #(n f/2 + n) for uncluttered environments as well as environments with small simplecover complexity. The maximum complexity of the free space of a robot moving in a threedimensional uncluttered environment is #(n 2f/3 +n). All these bounds fit nicely between the #(n) bound for the maximum freespace complexity for lowdensity environments and the #(n f ) bound for unrestricted environments. Surprisinglybecause contrary to the situation in the planethe maximum freespace complexity is #(n f ) for a threedimensional environment with small simplecover complexity.
A Randomized Algorithm for Triangulating a Simple Polygon in Linear Time
 DISCRETE & COMPUTATIONAL GEOMETRY
, 2001
"... We describe a randomized algorithm for computing the trapezoidal decomposition of a simple polygon. Its expected running time is linear in the size of the polygon. By a wellknown and simple linear time reduction, this implies a linear time algorithm for triangulating a simple polygon. Our algorithm ..."
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Cited by 8 (2 self)
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We describe a randomized algorithm for computing the trapezoidal decomposition of a simple polygon. Its expected running time is linear in the size of the polygon. By a wellknown and simple linear time reduction, this implies a linear time algorithm for triangulating a simple polygon. Our algorithm is considerably simpler than Chazelle's (1991) celebrated optimal deterministic algorithm. The new algorithm can be viewed as a combination of Chazelle's algorithm and of simple nonoptimal randomized algorithms due to Clarkson et al. (1991) and to Seidel (1991). As in Chazelle's algorithm, it is indispensable to include a bottomup preprocessing phase, in addition to the actual topdown construction. An essential new idea is the use of random sampling on subchains of the initial polygonal chain, rather than on individual edges as is normally done.
FixedDimensional Parallel Linear Programming via Relative ɛApproximations
"... We show that linear programming in IRd can be solved deterministically in O((loglogn)d) time using linear work in the PRAM model of computation, for any fixed constant d. Our method is developed for the CRCW variant of the PRAM parallel computation model, and can be easily implemented to run in O(lo ..."
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Cited by 7 (1 self)
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We show that linear programming in IRd can be solved deterministically in O((loglogn)d) time using linear work in the PRAM model of computation, for any fixed constant d. Our method is developed for the CRCW variant of the PRAM parallel computation model, and can be easily implemented to run in O(logn(loglogn)dl) time using linear work on an EREW PRAM. A key component in these algorithms is a new, efficient parallel method for constructing ɛnets and ɛapproximations (which have wide applicability in computational geometry). In addition, we introduce a new deterministic set approximation for range spaces with finite VCexponent, which we call the brelative ɛapproximation, and we show how such approximations can be efficiently constructed in parallel.