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Decision Trees For Geometric Models
, 1993
"... A fundamental problem in modelbased computer vision is that of identifying which of a given set of geometric models is present in an image. Considering a "probe" to be an oracle that tells us whether or not a model is present at a given point, we study the problem of computing efficien ..."
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Cited by 37 (4 self)
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A fundamental problem in modelbased computer vision is that of identifying which of a given set of geometric models is present in an image. Considering a "probe" to be an oracle that tells us whether or not a model is present at a given point, we study the problem of computing efficient strategies ("decision trees") for probing an image, with the goal to minimize the number of probes necessary (in the worst case) to determine which single model is present. We show that a dlg ke height binary decision tree always exists for k polygonal models (in fixed position), provided (1) they are nondegenerate (do not share boundaries) and (2) they share a common point of intersection. Further, we give an efficient algorithm for constructing such decision tress when the models are given as a set of polygons in the plane. We show that constructing a minimum height tree is NPcomplete if either of the two assumptions is omitted. We provide an efficient greedy heuristic strategy and show ...
Randomized Parallel Algorithms For Trapezoidal Diagrams
, 1992
"... We describe randomized parallel algorithms for building trapezoidal diagrams of line segments in the plane. The algorithms are designed for a CRCW PRAM. For general segments, we give an algorithm requiring optimal O(A + n log n) expected work and optimal O(logn) time, where A is the number of inters ..."
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Cited by 24 (0 self)
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We describe randomized parallel algorithms for building trapezoidal diagrams of line segments in the plane. The algorithms are designed for a CRCW PRAM. For general segments, we give an algorithm requiring optimal O(A + n log n) expected work and optimal O(logn) time, where A is the number of intersecting pairs of segments. If the segments form a simple chain, we give an algorithm requiring optimal O(n) expected work and O(logn log log n log n) expected time a , and a simpler algorithm requiring O(n log n) expected work. The serial algorithm corresponding to the latter is among the simplest known algorithms requiring O(n log n) expected operations. For a set of segments forming K chains, we give an algorithm requiring O(A + n log n + K log n) expected work and O(logn log log n log n) expected time. The parallel time bounds require the assumption that enough processors are available, with processor allocations every log n steps. Keywords: randomized, parallel, trapez...
Efficient ExpectedCase Algorithms for Planar Point Location
, 2000
"... . Planar point location is among the most fundamental search problems in computational geometry. Although this problem has been heavily studied from the perspective of worstcase query time, there has been surprisingly little theoretical work on expectedcase query time. We are given an nvertex ..."
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Cited by 15 (4 self)
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. Planar point location is among the most fundamental search problems in computational geometry. Although this problem has been heavily studied from the perspective of worstcase query time, there has been surprisingly little theoretical work on expectedcase query time. We are given an nvertex planar polygonal subdivision S satisfying some weak assumptions (satisfied, for example, by all convex subdivisions). We are to preprocess this into a data structure so that queries can be answered efficiently. We assume that the two coordinates of each query point are generated independently by a probability distribution also satisfying some weak assumptions (satisfied, for example, by the uniform distribution). In the decision tree model of computation, it is wellknown from information theory that a lower bound on the expected number of comparisons is entropy(S). We provide two data structures, one of size O(n 2 ) that can answer queries in 2 entropy(S) + O(1) expected number...
On the Complexity of Some Geometric Intersection Problems
 Journal of Computing and Information
, 1995
"... : A classification of polygons is proposed together with a new class of connected polygons, called ordinary polygons. Ordinary polygons include simple polygons possibly with holes. The determination of the intersection of a line segment and an ordinary polygon with N edges requires\Omega\Gamma N lo ..."
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: A classification of polygons is proposed together with a new class of connected polygons, called ordinary polygons. Ordinary polygons include simple polygons possibly with holes. The determination of the intersection of a line segment and an ordinary polygon with N edges requires\Omega\Gamma N log N) time in the worst case. A lineartime algorithm is given, however, if a planar subdivision of the polygon in trapezoids is allowed as a preprocessing. As the minimal trapezoidal subdivision of an ordinary polygon is NPcomplete, we propose a subdivision that, although not minimal, has at most 3N vertices and 5N edges, and can be computed in optimal \Theta(N log N) time in the worst case. The intersection of an Medge ordinary polygon with an Nedge ordinary polygon can be obtained in \Theta(M log M + MN + N log N) time, which is also worstcase optimal. Applications to worstcase optimal clipping and scanconversion algorithms and efficient hiddenline and hiddensurface algorithms th...
Quadtree Decomposition, Steiner Triangulation, and Ray shooting
 IN PROC. 9TH INTERNATIONAL SYMPOSIUM ON ALGORITHMS AND COMPUTATION
, 1998
"... We present a new quadtreebased decomposition of a polygon possibly with holes. For a polygon of n vertices, a truncated decomposition can be computed in O(n log n) time which yields a Steiner triangulation of the interior of the polygon that has O(n log n) size and approximates the minimum weig ..."
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Cited by 1 (1 self)
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We present a new quadtreebased decomposition of a polygon possibly with holes. For a polygon of n vertices, a truncated decomposition can be computed in O(n log n) time which yields a Steiner triangulation of the interior of the polygon that has O(n log n) size and approximates the minimum weight Steiner triangulation (MWST) to within a constant factor. An approximate MWST is good for ray shooting in the average case as defined by Aronov and Fortune. The untruncated decomposition also yields an approximate MWST. Moreover, we show that this triangulation supports querysensitive ray shooting as defined by Mitchell, Mount, and Suri. Hence, there exists a Steiner triangulation that is simultaneously good for ray shooting in the querysensitive sense and in the average case.
Three problems about simple polygons \Lambda
, 2004
"... Demaine, Kim, Kranakis, Lubiw, Maheshwari, Morin, Shin, Toussaint, Vigneron, and Yang, we show how to find a largest pair of disjoint congruent disks inside P in linear expected time. 2. As a subroutine for the above result, we show how to find the convex hull of any given subset of the vertices of ..."
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Demaine, Kim, Kranakis, Lubiw, Maheshwari, Morin, Shin, Toussaint, Vigneron, and Yang, we show how to find a largest pair of disjoint congruent disks inside P in linear expected time. 2. As a subroutine for the above result, we show how to find the convex hull of any given subset of the vertices of P in linear worstcase time. 3. More generally, we show how to compute a triangulation of any given subset of the vertices or edges of P in almost linear time.
Quadtree, Ray Shooting and Approximate Minimum Weight Steiner Triangulation
, 2001
"... We present a quadtreebased decomposition of the interior of a polygon with holes. The complete decomposition yields a constant factor approximation of the minimum weight Steiner triangulation (MWST) of the polygon. We show that this approximate MWST supports ray shooting queries in the queryse ..."
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We present a quadtreebased decomposition of the interior of a polygon with holes. The complete decomposition yields a constant factor approximation of the minimum weight Steiner triangulation (MWST) of the polygon. We show that this approximate MWST supports ray shooting queries in the querysensitive sense as defined by Mitchell, Mount and Suri. A proper truncation of our quadtreebased decomposition yields another constant factor approximation of the MWST. For a polygon with n vertices, the complexity of this approximate MWST is O(n log n) and it can be constructed in O(n log n) time. The running time is optimal in the algebraic decision tree model.