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SemiOnline Maintenance of Geometric Optima and Measures
, 2003
"... We give the first nontrivial worstcase results for dynamic versions of various basic geometric optimization and measure problems under the semionline model, where during the insertion of an object we are told when the object is to be deleted. Problems that we can solve with sublinear update time i ..."
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Cited by 23 (7 self)
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We give the first nontrivial worstcase results for dynamic versions of various basic geometric optimization and measure problems under the semionline model, where during the insertion of an object we are told when the object is to be deleted. Problems that we can solve with sublinear update time include the Hausdorff distance of two point sets, discrete 1center, largest empty circle, convex hull volume in three dimensions, volume of the union of axisparallel cubes, and minimum enclosing rectangle. The decision versions of the Hausdorff distance and discrete 1center problems can be solved fully dynamically. Some applications are mentioned.
Dynamic Subgraph Connectivity with Geometric Applications
 Proc. 34th ACM Sympos. Theory Comput
, 2002
"... Inspired by dynamic connectivity applications in computational geometry, we consider a problem we call dynamic subgraph connectivity : design a data structure for an undirected graph G = (V, E) and a subset of vertices S # V , to support insertions and deletions in S and connectivity queries (are ..."
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Cited by 19 (4 self)
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Inspired by dynamic connectivity applications in computational geometry, we consider a problem we call dynamic subgraph connectivity : design a data structure for an undirected graph G = (V, E) and a subset of vertices S # V , to support insertions and deletions in S and connectivity queries (are two vertices connected?) in the subgraph induced by S. We develop the first sublinear, fully dynamic method for this problem for general sparse graphs, using an elegant combination of several simple ideas. Our method requires linear space, # O(E 4#/(3#+3) ) = O(E 0.94 ) amortized update time, and # O(E 1/3 ) query time, where # is the matrix multiplication exponent and # O hides polylogarithmic factors.
RangeAggregate Queries for Geometric Extent Problems Peter Brass 1 Christian Knauer 2 ChanSu Shin 3 Michiel Smid 4
"... Let S be a set of n points in the plane. We present data structures that solve rangeaggregate query problems on three geometric extent measure problems. Using these data structures, we can report, for any axisparallel query rectangle Q, the area/perimeter of the convex hull, the width, and the rad ..."
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Cited by 1 (0 self)
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Let S be a set of n points in the plane. We present data structures that solve rangeaggregate query problems on three geometric extent measure problems. Using these data structures, we can report, for any axisparallel query rectangle Q, the area/perimeter of the convex hull, the width, and the radius of the smallest enclosing disk of the points in S ∩ Q.
Dynamic Coresets
, 2008
"... Abstract We give a dynamic data structure that can maintain an "coreset of n points, with respect to the extent measure, in O(log n) time for any constant " ? 0 and any constant dimension. The previous method by Agarwal, HarPeled, and Varadarajan requires polylogarithmic update t ..."
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Abstract We give a dynamic data structure that can maintain an &quot;coreset of n points, with respect to the extent measure, in O(log n) time for any constant &quot; ? 0 and any constant dimension. The previous method by Agarwal, HarPeled, and Varadarajan requires polylogarithmic update time. For points with integer coordinates bounded by U, we alternatively get O(log log U) time. Numerous applications follow, for example, on dynamically approximating the width, smallest enclosing cylinder, minimum bounding box, or minimumwidth annulus. We can also use the same approach to maintain approximate kcenters in O(minflog n; log log U g) randomized amortized time for any constant k and any constant dimension. For the smallest enclosing cylinder problem, we also show that a constantfactor approximation can be maintained in O(1) randomized amortized time on the word RAM.
Dynamic Subgraph Connectivity with Geometric Applications\Lambda
, 2006
"... Key words. Data structures, dynamic graph algorithms, connectivity, computational geometry AMS subject classifications. 68Q25, 68P05, 68U05 Abbreviated title. Dynamic subgraph connectivity 1 Introduction Geometric motivation. Dynamic graph connectivitymaintaining an undirected graph under edge ins ..."
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Key words. Data structures, dynamic graph algorithms, connectivity, computational geometry AMS subject classifications. 68Q25, 68P05, 68U05 Abbreviated title. Dynamic subgraph connectivity 1 Introduction Geometric motivation. Dynamic graph connectivitymaintaining an undirected graph under edge insertions and deletions, to answer queries of the form, &quot;are two vertices connected?&quot;is a basic problem in the area of graph data structures. In the same way, connectivity problems concerning a dynamic collection of geometric objects are fundamental in computational geometry. However, unlike dynamic graph connectivity, which has been extensively studied and has enjoyed much recent success with the discovery of nearlogarithmic algorithms [17, 21, 32], progress in dynamic geometric connectivity has been scarce. For this reason, we decide to start our investigation with a simple version of the problem: Maintain a set of n axisparallel rectangles in the plane, under insertions and deletions, to answer queries of the form, &quot;given two points a and b, is there a path from a to b that lies inside the union of these rectangles?&quot;
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"... The min # problem, a hybrid error criterion for nearlinear time performance For a given polygonal chain and a family of tolerance regions, we study the min # problem, which consists in finding an approximate and ordered subchain with a minimum number of vertices. Our algorithm computes an approxi ..."
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The min # problem, a hybrid error criterion for nearlinear time performance For a given polygonal chain and a family of tolerance regions, we study the min # problem, which consists in finding an approximate and ordered subchain with a minimum number of vertices. Our algorithm computes an approximation that retains the shape of the original chain. Moreover, our method reaches a nearlinear time complexity and its implementation is based on classical functions. To our knowledge, this is the first algorithm providing all these advantages. 1