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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 ..."
Abstract

Cited by 19 (6 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 12 (3 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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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.