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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.
TwoDimensional Range Diameter Queries
"... Abstract. Given a set of n points in the plane, range diameter queries ask for the furthest pair of points in a given axisparallel rectangular range. We provide evidence for the hardness of designing spaceefficient data structures that support range diameter queries by giving a reduction from the ..."
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Abstract. Given a set of n points in the plane, range diameter queries ask for the furthest pair of points in a given axisparallel rectangular range. We provide evidence for the hardness of designing spaceefficient data structures that support range diameter queries by giving a reduction from the set intersection problem. The difficulty of the latter problem is widely acknowledged and is conjectured to require nearly quadratic space in order to obtain constant query time, which is matched by known data structures for both problems, up to polylogarithmic factors. We strengthen the evidence by giving a lower bound for an important subproblem arising in solutions to the range diameter problem: computing the diameter of two convex polygons, that are separated by a vertical line and are preprocessed independently, requires almost linear time in the number of vertices of the smaller polygon, no matter how much space is used. We also show that range diameter queries can be answered much more efficiently for the case of points in convex position by describing a data structure of size O(n log n) that supports queries in O(log n) time. 1
Improved bounds for Smallest Enclosing Disk Range Queries
"... Let S be a set of n points in the plane. We present a method where, using O(n log2 n) time and space, S can be preprocessed into a data structure such that given an axisparallel query rectangle q, we can report the radius of the smallest enclosing disk of the points lying in S ∩ q in O(log6 n) tim ..."
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Let S be a set of n points in the plane. We present a method where, using O(n log2 n) time and space, S can be preprocessed into a data structure such that given an axisparallel query rectangle q, we can report the radius of the smallest enclosing disk of the points lying in S ∩ q in O(log6 n) time per query. 1