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On the power of the semiseparated pair decomposition
 WADS, LNCS Volume
"... s> 1, of a set S ⊂ R d is a set {(Ai, Bi)} of pairs of subsets of S such that for each i, there are balls DA i and DB i containing Ai and Bi respectively such that d(DA i, DB i) ≥ s · min(radius(DA i), radius(DB i and for any two points p, q ∈ S there is a unique index i such that p ∈ Ai and q ∈ ..."
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s> 1, of a set S ⊂ R d is a set {(Ai, Bi)} of pairs of subsets of S such that for each i, there are balls DA i and DB i containing Ai and Bi respectively such that d(DA i, DB i) ≥ s · min(radius(DA i), radius(DB i and for any two points p, q ∈ S there is a unique index i such that p ∈ Ai and q ∈ Bi or viceversa. In this paper, we use the SSPD to obtain the following results: First, we consider the construction of geometric tspanners in the context of imprecise points and we prove that any set S ⊂ R d of n imprecise points, modeled as pairwise disjoint balls, admits a tspanner with O(n log n/(t − 1) d) edges which can be computed in O(n log n/(t − 1) d) time. If all balls have the same radius, the number of edges reduces to O(n/(t − 1) d). Secondly, for a set of n points in the plane, we design a query data structure for halfplane closestpair queries that can be built in O(n 2 log 2 n) time using O(n log n) space and answers a query in O(n 1/2+ε) time, for any ε> 0. By reducing the preprocessing time to O(n 1+ε) and using O(n log 2 n) space, the query can be answered in O(n 3/4+ε) time. Moreover, we improve the preprocessing time of an existing axisparallel rectangle closestpair query data structure from quadratic to nearlinear. Finally, we revisit some previously studied problems, namely spanners for complete kpartite graphs and lowdiameter spanners, and show how to use the SSPD to obtain simple algorithms for these problems. 1
Data structures for rangeaggregate extent queries
 In Proc. 20th CCCG
, 2008
"... A fundamental and wellstudied problem in computational geometry is range searching, where the goal is to preprocess a set, S, of geometric objects (e.g., points in the plane) so that the subset S ′ ⊆ S that is contained in a query range (e.g., an axesparallel rectangle) can be reported efficientl ..."
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Cited by 5 (2 self)
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A fundamental and wellstudied problem in computational geometry is range searching, where the goal is to preprocess a set, S, of geometric objects (e.g., points in the plane) so that the subset S ′ ⊆ S that is contained in a query range (e.g., an axesparallel rectangle) can be reported efficiently. However, in many situations, what is of interest is to generate a more informative “summary ” of the output, obtained by applying a suitable aggregation function on S ′. Examples of such aggregation functions include count, sum, min, max, mean, median, mode, and topk that are usually computed on a set of weights defined suitably on the objects. Such rangeaggregate query problems have been the subject of much recent research in both the database and the computational geometry communities. In this paper, we further generalize this line of work by considering aggregation functions on pointsets that measure the extent or “spread ” of the objects in the retrieved set S ′. The functions considered here include closest pair, diameter, and width. The challenge here is that these aggregation functions (unlike, say, count) are not efficiently decomposable in the sense that the answer to S ′ cannot be inferred easily from answers to subsets that induce a partition
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.