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Approximating RangeAggregate Queries using Coresets
"... Let µ be a function that assigns a real number µ(P) ≥ 0 to any point set P in R d; for example, µ(P) can be the diameter or radius of the smallest enclosing ball of P. Let S be a set of n points in R d. We consider the problem of storing S in a data structure, such that for any query rectangle Q, w ..."
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Let µ be a function that assigns a real number µ(P) ≥ 0 to any point set P in R d; for example, µ(P) can be the diameter or radius of the smallest enclosing ball of P. Let S be a set of n points in R d. We consider the problem of storing S in a data structure, such that for any query rectangle Q, we can efficiently compute an approximation to the value µ(S ∩Q). Our solutions are obtained by combining rangesearching techniques with coresets. 1
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.
Data Structures for RangeAggregate Extent Queries ✩
"... 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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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
New Constructions of SSPDs and their Applications∗
, 2011
"... We present a new optimal construction of a semiseparated pair decomposition (i.e., SSPD) for a set of n points in IRd. In the new construction each point participates in a few pairs, and it extends easily to spaces with low doubling dimension. This is the first optimal construction with these prope ..."
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We present a new optimal construction of a semiseparated pair decomposition (i.e., SSPD) for a set of n points in IRd. In the new construction each point participates in a few pairs, and it extends easily to spaces with low doubling dimension. This is the first optimal construction with these properties. As an application of the new construction, for a fixed t> 1, we present a new construction of a tspanner with O(n) edges and maximum degree O(log2 n) that has a separator of size O n1−1/d
A Linear Time Euclidean Spanner on Imprecise Points
"... An sspanner on a set S of n points in Rd is a graph on S where for every two points p, q ∈ S, there exists a path between them in G whose length is less than or equal to s · pq  where pq  is the Euclidean distance between p and q. In this paper, we consider the construction of a Euclidean spann ..."
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An sspanner on a set S of n points in Rd is a graph on S where for every two points p, q ∈ S, there exists a path between them in G whose length is less than or equal to s · pq  where pq  is the Euclidean distance between p and q. In this paper, we consider the construction of a Euclidean spanner for imprecise points where we take advantage of prior, inexact knowledge of our input. In particular, in the first phase, we preprocess n ddimensional balls with radius r that are approximations of the input points with the guarantee that each input point lies within its respective ball. In the second phase, the specific points are revealed and we quickly compute a spanner using data from the preprocessing phase. We can compute (or update) the (1 + ε)spanner in time O(n(r+ 1ε) d log(r+ 1ε)) after O(n(r+
Chapter 3 Well Separated Pairs Decomposition
, 2010
"... The fact remains that getting people right is not what living is all about anyway. It’s getting them wrong that is living, getting them wrong and wrong and wrong and then, on careful reconsideration, getting them wrong again. That’s how we known we’re alive: we’re wrong. Maybe the best thing would b ..."
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The fact remains that getting people right is not what living is all about anyway. It’s getting them wrong that is living, getting them wrong and wrong and wrong and then, on careful reconsideration, getting them wrong again. That’s how we known we’re alive: we’re wrong. Maybe the best thing would be to forget being right or wrong about people and just go along for the ride. But if you can do that well, lucky you. – American Pastoral, Philip Roth. In this chapter, we will investigate how to represent distances between points efficiently. Naturally, an explicit description of the distances between n points requires listing all the () n distances.