Computing the Expected Value and Variance of Geometric Measures

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  • Frank Staals
    Frank Staals
  • Constantinos Tsirogiannis
    Constantinos Tsirogiannis
Let P be a set of points in R^d, and let M be a function that maps any subset of P to a positive real number. We examine the problem of computing the exact mean and variance of M when a subset of points in P is selected according to a well-defined random distribution. We consider two distributions; in the first distribution (which we call the Bernoulli distribution), each point p in P is included in the random subset independently, with probability pi(p). In the second distribution (the fixed-size distribution), a subset of exactly s points is selected uniformly at random among all possible subsets of s points in P. This problem is a crucial part of modern ecological analyses; each point in P represents a species in d-dimensional trait space, and the goal is to compute the statistics of a geometric measure on this trait space, when subsets of species are selected under random processes. We present efficient exact algorithms for computing the mean and variance of several geometric measures when point sets are selected under one of the described random distributions. More specifically, we provide algorithms for the following measures: the bounding box volume, the convex hull volume, the mean pairwise distance (MPD), the squared Euclidean distance from the centroid, and the diameter of the minimum enclosing disk. We also describe an efficient (1-e)-approximation algorithm for computing the mean and variance of the mean pairwise distance. We implemented three of our algorithms and we show that our implementations can provide major speedups compared to the existing imprecise approaches.
Original languageEnglish
Title of host publication2017 Proceedings of the Ninteenth Workshop on Algorithm Engineering and Experiments (ALENEX)
EditorsSándor Fekete, Vijaya Ramachandran
Number of pages15
PublisherSociety for Industriel & Applied Mathematics
Publication year2017
Pages232 - 246
ISBN (Electronic)978-1-61197-476-8
DOIs
StatePublished - 2017
EventALENEX17 - Hotel Porta Fira, Barcelona, Spain
Duration: 17 Jan 201718 Jan 2017
Conference number: 19
http://epubs.siam.org/doi/book/10.1137/1.9781611974768

Workshop

WorkshopALENEX17
Nummer19
LocationHotel Porta Fira
LandSpain
ByBarcelona
Periode17/01/201718/01/2017
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