Algorithms for ε-Approximations of Terrains

  • Jeff M. Phillips
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5125)

Abstract

Consider a point set \({\mathcal{D}}\) with a measure function \(\mu : {\mathcal{D}} \to \mathcal{R}\). Let \({\mathcal{A}}\) be the set of subsets of \(\mathcal{D}\) induced by containment in a shape from some geometric family (e.g. axis-aligned rectangles, half planes, balls, k-oriented polygons). We say a range space \((\mathcal{D}, \mathcal{A})\) has an ε-approximation P if

$$\max_{R \in \mathcal{A}} \left| \frac{\mu(R \cap P)}{ \mu(P)} - \frac{\mu(R \cap \mathcal{D})}{ \mu(\mathcal{D})} \right| \leq \varepsilon.$$

We describe algorithms for deterministically constructing discrete ε-approximations for continuous point sets such as distributions or terrains. Furthermore, for certain families of subsets \(\mathcal{A}\), such as those described by axis-aligned rectangles, we reduce the size of the ε-approximations by almost a square root from \(O(\frac{1}{\varepsilon^2} \log \frac{1}{\varepsilon})\) to Open image in new window. This is often the first step in transforming a continuous problem into a discrete one for which combinatorial techniques can be applied. We describe applications of this result in geospatial analysis, biosurveillance, and sensor networks.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2008

Authors and Affiliations

  • Jeff M. Phillips
    • 1
  1. 1.Department of Computer ScienceDuke UniversityDurham

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