Algorithms for ε-Approximations of Terrains

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


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.


Sensor Network Height Function Combinatorial Discrepancy Range Space Polygonal Domain 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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© Springer-Verlag Berlin Heidelberg 2008

Authors and Affiliations

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

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