Algorithms for ε-Approximations of Terrains

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 . 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.

Work on this paper is supported by a James B. Duke Fellowship, by NSF under a Graduate Research Fellowship and grants CNS-05-40347, CFF-06-35000, and DEB-04-25465, by ARO grants W911NF-04-1-0278 and W911NF-07-1-0376, by an NIH grant 1P50-GM-08183-01, by a DOE grant OEGP200A070505, and by a grant from the U.S. Israel Binational Science Foundation.