Algorithms for ε-approximations of terrains

Abstract

Consider a point set with a measure function μ : D → ℝ. Let A be the set of subsets of 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 (D, A) has an ε-approximation P if max R∈A|μ(R∩P)/μ(P)-μ(R∩D)/μ(D)|≤ε. We describe algorithms for deterministically constructing discrete ε-approximations for continuous point sets such as distributions or terrains. Furthermore, for certain families of subsets A, such as those described by axis-aligned rectangles, we reduce the size of the ε-approximations by almost a square root from O(1/ε2 log 1/ε) to O(1/ε polylog1/ε). 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. © 2008 Springer-Verlag.