Math @ Duke

Publications [#235598] of Pankaj K. Agarwal
Papers Published
 Agarwal, PK, Surface approximation and geometric partitions,
SIAM Journal on Computing, vol. 27 no. 4
(1998),
pp. 10161035
(last updated on 2017/12/17)
Abstract: Motivated by applications in computer graphics, visualization, and scientific computation, we study the computational complexity of the following problem: given a set S of n points sampled from a bivariate function f(x, y) and an input parameter ε > 0, compute a piecewiselinear function ∑(x, y) of minimum complexity (that is, an xymonotone polyhedral surface, with a minimum number of vertices, edges, or faces) such that ∑(xp,yp)  zp ≤ ε for all (xp, yp, zp) ∈ S. We give hardness evidence for this problem, by showing that a closely related problem is NPhard. The main result of our paper is a polynomialtime approximation algorithm that computes a piecewiselinear surface of size O(Ko log Ko), where Ko is the complexity of an optimal surface satisfying the constraints of the problem. The technique developed in our paper is more general and applies to several other problems that deal with partitioning of points (or other objects) subject to certain geometric constraints. For instance, we get the same approximation bound for the following problem arising in machine learning: given n "red" and m "blue" points in the plane, find a minimum number of pairwise disjoint triangles such that each blue point is covered by some triangle and no red point lies in any of the triangles. © 1998 Society for Industrial and Applied Mathematics.


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