Surface Approximation and Geometric Partitions

Motivated by applications in computer graphics, visualization, and scienti c 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 " > 0, compute a piecewise linear function (x;y) of minimum complexity (that is, a xy-monotone polyhedral surface, with a minimum number of vertices, edges, or faces) such that j (xp;yp) ? zpj "; for all (xp;yp;zp) 2 S: We prove that the decision version of this problem is NP-Hard. The main result of our paper is a polynomial-time approximation algorithm that computes a piecewise linear surface of size O(Ko logKo), where Ko is the complexityof an optimalsurface 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...

• Type: Journal
• Year: 1998
• Where: SIAMCOMP
• Authors: Pankaj K. Agarwal, Subhash Suri
• Keywords: Minimum Number, Piecewise Linear Function, SIAMCOMP 1998, Xy-monotone Polyhedral Surface