Computing Similarity between Piecewise-Linear Functions

Maarten Löffler

We study the problem of computing the similarity of two piecewise-linear bivariate functions defined over a common domain, where the surfaces they define in 3D---polyhedral terrains---can be transformed vertically by a linear transformation of the third coordinate (scaling and translation). We give an algorithm that minimizes the maximum vertical distance between the graphs of the two functions, over all linear transformations, in O(n^4/3 polylog n) expected time, where n is the total number of vertices in the graphs of the two functions. We also study the computation of similarity of two univariate or bivariate functions by minimizing the area or volume in between their graphs. For univariate functions we give a (1+ε)-approximation algorithm for minimizing the area that runs in O(n/ε^1/2) time, for any fixed ε>0. The (1+ε)-approximation algorithm for the bivariate version, where volume is minimized, runs in O(n/ε²) time, for any fixed ε>0, provided the two functions are defined over the same triangulation of their domain.