A polynomial time algorithm for multivariate interpolation in arbitrary dimension via the Delaunay triangulation

Abstract

The Delaunay triangulation is a fundamental construct from computational geometry, which finds wide use as a model for multivariate piecewise linear interpolation in fields such as geographic information systems, civil engineering, physics, and computer graphics. Though efficient solutions exist for computation of two- and three-dimensional Delaunay triangulations, the computational complexity for constructing the complete Delaunay triangulation grows exponentially in higher dimensions. Therefore, usage of the Delaunay triangulation as a model for interpolation in high-dimensional domains remains computationally infeasible by standard methods. In this paper, a polynomial time algorithm is presented for interpolating at a finite set of points in arbitrary dimension via the Delaunay triangulation. This is achieved by computing a small subset of the simplices in the complete triangulation, such that all interpolation points lie in the support of the subset. An empirical study on the runtime of the proposed algorithm is presented, demonstrating its scalability to high-dimensional spaces.