Voronoi Diagram and Delaunay Triangulation with Independent and Dependent Geometric Uncertainties

We address the problems of constructing the Voronoi diagram (VD) and Delaunay triangulation (DT) of points in the plane with mutually dependent location uncertainties, testing their stability, and computing their components. Point coordinate uncertainties are modeled with the Linear Parametric Geometric Uncertainty Model (LPGUM), a deterministic, expressive and computationally efficient first order linear approximation of geometric uncertainty that supports parametric dependencies between point locations. We define an uncertain three-point circle and its properties and present in-circle-test algorithms for the dependent and the independent cases whose respective time complexity is O(P4(k)) and O(klog k), where k is the number of parameters that describe the points location uncertainty and P4(k) is the complexity of quartic k-variable optimization. We define the uncertain VD and DT of n LPGUM points and show that an unstable VD may have an exponential number of topologically different instances. We describe algorithms for testing VD and DT stability whose time complexity is O(nP4(k)) and O(nklog k) for the dependent and independent cases when the nominal (exact) VD is given. We present algorithms to compute the vertices, edges, and faces of uncertain VD and DT for the independent case whose complexity is O(nk2log k + nlog n) time and O(nk) space. Finally, we describe algorithms for answering exact and uncertain point location queries in a stable uncertain VD and for dynamically updating it in O(k2log k +log n) and O(k2log k +log n + P 4(k)) average case time complexity for the independent and dependent cases.