Euclidean minimum spanning trees with independent and dependent geometric uncertainties

Abstract

We address the problems of constructing the Euclidean Minimum Spanning Tree (EMST) of points in the plane with mutually dependent location uncertainties, testing its stability, and computing its total weight. Point coordinate uncertainties are modeled with the Linear Parametric Geometric Uncertainty Model (LPGUM), an expressive and computationally efficient worst-case, first order linear approximation of geometric uncertainty that supports parametric dependencies between point locations. We define uncertain EMST stability of n LPGUM points modeled with k real valued uncertainty parameters. We prove that when the uncertain EMST is unstable, it may have an exponential number of topologically different instances, thus precluding its polynomial-time computation. We present algorithms for comparing two edge weights defined by the distance between the edge endpoints for the independent and dependent cases with time complexity of O(klog⁡k) and O(T(k)) respectively, where T(k) is the time required to solve a quadratic optimization problem with k parameters. We describe an uncertain EMST stability test algorithm whose time complexity is O(n3klog⁡k) and O(nk+T(k)) for the independent and dependent case, respectively. We then present a more efficient O(nklog⁡nklog⁡n) time algorithm for the independent case and a method for computing the minimum and maximum total weight whose complexity is O(N3) time, where N=max⁡{k,n}.