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 $O(k\log k)$ and $O\left(T\left(k\right)\right)$ respectively, where $T\left(k\right)$ 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 $O(n^{3}k\log k)$ and $O(nk+T\left(k\right))$ for the independent and dependent case, respectively. We then present a more efficient $O(nk\log nk\log n)$ time algorithm for the independent case and a method for computing the minimum and maximum total weight whose complexity is $O\left(N^3\right)$ time, where $N=\max \{k,n\}$.

我们解决了以下问题：在具有相互依赖的位置不确定性的平面上构造点的欧几里得最小生成树（EMST），测试其稳定性并计算其总权重。点坐标不确定性使用线性参数几何不确定性模型（LPGUM）建模，该模型是表达性和计算效率最差的几何不确定性一阶线性逼近，支持点位置之间的参数依赖性。我们定义了用k建模的n个LPGUM点的不确定EMST稳定性实值不确定性参数。我们证明，当不确定的EMST不稳定时，它可能具有指数数量的拓扑不同实例，从而排除了其多项式时间计算。我们提出了用于比较两个边缘权重的算法，这些权重由独立案例和从属案例的边缘端点之间的距离定义，时间复杂度为$Ø（ķ\mathrm{日志}ķ）$ 和 $Ø（Ť（ķ））$ 分别在哪里 $Ť（ķ）$是解决具有k个参数的二次优化问题所需的时间。我们描述了一种不确定的EMST稳定性测试算法，其时间复杂度为$Ø（{ñ}^3ķ\mathrm{日志}ķ）$ 和 $Ø（ñķ+Ť（ķ））$分别针对独立案件和依存案件。然后我们提出一个更有效的$Ø（ñķ\mathrm{日志}ñķ\mathrm{日志}ñ）$ 独立案例的时间算法和计算最小和最大总权重的方法，其复杂度为 $Ø（{ñ}^3）$ 时间，地点 $ñ=\mathrm{最高}\{ķ，ñ\}$。