Minimum Spanning Trees

Years and Authors of Summarized Original Work

• 2002; Pettie, Ramachandran

Problem Definition

The minimum spanning tree (MST) problem is, given a connected, weighted, and undirected graph G = (V, E, w), to find the tree with minimum total weight spanning all the vertices V . Here, $w : E \rightarrow \mathbb{R}$ is the weight function. The problem is frequently defined in geometric terms, where V is a set of points in d-dimensional space and w corresponds to Euclidean distance. The main distinction between these two settings is the form of the input. In the graph setting, the input has size O(m + n) and consists of an enumeration of the $n =\vert V \vert$ vertices and $m =\vert E\vert$ edges and edge weights. In the geometric setting, the input consists of an enumeration of the coordinates of each point (O(dn) space): all $\left (\begin{array}{*{20}c} V \\ 2\\ \end{array} \right )$ edges are implicitly present and their weights implicit in the point coordinates. See [16] for a...