Minimum Geometric Spanning Trees

Years and Authors of Summarized Original Work

• 1999; Krznaric, Levcopoulos, Nilsson

Problem Definition

Let S be a set of n points in d-dimensional real space where d ≥ 1 is an integer constant. A minimum spanning tree (MST) of S is a connected acyclic graph with vertex set S of minimum total edge length. The length of an edge equals the distance between its endpoints under some metric. Under the so-called L _p metric, the distance between two points x and y with coordinates $(x_{1},x_{2},\ldots ,x_{d})$ and $(y_{1},y_{2},\ldots ,y_{d})$, respectively, is defined as the pth root of the sum $\sum \limits _{i=1}^{d}\left \vert x_{i} - y_{i}\right \vert ^{p}$.

Key Results

Since there is a very large number of papers concerned with geometric MSTs, only a few of them will be mentioned here.

In the common Euclidean L₂ metric, which simply measures straight-line distances, the MST problem in two dimensions can be solved optimally in time O(nlog n), by using the fact that the MST is a...