Minimum Geometric Spanning Trees

Keywords and Synonyms

Minimum length spanning trees; Minimum weight spanning trees; Euclidean minimum spanning trees; MST; EMST      

Problem Definition

Let S be a set of n points in d‑dimensional real space where ${ d \geq 1 }$ 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 (${ x_1, x_2, \dots, x_d }$) and ($y_1, y_2, \dots, y_d$), respectively, is defined as the pth root of the sum ${ \sum_{i=1}^d | x_i - y_i | ^ p }$.

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 L ₂ metric, which simply measures straight-line distances, the MST problem in two dimensions can be solved optimally in time ${ O(n \log n) }$,...