# mst: Minimum Spanning Tree of the Pairwise Distance Graph

## Description

An parallelised implementation of a Jarnik (Prim/Dijkstra)-like algorithm for determining a(*) minimum spanning tree (MST) of a complete undirected graph representing a set of n points with weights given by a pairwise distance matrix.

(*) Note that there might be multiple minimum trees spanning a given graph.

## Usage

mst(d, ...)

## Default S3 method:
mst(
d,
distance = c("euclidean", "l2", "manhattan", "cityblock", "l1", "cosine"),
M = 1L,
cast_float32 = TRUE,
verbose = FALSE,
...
)

## S3 method for class 'dist'
mst(d, M = 1L, verbose = FALSE, ...)


## Arguments

d

either a numeric matrix (or an object coercible to one, e.g., a data frame with numeric-like columns) or an object of class dist, see dist

...

further arguments passed to or from other methods

distance

metric used to compute the linkage, one of: "euclidean" (synonym: "l2"), "manhattan" (a.k.a. "l1" and "cityblock"), "cosine"

M

smoothing factor; M = 1 gives the selected distance; otherwise, the mutual reachability distance is used

cast_float32

logical; whether to compute the distances using 32-bit instead of 64-bit precision floating-point arithmetic (up to 2x faster)

verbose

logical; whether to print diagnostic messages and progress information

## Details

If d is a numeric matrix of size $$n p$$, the $$n (n-1)/2$$ distances are computed on the fly, so that $$O(n M)$$ memory is used.

The algorithm is parallelised; set the OMP_NUM_THREADS environment variable Sys.setenv to control the number of threads used.

Time complexity is $$O(n^2)$$ for the method accepting an object of class dist and $$O(p n^2)$$ otherwise.

If M >= 2, then the mutual reachability distance $$m(i,j)$$ with smoothing factor M (see Campello et al. 2013) is used instead of the chosen “raw” distance $$d(i,j)$$. It holds $$m(i, j)=\max(d(i,j), c(i), c(j))$$, where $$c(i)$$ is $$d(i, k)$$ with $$k$$ being the (M-1)-th nearest neighbour of $$i$$. This makes “noise” and “boundary” points being “pulled away” from each other. Genie++ clustering algorithm (see gclust) with respect to the mutual reachability distance gains the ability to identify some observations are noise points.

Note that the case M = 2 corresponds to the original distance, but we are determining the 1-nearest neighbours separately as well, which is a bit suboptimal; you can file a feature request if this makes your data analysis tasks too slow.

## Value

Matrix of class mst with n-1 rows and 3 columns: from, to and dist. It holds from < to. Moreover, dist is sorted nondecreasingly. The i-th row gives the i-th edge of the MST. (from[i], to[i]) defines the vertices (in 1,…,n) and dist[i] gives the weight, i.e., the distance between the corresponding points.

The method attribute gives the name of the distance used. The Labels attribute gives the labels of all the input points.

If M > 1, the nn attribute gives the indices of the M-1 nearest neighbours of each point.

## Author(s)

Marek Gagolewski and other contributors

## References

Jarnik V., O jistem problemu minimalnim, Prace Moravske Prirodovedecke Spolecnosti 6, 1930, 57-63.

Olson C.F., Parallel algorithms for hierarchical clustering, Parallel Comput. 21, 1995, 1313-1325.

Prim R., Shortest connection networks and some generalisations, Bell Syst. Tech. J. 36, 1957, 1389-1401.

Campello R.J.G.B., Moulavi D., Sander J., Density-based clustering based on hierarchical density estimates, Lecture Notes in Computer Science 7819, 2013, 160-172, doi:10.1007/978-3-642-37456-2_14.

The official online manual of genieclust at https://genieclust.gagolewski.com/

Gagolewski M., genieclust: Fast and robust hierarchical clustering, SoftwareX 15:100722, 2021, doi:10.1016/j.softx.2021.100722.

emst_mlpack() for a very fast alternative in case of (very) low-dimensional Euclidean spaces (and M = 1).

## Examples

library("datasets")
data("iris")
X <- iris[1:4]
tree <- mst(X)