# BUILDING MINIMUM SPANNING TREES BY LIMITED NUMBER OF NODES OVER TRIANGULATED SET OF INITIAL NODES

## Authors

• Vadim Romanuke Vinnytsia Institute of Trade and Economics of State University of Trade and Economics, Ukraine

## Keywords:

minimum spanning tree, triangulation, edge lengths, redundant nodes, root node

## Abstract

Background. The common purpose of modelling and using minimum spanning trees is to ensure efficient coverage. In many tasks of designing efficient telecommunication networks, the number of network nodes is usually limited. In terms of rational allocation, there are more possible locations than factually active tools to be settled to those locations.

Objective. Given an initial set of planar nodes, the problem is to build a minimum spanning tree connecting a given number of the nodes, which can be less than the cardinality of the initial set. The root node is primarily assigned, but it can be changed if needed.

Methods. To obtain a set of edges, a Delaunay triangulation is performed over the initial set of nodes. Distances between every pair of the nodes in respective edges are calculated. These distances being the lengths of the respective edges are used as graph weights, and a minimum spanning tree is built over this graph.

Results. The problem always has a solution if the desired number of nodes (the number of available recipient nodes) is equal to the number of initially given nodes. If the desired number is lesser, the maximal edge length is found and the edges of the maximal length are excluded while the number of minimum spanning tree nodes is greater than the desired number of nodes.

Conclusions. To build a minimum spanning tree by a limited number of nodes it is suggested to use the Delaunay triangulation and an iterative procedure in order to meet the desired number of nodes. Planar nodes of an initial set are triangulated, whereupon the edge lengths are used as weights of a graph. The iterations to reduce nodes are done only if there are redundant nodes. When failed, the root node must be changed before the desired number of nodes is changed.

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