Balancing minimum spanning trees and multiple-source minimum routing cost spanning trees on metric graphs

The building cost of a spanning tree is the sum of weights of the edges used to construct the spanning tree. The routing cost of a source vertex s on a spanning tree T is the total summation of distances between the source vertex s and all the vertices d in T . Given a source vertices set S, the multiple-source routing cost of a spanning tree T is the summation of the routing costs for source vertices in S. Both the building cost and the multiple-source routing cost are important considerations in construction of a network system. A spanning tree with minimum building cost among all spanning trees is called a minimum spanning tree (MST), and a spanning tree with minimum k-source routing cost among all spanning trees is called a k-source minimum routing cost spanning tree (k-MRCT). Usually a k-MRCT of a graph G with respect to k sources is not a MST of G, and vice versa. This paper proposes an algorithm to construct a spanning tree T for a metric graph G with a source vertex set S such that the building cost of T is less than 1 + 2/(α − 1) times of that of a MST of G, and the k-source routing cost of T is less than α(1 + 2(k−1)(n−2) k(n+k−2) ) times of that of a k-MRCT of G with respect to S, where α > 1, k = |S| and n is the number of vertices of G.