BALANCING MINIMUM SPANNING-TREES AND SHORTEST-PATH TREES

We give a simple algorithm to find a spanning tree that simultaneously approximates a shortest-path tree and a minimum spanning tree. The algorithm provides a continuous tradeoff: given the two trees and a gamma > 0, the algorithm returns a spanning tree in which the distance between any vertex and the root of the shortest-path tree is at most 1 + root 2 gamma times the shortest-path distance, and yet the total weight of the tree is at most 1 + root 2/gamma times the weight of a minimum spanning tree. Our algorithm runs in linear time and obtains the best-possible tradeoff. It can be implemented on a CREW PRAM to run a logarithmic time using one processor per vertex.