Constructing light spanning trees with small routing cost

Let G = (V, E, w) be an undirected graph with nonnegative edge weight. For any spanning tree T of G, the weight of T is the total weight of its tree edges and the routing cost of T is Sigma(u,v is an element of V) d(T)(u, v), where d(T)(u,v) is the distance between u and v on T. In this paper, we present an algorithm providing a trade off among tree weight, routing cost and time complexity. For any real number alpha > 1 and an integer 1 less than or equal to k less than or equal to 6 alpha - 3, in O(n(k+1) + n(3)) time, the algorithm finds a spanning tree whose routing cost is at most (1 + 2/(k + 1)) alpha times the one of the minimum routing cost tree, and the tree weight is at most (f(k) + 2/(alpha - 1)) times the one of the minimum spanning tree, where f(k) = 1 if k = 1 and f(k) = 2 if k > 1.