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 ∑ u,v∈V dT (u, v), where dT (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 α > 1 and an integer 1 ≤ k ≤ 6α−3, in O(n+n) time, the algorithm finds a spanning tree whose routing cost is at most (1 + 2/(k + 1))α times the one of the minimum routing cost tree, and the tree weight is at most (f(k) + 2/(α − 1)) times the one of the minimum spanning tree, where f(k) = 1 if k = 1 and f(k) = 2 if k > 1.