Algorithms of an optimal integer tree labeling

Suppose we label the vertices of a tree by positive integers. The weight of an edge is defined by a monotonically increasing function of the absolute value of the difference of the labels of its endpoints. We define the total cost of the labeling to be the sum of weights of all the edges. The problem we consider is that of determining for a given tree G and given a labeling of the leaves of G the minimum total cost labelings of G. In this paper we present an algorithm that works for any cost function satisfies the condition of monotony mentioned above. In a case of the function defined as the absolute value of the difference of the labels the fast algorithm is presented.