Algorithms of an optimal integer tree labeling

TL;DR

This paper introduces algorithms for optimally labeling tree vertices with positive integers to minimize total edge weights, applicable to any monotonic cost function, with a fast solution for the absolute difference case.

Contribution

It presents a general algorithm for minimal total cost labelings of trees under monotonic cost functions, including a fast method for the absolute difference case.

Findings

Algorithm works for any monotonic cost function

Fast algorithm developed for absolute difference cost function

Optimal labelings minimize total edge weight

Abstract

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 weight 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 labellings 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.