Minimum Linear Arrangement of Trees

Abstract

In the minimum linear arrangement problem one is given a graph, and wishes to assign distinct integers to the vertices of the graph so that the sum of the differences (in absolute value) across the edges of the graph is minimized. This problem is known to be NP-complete for the class of all graphs, but polynomial for special classes of graphs, one of which is the class of trees. For trees on n vertices, algorithms of time complexity O(n2.2) and O(n1.6) were given by Shiloach in 1979 and Chung in 1983 respectively, with no improvement since then. In this thesis, we present a linear-time algorithm for finding the optimal embedding among those embeddings which have no "crossings," and we describe a C++ implementation of that algorithm as well as Shiloach's algorithm which we make available to the research community.