Optimal one-page tree embeddings in linear time

In the minimum linear arrangement problem one wishes to assign distinct integers to the vertices of a given 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 trees—algorithms of time complexity O( n 2.2) and O( n 1.6) were given by Shiloach [SIAM J. Comput. 8 (1979) 15–32] and Chung [Comput. Math. Appl. 10 (1984) 43–60], respectively. We present a linear-time algorithm for finding the optimal embedding (arrangement) in a restricted but important class of embeddings called one-page embeddings. 1 1 A more extensive report is in Robert Hochberg's thesis [7].