Optimal arrangement of data in a tree directory

We consider a graph problem GRAPH EMBEDDING which is a variation on OPTIMAL LINEAR ARRANGEMENT. We show that various classical graph problems can be obtained as special cases of GRAPH EMBEDDING. This implies that the problem is NP-complete in very restricted versions. We define the decision problem DATA ARRANGEMENT, which involves arranging the vertices of a graph G at the leaves of a d-ary tree so that a weighted sum of the distances between pairs of vertices measured with respect to the tree topology is at most a given value. We show that DATA ARRANGEMENT is strongly NP-complete for any fixed d≥2 and explain the connection between DATA ARRANGEMENT and arranging data in a particular form of distributed directory.