A linear-space algorithm for distance preserving graph embedding

The distance preserving graph embedding problem is to embed vertices of a given weighted graph into points in d-dimensional Euclidean space for a constant d so that for each edge the distance between their corresponding endpoints is as close to the weight of the edge as possible. If the given graph is complete, that is, if distance constraints are given as a full matrix, then multi-dimensional scaling can minimize the sum of squared embedding errors in polynomial time. A serious disadvantage is its quadratic space requirement. In this paper we develop a linear-space algorithm for this problem. A key idea is to partition a set of n objects into O( √ n) disjoint subsets (clusters) of size O( √ n) such that the minimum inter cluster distance is maximized among all possible such partitions. Experimental results and applications to canonical representations of manifold meshes are included.