The algorithm we present is a natural next step to well-known algorithms for finding optimal graph realizations of tree-realizable distance matrices. It is based on the fact, which we prove first, that the quest for optimal realizations of nontree-realizable distance matrices can be narrowed to a proper subclass * of the class of all such matrices. The matrices in * are those which satisfy the following condition: for each pair of indices {h, i}, there is another pair {j, k} such that the submatrix {h, i, j, k} is nontree-realizable. Given an arbitrary distance matrix D in , the algorithm associates to D a matrix D* in the subclass *, whose optimal realization, if known, easily yields the optimal realization of D. The practical usefulness of this algorithm is underscored by a growing number of distance matrices whose optimal realizations are known [4, 7]. Time and space requirements of the algorithm are also discussed.