Abstract
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 D ∗ of the class D of all such matrices. The matrices in D ∗ 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 D , the algorithm associates to D a matrix D∗ in the subclass D ∗, 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.
Published Version
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