The distance matrix of a graph and its tree realization

Abstract

Abstract.The results of Hakimi and Yau and others in the realization of a distance matrix are generalized to graphs (digraphs) whose branches (arcs) may have negative weights. Conditions under which such matrices have a tree, hypertree or directed tree realization are given, uniqueness of these realizations is discussed and algorithms for their construction are indicated.1. Notation. A number of definitions are given so that results will be presented in The degree of a vertex Vi in G, denoted deg (y, , G), is the number of branches (arcs) incident at i>, in G. The outdegree of a vertex v{ in digraph G, denoted outdeg(y, , G), is equal to the number of arcs incident at v, in G and directed away from v{. The indegree of Vi, denoted indeg(z\-, G), is equal to the number of arcs incident at v{ in G and directed towards v{ . A weighted graph (digraph) is a graph (digraph) together with a function which assigns a real number wx to each branch h, (arc a,). All graphs (digraphs) presented here are weighted.An edge-sequence in a graph (digraph) between two vertices v, and vf is an alternating sequence of vertices and branches (arcs)• • • M,-beginning and ending with Vi and Vj , in which each branch (arc) is incident at the vertex preceding and the vertex following it. A path from to v,-is the set of all branches (arcs) in an edge-sequence between v{ and v, . A directed path in a digraph is a path in which each arc is directed from the vertex preceding it to the vertex following it in the corresponding edge-sequence. A path or directed path is called elementary if all vertices in the edge-sequence are distinct. A path (directed path) is a circuit {cycle) if the first and last vertex in the edgesequence are the same and all others distinct. The length of a path (directed path) is the sum of the weights of the branches (arcs) in it. A connected graph (digraph) is a graph (digraph) in which every pair of vertices is joined by a path.

Optimal variable weighting for hierarchical clustering: An alternating least-squares algorithm

“…An alternative method proposed by De Soete (1983) involves modifying the loss functions in expressions (1), (6), (9), and (10) to reflect path length (Carroll and Chang 1973), free (Cunningham 1974, t978), or additive trees (Sattath and Tversky 1977) as opposed to ultrametric trees. Bunemann (1974), Dobson (1974), and Patrinos and Hakimi (1972) devised a four-point condition that must be satisfied by path length distances. One way of stating this condition is: if 6;j + 8k~ >/ 6,-k + 6il /> 6jk + 6;Z and perform analogous Phases A, B, and C for additive trees.…”

Distance Realization Problems with Applications to Internet Tomography

“…Special attention has been given to the case of tree realizations, i.e., when the graph that realized the distance matrix is a tree. Necessary and sufficient condition for a distance matrix realizable by a tree were given in several papers [4,16,[26][27][28]. An O(n 2 ) time algorithm for testing and constructing a tree realization from a distance matrix was described in [15].…”

Tree structures for proximity data

“…There exist several parallel proofs in the literature that the so-called four-point condition is necessary and sufficient for an additive tree representation (Simoes-Pereira, 1969;Patrinos & Hakimi, 1972;Dobson, 1974;Buneman, 1974). This condition is defined as follows: The problem of finding an optimal tree that gives the minimum discrepancy between the dissimilarity index and the tree distance has not yet been completely solved.…”