Properties of the distance matrix of a tree

Boesch, Francis T.

doi:10.1090/qam/265210

Cited by 29 publications

(9 citation statements)

References 3 publications

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“…Therefore there is a finite (but exponential) algorithm to find an optimal realization for a given distance matrix. Various heuristics are discussed in many papers [7,25,22,[28][29][30]. However, solutions to this problem seem to be elusive, and, in fact, computing optimal realizations for distance matrices with a small number of terminal nodes is already quite complicated.…”

Section: Several Distance Realization Problemsmentioning

confidence: 99%

Distance Realization Problems with Applications to Internet Tomography

Chung¹,

Garrett²,

Graham

et al. 2001

Journal of Computer and System Sciences

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In recent years, a variety of graph optimization problems have arisen in which the graphs involved are much too large for the usual algorithms to be effective. In these cases, even though we are not able to examine the entire graph (which may be changing dynamically), we would still like to deduce various properties of it, such as the size of a connected component, the set of neighbors of a subset of vertices, etc. In this paper, we study a class of problems, called distance realization problems, which arise in the study of Internet data traffic models. Suppose we are given a set S of terminal nodes, taken from some (unknown) weighted graph. A basic problem is to reconstruct a weighted graph G including S, with possibly additional vertices, that realizes the given distance matrix for S. We will first show that this problem is not only difficult bu the solution is often unstable in the sense that even if all distances between nodes in S decrease, the solution can increase by a factor proportional to the size of S in the worst case. We then proceed to consider a weaker version of the realization problem that only requires the distances in G to upper bound the given distances. We will show that this weak realization problem is NP-complete and that its optimum solutions can be approximated to within a factor of 2. We also consider several variants of these problems and a number of heuristics are presented. These problems are of interest for monitoring large-scale networks and for supplementing network management techniques.

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Section: Several Distance Realization Problemsmentioning

confidence: 99%

Distance Realization Problems with Applications to Internet Tomography

Chung¹,

Garrett²,

Graham

et al. 2001

Journal of Computer and System Sciences

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“…Theorem 2 also provides a generalization of Zaretskii's results to the weighted case. Boesch [8], considering strictly non-negative weighted graphs, gave some properties of the distance matrix of a tree and suggested two algorithms for a tree realization. We indicate here that one of these algorithms (the one derived from theorem II of his work) can be successfully used in the general case.…”

Section: A N Patrixos and S L Hakimimentioning

confidence: 99%

The distance matrix of a graph and its tree realization

Patrinos¹,

Hakimi²

1972

Quart. Appl. Math.

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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.

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“…A lot of authors have considered the special case where the distance matrix is a tree metric, also called an additive metric (see for example [1][2][3][4][5][6][7][8][9][10][11][12][13][14]). Efficient algorithms that construct such trees were published in [15] or [16].…”

Section: Introductionmentioning

confidence: 99%

A constructive algorithm for realizing a distance matrix

Varone

2006

European Journal of Operational Research

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The natural metric of a weighted graph is the length of the shortest paths between all pairs of vertices. The investigated problem consists in a representation of a given metric by a graph, such that the total length of the graph is minimized. For that purpose, we give a constructive algorithm based on a technique of reduction, fusion and deletion. We then show some results on a set of various distance matrices whose optimal realization is known.

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Properties of the distance matrix of a tree

Cited by 29 publications

References 3 publications

Distance Realization Problems with Applications to Internet Tomography

Distance Realization Problems with Applications to Internet Tomography

The distance matrix of a graph and its tree realization

A constructive algorithm for realizing a distance matrix

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