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Concerning the relationship between realizations and tight spans of finite metrics
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics. Uppsala University, Disciplinary Domain of Science and Technology, Biology, Department of Cell and Molecular Biology, The Linnaeus Centre for Bioinformatics.
2007 (English)In: Discrete & Computational Geometry, ISSN 0179-5376, E-ISSN 1432-0444, Vol. 38, no 3, 605-614 p.Article in journal (Refereed) Published
##### Abstract [en]

Given a metric d on a finite set X, a realization of d is a weighted graph $G=(V,E,w\colon \ E \to {\Bbb R}_{>0})$ with $X \subseteq V$ such that for all $x,y \in X$ the length of any shortest path in G between x and y equals d(x,y). In this paper we consider two special kinds of realizations, optimal realizations and hereditarily optimal realizations, and their relationship with the so-called tight span. In particular, we present an infinite family of metrics {dk}k≥1, and—using a new characterization for when the so-called underlying graph of a metric is an optimal realization that we also present—we prove that dk has (as a function of k) exponentially many optimal realizations with distinct degree sequences. We then show that this family of metrics provides counter-examples to a conjecture made by Dress in 1984 concerning the relationship between optimal realizations and the tight span, and a negative reply to a question posed by Althofer in 1988 on the relationship between optimal and hereditarily optimal realizations.

##### Place, publisher, year, edition, pages
2007. Vol. 38, no 3, 605-614 p.
Mathematics
##### Identifiers
ISI: 000249696700007OAI: oai:DiVA.org:uu-96396DiVA: diva2:170956
Available from: 2007-11-07 Created: 2007-11-07 Last updated: 2011-01-25Bibliographically approved
##### In thesis
1. Optimal and Hereditarily Optimal Realizations of Metric Spaces
Open this publication in new window or tab >>Optimal and Hereditarily Optimal Realizations of Metric Spaces
2007 (English)Doctoral thesis, comprehensive summary (Other academic)
##### Alternative title[sv]
Optimala och ärftligt optimala realiseringar av metriker
##### Abstract [en]

This PhD thesis, consisting of an introduction, four papers, and some supplementary results, studies the problem of finding an optimal realization of a given finite metric space: a weighted graph which preserves the metric's distances and has minimal total edge weight. This problem is known to be NP-hard, and solutions are not necessarily unique.

It has been conjectured that extremally weighted optimal realizations may be found as subgraphs of the hereditarily optimal realization Γd, a graph which in general has a higher total edge weight than the optimal realization but has the advantages of being unique, and possible to construct explicitly via the tight span of the metric.

In Paper I, we prove that the graph Γd is equivalent to the 1-skeleton of the tight span precisely when the metric considered is totally split-decomposable. For the subset of totally split-decomposable metrics known as consistent metrics this implies that Γd is isomorphic to the easily constructed Buneman graph.

In Paper II, we show that for any metric on at most five points, any optimal realization can be found as a subgraph of Γd.

In Paper III we provide a series of counterexamples; metrics for which there exist extremally weighted optimal realizations which are not subgraphs of Γd. However, for these examples there also exists at least one optimal realization which is a subgraph.

Finally, Paper IV examines a weakened conjecture suggested by the above counterexamples: can we always find some optimal realization as a subgraph in Γd? Defining extremal optimal realizations as those having the maximum possible number of shortest paths, we prove that any embedding of the vertices of an extremal optimal realization into Γd is injective. Moreover, we prove that this weakened conjecture holds for the subset of consistent metrics which have a 2-dimensional tight span

##### Place, publisher, year, edition, pages
Uppsala: Matematiska institutionen, 2007. 70 p.
##### Series
Uppsala Dissertations in Mathematics, ISSN 1401-2049 ; 52
##### Keyword
Applied mathematics, optimal realization, hereditarily optimal realization, tight span, phylogenetic network, Buneman graph, split decomposition, T-theory, finite metric space, topological graph theory, discrete geometry, Tillämpad matematik
##### Identifiers
urn:nbn:se:uu:diva-8297 (URN)978-91-506-1967-6 (ISBN)
##### Public defence
2007-11-30, Polhemssalen, Ångströmlaboratoriet, Lägerhyddsvägen 1, Uppsala, 10:15
##### Supervisors
Available from: 2007-11-07 Created: 2007-11-07Bibliographically approved

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##### By organisation
Department of MathematicsThe Linnaeus Centre for Bioinformatics
##### In the same journal
Discrete & Computational Geometry
Mathematics

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