1,526 research outputs found
Trees, Tight-Spans and Point Configuration
Tight-spans of metrics were first introduced by Isbell in 1964 and
rediscovered and studied by others, most notably by Dress, who gave them this
name. Subsequently, it was found that tight-spans could be defined for more
general maps, such as directed metrics and distances, and more recently for
diversities. In this paper, we show that all of these tight-spans as well as
some related constructions can be defined in terms of point configurations.
This provides a useful way in which to study these objects in a unified and
systematic way. We also show that by using point configurations we can recover
results concerning one-dimensional tight-spans for all of the maps we consider,
as well as extend these and other results to more general maps such as
symmetric and unsymmetric maps.Comment: 21 pages, 2 figure
Hyperconvexity and Tight Span Theory for Diversities
The tight span, or injective envelope, is an elegant and useful construction
that takes a metric space and returns the smallest hyperconvex space into which
it can be embedded. The concept has stimulated a large body of theory and has
applications to metric classification and data visualisation. Here we introduce
a generalisation of metrics, called diversities, and demonstrate that the rich
theory associated to metric tight spans and hyperconvexity extends to a
seemingly richer theory of diversity tight spans and hyperconvexity.Comment: revised in response to referee comment
On the half-plane property and the Tutte group of a matroid
A multivariate polynomial is stable if it is non-vanishing whenever all
variables have positive imaginary parts. A matroid has the weak half-plane
property (WHPP) if there exists a stable polynomial with support equal to the
set of bases of the matroid. If the polynomial can be chosen with all of its
nonzero coefficients equal to one then the matroid has the half-plane property
(HPP). We describe a systematic method that allows us to reduce the WHPP to the
HPP for large families of matroids. This method makes use of the Tutte group of
a matroid. We prove that no projective geometry has the WHPP and that a binary
matroid has the WHPP if and only if it is regular. We also prove that T_8 and
R_9 fail to have the WHPP.Comment: 8 pages. To appear in J. Combin. Theory Ser.
Computing the blocks of a quasi-median graph
Quasi-median graphs are a tool commonly used by evolutionary biologists to
visualise the evolution of molecular sequences. As with any graph, a
quasi-median graph can contain cut vertices, that is, vertices whose removal
disconnect the graph. These vertices induce a decomposition of the graph into
blocks, that is, maximal subgraphs which do not contain any cut vertices. Here
we show that the special structure of quasi-median graphs can be used to
compute their blocks without having to compute the whole graph. In particular
we present an algorithm that, for a collection of aligned sequences of
length , can compute the blocks of the associated quasi-median graph
together with the information required to correctly connect these blocks
together in run time , independent of the size of the
sequence alphabet. Our primary motivation for presenting this algorithm is the
fact that the quasi-median graph associated to a sequence alignment must
contain all most parsimonious trees for the alignment, and therefore
precomputing the blocks of the graph has the potential to help speed up any
method for computing such trees.Comment: 17 pages, 2 figure
Embedding into the rectilinear plane in optimal O*(n^2)
We present an optimal O*(n^2) time algorithm for deciding if a metric space
(X,d) on n points can be isometrically embedded into the plane endowed with the
l_1-metric. It improves the O*(n^2 log^2 n) time algorithm of J. Edmonds
(2008). Together with some ingredients introduced by J. Edmonds, our algorithm
uses the concept of tight span and the injectivity of the l_1-plane. A
different O*(n^2) time algorithm was recently proposed by D. Eppstein (2009).Comment: 12 pages, 13 figure
On the relation between hyperrings and fuzzy rings
We construct a full embedding of the category of hyperfields into Dress's category of fuzzy rings and explicitly characterize the essential image --- it fails to be essentially surjective in a very minor way. This embedding provides an identification of Baker's theory of matroids over hyperfields with Dress's theory of matroids over fuzzy rings (provided one restricts to those fuzzy rings in the essential image). The embedding functor extends from hyperfields to hyperrings, and we study this extension in detail. We also analyze the relation between hyperfields and Baker's partial demifields
On Patchworks and Hierarchies
Motivated by questions in biological classification, we discuss some
elementary combinatorial and computational properties of certain set systems
that generalize hierarchies, namely, 'patchworks', 'weak patchworks', 'ample
patchworks' and 'saturated patchworks' and also outline how these concepts
relate to an apparently new 'duality theory' for cluster systems that is based
on the fundamental concept of 'compatibility' of clusters.Comment: 17 pages, 2 figure
Searching for Realizations of Finite Metric Spaces in Tight Spans
An important problem that commonly arises in areas such as internet
traffic-flow analysis, phylogenetics and electrical circuit design, is to find
a representation of any given metric on a finite set by an edge-weighted
graph, such that the total edge length of the graph is minimum over all such
graphs. Such a graph is called an optimal realization and finding such
realizations is known to be NP-hard. Recently Varone presented a heuristic
greedy algorithm for computing optimal realizations. Here we present an
alternative heuristic that exploits the relationship between realizations of
the metric and its so-called tight span . The tight span is a
canonical polytopal complex that can be associated to , and our approach
explores parts of for realizations in a way that is similar to the
classical simplex algorithm. We also provide computational results illustrating
the performance of our approach for different types of metrics, including
-distances and two-decomposable metrics for which it is provably possible
to find optimal realizations in their tight spans.Comment: 20 pages, 3 figure
A matroid associated with a phylogenetic tree
A (pseudo-)metric D on a finite set X is said to be a `tree metric' if there is a finite tree with leaf set X and non-negative edge weights so that, for all x,y ∈X, D(x,y) is the path distance in the tree between x and y. It is well known that not every metric is a tree metric. However, when some such tree exists, one can always find one whose interior edges have strictly positive edge weights and that has no vertices of degree 2, any such tree is 13; up to canonical isomorphism 13; uniquely determined by D, and one does not even need all of the distances in order to fully (re-)construct the tree's edge weights in this case. Thus, it seems of some interest to investigate which subsets of X, 2 suffice to determine (`lasso') these edge weights. In this paper, we use the results of a previous paper to discuss the structure of a matroid that can be associated with an (unweighted) X-tree T defined by the requirement that its bases are exactly the `tight edge-weight lassos' for T, i.e, the minimal subsets of X, 2 that lasso the edge weights of T
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