4,413 research outputs found
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
Diversities and the Geometry of Hypergraphs
The embedding of finite metrics in has become a fundamental tool for
both combinatorial optimization and large-scale data analysis. One important
application is to network flow problems in which there is close relation
between max-flow min-cut theorems and the minimal distortion embeddings of
metrics into . Here we show that this theory can be generalized
considerably to encompass Steiner tree packing problems in both graphs and
hypergraphs. Instead of the theory of metrics and minimal distortion
embeddings, the parallel is the theory of diversities recently introduced by
Bryant and Tupper, and the corresponding theory of diversities and
embeddings which we develop here.Comment: 19 pages, no figures. This version: further small correction
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