5,629 research outputs found
Computing the Distribution of a Tree Metric
The Robinson-Foulds (RF) distance is by far the most widely used measure of
dissimilarity between trees. Although the distribution of these distances has
been investigated for twenty years, an algorithm that is explicitly polynomial
time has yet to be described for computing this distribution (which is also the
distribution of trees around a given tree under the popular Robinson-Foulds
metric). In this paper we derive a polynomial-time algorithm for this
distribution. We show how the distribution can be approximated by a Poisson
distribution determined by the proportion of leaves that lie in `cherries' of
the given tree. We also describe how our results can be used to derive
normalization constants that are required in a recently-proposed maximum
likelihood approach to supertree construction.Comment: 16 pages, 3 figure
Constant distortion embeddings of Symmetric Diversities
Diversities are like metric spaces, except that every finite subset, instead
of just every pair of points, is assigned a value. Just as there is a theory of
minimal distortion embeddings of finite metric spaces into , there is a
similar, yet undeveloped, theory for embedding finite diversities into the
diversity analogue of spaces. In the metric case, it is well known that
an -point metric space can be embedded into with
distortion. For diversities, the optimal distortion is unknown. Here, we
establish the surprising result that symmetric diversities, those in which the
diversity (value) assigned to a set depends only on its cardinality, can be
embedded in with constant distortion.Comment: 14 pages, 3 figure
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
New Zealand’s Diaspora and Overseas-born Population
Many New Zealand-born people migrate overseas, creating a diaspora, and many overseas-born people migrate to New Zealand. Both the diaspora and the overseas-born population in New Zealand may facilitate the international exchange of goods and ideas. Much discussion of international linkages has, however, been limited by a lack of data on numbers of people involved. Based mainly on place-of-birth data from national censuses, this paper provides estimates of the size and structure of New Zealand’s diaspora and overseas-born population, as well as comparisons with selected OECD countries such as Australia. A tentative conclusion is that the potential contribution of New Zealand’s diaspora may have been overestimated, and the contribution of the overseas-born population underestimated.International migration; diaspora; measurement; New Zealand; Australia; population; emigration; immigration
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