1 research outputs found
Reeb Posets and Tree Approximations
A well known result in the analysis of finite metric spaces due to Gromov
says that given any there exists a \emph{tree metric} on
such that is bounded above by twice . Here is the \emph{hyperbolicity} of , a
quantity that measures the \emph{treeness} of -tuples of points in . This
bound is known to be asymptotically tight.
We improve this bound by restricting ourselves to metric spaces arising from
filtered posets. By doing so we are able to replace the cardinality appearing
in Gromov's bound by a certain poset theoretic invariant (the maximum length of
fences in the poset) which can be much smaller thus significantly improving the
approximation bound.
The setting of metric spaces arising from posets is rich: For example, save
for the possible addition of new vertices, every finite metric graph can be
induced from a filtered poset. Since every finite metric space can be
isometrically embedded into a finite metric graph, our ideas are applicable to
finite metric spaces as well.
At the core of our results lies the adaptation of the Reeb graph and Reeb
tree constructions and the concept of hyperbolicity to the setting of posets,
which we use to formulate and prove a tree approximation result for any
filtered poset.Comment: 18 pages, 2 figure