5 research outputs found
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
ATX-Approach to Some Results on Cuts and Metrics
AbstractWe give simple algorithmic proofs of some theorems of17and970549on the packing of metrics by cuts
Optimally fast incremental Manhattan plane embedding and planar tight span construction
We describe a data structure, a rectangular complex, that can be used to
represent hyperconvex metric spaces that have the same topology (although not
necessarily the same distance function) as subsets of the plane. We show how to
use this data structure to construct the tight span of a metric space given as
an n x n distance matrix, when the tight span is homeomorphic to a subset of
the plane, in time O(n^2), and to add a single point to a planar tight span in
time O(n). As an application of this construction, we show how to test whether
a given finite metric space embeds isometrically into the Manhattan plane in
time O(n^2), and add a single point to the space and re-test whether it has
such an embedding in time O(n).Comment: 39 pages, 15 figure