2,036 research outputs found
Ramified rectilinear polygons: coordinatization by dendrons
Simple rectilinear polygons (i.e. rectilinear polygons without holes or
cutpoints) can be regarded as finite rectangular cell complexes coordinatized
by two finite dendrons. The intrinsic -metric is thus inherited from the
product of the two finite dendrons via an isometric embedding. The rectangular
cell complexes that share this same embedding property are called ramified
rectilinear polygons. The links of vertices in these cell complexes may be
arbitrary bipartite graphs, in contrast to simple rectilinear polygons where
the links of points are either 4-cycles or paths of length at most 3. Ramified
rectilinear polygons are particular instances of rectangular complexes obtained
from cube-free median graphs, or equivalently simply connected rectangular
complexes with triangle-free links. The underlying graphs of finite ramified
rectilinear polygons can be recognized among graphs in linear time by a
Lexicographic Breadth-First-Search. Whereas the symmetry of a simple
rectilinear polygon is very restricted (with automorphism group being a
subgroup of the dihedral group ), ramified rectilinear polygons are
universal: every finite group is the automorphism group of some ramified
rectilinear polygon.Comment: 27 pages, 6 figure
Manhattan orbifolds
We investigate a class of metrics for 2-manifolds in which, except for a
discrete set of singular points, the metric is locally isometric to an L_1 (or
equivalently L_infinity) metric, and show that with certain additional
conditions such metrics are injective. We use this construction to find the
tight span of squaregraphs and related graphs, and we find an injective metric
that approximates the distances in the hyperbolic plane analogously to the way
the rectilinear metrics approximate the Euclidean distance.Comment: 17 pages, 15 figures. Some definitions and proofs have been revised
since the previous version, and a new example has been adde
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
Factorization homology and calculus à la Kontsevich Soibelman
International audienc
Fast Clustering with Lower Bounds: No Customer too Far, No Shop too Small
We study the \LowerBoundedCenter (\lbc) problem, which is a clustering
problem that can be viewed as a variant of the \kCenter problem. In the \lbc
problem, we are given a set of points P in a metric space and a lower bound
\lambda, and the goal is to select a set C \subseteq P of centers and an
assignment that maps each point in P to a center of C such that each center of
C is assigned at least \lambda points. The price of an assignment is the
maximum distance between a point and the center it is assigned to, and the goal
is to find a set of centers and an assignment of minimum price. We give a
constant factor approximation algorithm for the \lbc problem that runs in O(n
\log n) time when the input points lie in the d-dimensional Euclidean space
R^d, where d is a constant. We also prove that this problem cannot be
approximated within a factor of 1.8-\epsilon unless P = \NP even if the input
points are points in the Euclidean plane R^2.Comment: 14 page
Grid Diagrams and Legendrian Lens Space Links
Grid diagrams encode useful geometric information about knots in S^3. In
particular, they can be used to combinatorially define the knot Floer homology
of a knot K in S^3, and they have a straightforward connection to Legendrian
representatives of K in (S^3, \xi_\st), where \xi_\st is the standard, tight
contact structure. The definition of a grid diagram was extended to include a
description for links in all lens spaces, resulting in a combinatorial
description of the knot Floer homology of a knot K in L(p, q) for all p > 0. In
the present article, we explore the connection between lens space grid diagrams
and the contact topology of a lens space. Our hope is that an understanding of
grid diagrams from this point of view will lead to new approaches to the Berge
conjecture, which claims to classify all knots in S^3 upon which surgery yields
a lens space.Comment: 27 pages, 20 figure
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