2,367 research outputs found
Happy endings for flip graphs
We show that the triangulations of a finite point set form a flip graph that
can be embedded isometrically into a hypercube, if and only if the point set
has no empty convex pentagon. Point sets of this type include convex subsets of
lattices, points on two lines, and several other infinite families. As a
consequence, flip distance in such point sets can be computed efficiently.Comment: 26 pages, 15 figures. Revised and expanded for journal publicatio
Operads, configuration spaces and quantization
We review several well-known operads of compactified configuration spaces and
construct several new such operads, C, in the category of smooth manifolds with
corners whose complexes of fundamental chains give us (i) the 2-coloured operad
of A-infinity algebras and their homotopy morphisms, (ii) the 2-coloured operad
of L-infinity algebras and their homotopy morphisms, and (iii) the 4-coloured
operad of open-closed homotopy algebras and their homotopy morphisms. Two
gadgets - a (coloured) operad of Feynman graphs and a de Rham field theory on C
- are introduced and used to construct quantized representations of the
(fundamental) chain operad of C which are given by Feynman type sums over
graphs and depend on choices of propagators.Comment: 58 page
06481 Abstracts Collection -- Geometric Networks and Metric Space Embeddings
The Dagstuhl Seminar 06481 ``Geometric Networks and Metric Space
Embeddings\u27\u27 was held from November~26 to December~1, 2006 in the
International Conference and Research Center (IBFI), Schloss
Dagstuhl. During the seminar, several participants presented their
current research, and ongoing work and open problems were discussed.
In this paper we describe the seminar topics, we have compiled a
list of open questions that were posed during the seminar, there is
a list of all talks and there are abstracts of the presentations
given during the seminar. Links to extended abstracts or full
papers are provided where available
Oriented Spanners
Given a point set P in the Euclidean plane and a parameter t, we define an oriented t-spanner as an oriented subgraph of the complete bi-directed graph such that for every pair of points, the shortest cycle in G through those points is at most a factor t longer than the shortest oriented cycle in the complete bi-directed graph. We investigate the problem of computing sparse graphs with small oriented dilation.
As we can show that minimising oriented dilation for a given number of edges is NP-hard in the plane, we first consider one-dimensional point sets. While obtaining a 1-spanner in this setting is straightforward, already for five points such a spanner has no plane embedding with the leftmost and rightmost point on the outer face. This leads to restricting to oriented graphs with a one-page book embedding on the one-dimensional point set. For this case we present a dynamic program to compute the graph of minimum oriented dilation that runs in ?(n?) time for n points, and a greedy algorithm that computes a 5-spanner in ?(nlog n) time.
Expanding these results finally gives us a result for two-dimensional point sets: we prove that for convex point sets the greedy triangulation results in an oriented ?(1)-spanner
Oriented Spanners
Given a point set in the Euclidean plane and a parameter , we define
an \emph{oriented -spanner} as an oriented subgraph of the complete
bi-directed graph such that for every pair of points, the shortest cycle in
through those points is at most a factor longer than the shortest oriented
cycle in the complete bi-directed graph. We investigate the problem of
computing sparse graphs with small oriented dilation.
As we can show that minimising oriented dilation for a given number of edges
is NP-hard in the plane, we first consider one-dimensional point sets. While
obtaining a -spanner in this setting is straightforward, already for five
points such a spanner has no plane embedding with the leftmost and rightmost
point on the outer face.
This leads to restricting to oriented graphs with a one-page book embedding
on the one-dimensional point set. For this case we present a dynamic program to
compute the graph of minimum oriented dilation that runs in time for
points, and a greedy algorithm that computes a -spanner in
time.
Expanding these results finally gives us a result for two-dimensional point
sets: we prove that for convex point sets the greedy triangulation results in
an oriented -spanner.Comment: conference version: ESA '2
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