603 research outputs found
Partitioning Graph Drawings and Triangulated Simple Polygons into Greedily Routable Regions
A greedily routable region (GRR) is a closed subset of , in
which each destination point can be reached from each starting point by
choosing the direction with maximum reduction of the distance to the
destination in each point of the path.
Recently, Tan and Kermarrec proposed a geographic routing protocol for dense
wireless sensor networks based on decomposing the network area into a small
number of interior-disjoint GRRs. They showed that minimum decomposition is
NP-hard for polygons with holes.
We consider minimum GRR decomposition for plane straight-line drawings of
graphs. Here, GRRs coincide with self-approaching drawings of trees, a drawing
style which has become a popular research topic in graph drawing. We show that
minimum decomposition is still NP-hard for graphs with cycles, but can be
solved optimally for trees in polynomial time. Additionally, we give a
2-approximation for simple polygons, if a given triangulation has to be
respected.Comment: full version of a paper appearing in ISAAC 201
Discrete Riemann Surfaces and the Ising model
We define a new theory of discrete Riemann surfaces and present its basic
results. The key idea is to consider not only a cellular decomposition of a
surface, but the union with its dual. Discrete holomorphy is defined by a
straightforward discretisation of the Cauchy-Riemann equation. A lot of
classical results in Riemann theory have a discrete counterpart, Hodge star,
harmonicity, Hodge theorem, Weyl's lemma, Cauchy integral formula, existence of
holomorphic forms with prescribed holonomies. Giving a geometrical meaning to
the construction on a Riemann surface, we define a notion of criticality on
which we prove a continuous limit theorem. We investigate its connection with
criticality in the Ising model. We set up a Dirac equation on a discrete
universal spin structure and we prove that the existence of a Dirac spinor is
equivalent to criticality
Toric topology
We survey some results on toric topology.Comment: English translation of the Japanese article which appeared in
"Sugaku" vol. 62 (2010), 386-41
Splitting Polytopes
A split of a polytope is a (regular) subdivision with exactly two maximal
cells. It turns out that each weight function on the vertices of admits a
unique decomposition as a linear combination of weight functions corresponding
to the splits of (with a split prime remainder). This generalizes a result
of Bandelt and Dress [Adv. Math. 92 (1992)] on the decomposition of finite
metric spaces.
Introducing the concept of compatibility of splits gives rise to a finite
simplicial complex associated with any polytope , the split complex of .
Complete descriptions of the split complexes of all hypersimplices are
obtained. Moreover, it is shown that these complexes arise as subcomplexes of
the tropical (pre-)Grassmannians of Speyer and Sturmfels [Adv. Geom. 4 (2004)].Comment: 25 pages, 7 figures; minor corrections and change
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