14,096 research outputs found
Spanners for Geometric Intersection Graphs
Efficient algorithms are presented for constructing spanners in geometric
intersection graphs. For a unit ball graph in R^k, a (1+\epsilon)-spanner is
obtained using efficient partitioning of the space into hypercubes and solving
bichromatic closest pair problems. The spanner construction has almost
equivalent complexity to the construction of Euclidean minimum spanning trees.
The results are extended to arbitrary ball graphs with a sub-quadratic running
time.
For unit ball graphs, the spanners have a small separator decomposition which
can be used to obtain efficient algorithms for approximating proximity problems
like diameter and distance queries. The results on compressed quadtrees,
geometric graph separators, and diameter approximation might be of independent
interest.Comment: 16 pages, 5 figures, Late
Squarepants in a Tree: Sum of Subtree Clustering and Hyperbolic Pants Decomposition
We provide efficient constant factor approximation algorithms for the
problems of finding a hierarchical clustering of a point set in any metric
space, minimizing the sum of minimimum spanning tree lengths within each
cluster, and in the hyperbolic or Euclidean planes, minimizing the sum of
cluster perimeters. Our algorithms for the hyperbolic and Euclidean planes can
also be used to provide a pants decomposition, that is, a set of disjoint
simple closed curves partitioning the plane minus the input points into subsets
with exactly three boundary components, with approximately minimum total
length. In the Euclidean case, these curves are squares; in the hyperbolic
case, they combine our Euclidean square pants decomposition with our tree
clustering method for general metric spaces.Comment: 22 pages, 14 figures. This version replaces the proof of what is now
Lemma 5.2, as the previous proof was erroneou
Paper folding, Riemann surfaces, and convergence of pseudo-Anosov sequences
A method is presented for constructing closed surfaces out of Euclidean
polygons with infinitely many segment identifications along the boundary. The
metric on the quotient is identified. A sufficient condition is presented which
guarantees that the Euclidean structure on the polygons induces a unique
conformal structure on the quotient surface, making it into a closed Riemann
surface. In this case, a modulus of continuity for uniformizing coordinates is
found which depends only on the geometry of the polygons and on the
identifications. An application is presented in which a uniform modulus of
continuity is obtained for a family of pseudo-Anosov homeomorphisms, making it
possible to prove that they converge to a Teichm\"uller mapping on the Riemann
sphere.Comment: 75 pages, 18 figure
Cannon-Thurston Maps for Kleinian Groups
We show that Cannon-Thurston maps exist for degenerate free groups without
parabolics, i.e. for handlebody groups. Combining these techniques with earlier
work proving the existence of Cannon-Thurston maps for surface groups, we show
that Cannon-Thurston maps exist for arbitrary finitely generated Kleinian
groups without parabolics, proving conjectures of Thurston and McMullen. We
also show that point pre-images under Cannon-Thurston maps for degenerate free
groups without parabolics correspond to end-points of leaves of an ending
lamination in the Masur domain, whenever a point has more than one pre-image.
This proves a conjecture of Otal. We also prove a similar result for point
pre-images under Cannon-Thurston maps for arbitrary finitely generated Kleinian
groups without parabolics.Comment: 39 pgs 1 fig. Final version incorporating referee comments. To appear
in Forum of Mathematics, P
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