21 research outputs found
Pants Decomposition of the Punctured Plane
A pants decomposition of an orientable surface S is a collection of simple
cycles that partition S into pants, i.e., surfaces of genus zero with three
boundary cycles. Given a set P of n points in the plane, we consider the
problem of computing a pants decomposition of the surface S which is the plane
minus P, of minimum total length. We give a polynomial-time approximation
scheme using Mitchell's guillotine rectilinear subdivisions. We give a
quartic-time algorithm to compute the shortest pants decomposition of S when
the cycles are restricted to be axis-aligned boxes, and a quadratic-time
algorithm when all the points lie on a line; both exact algorithms use dynamic
programming with Yao's speedup.Comment: 5 pages, 1 grayscale figur
Shortest path embeddings of graphs on surfaces
The classical theorem of F\'{a}ry states that every planar graph can be
represented by an embedding in which every edge is represented by a straight
line segment. We consider generalizations of F\'{a}ry's theorem to surfaces
equipped with Riemannian metrics. In this setting, we require that every edge
is drawn as a shortest path between its two endpoints and we call an embedding
with this property a shortest path embedding. The main question addressed in
this paper is whether given a closed surface S, there exists a Riemannian
metric for which every topologically embeddable graph admits a shortest path
embedding. This question is also motivated by various problems regarding
crossing numbers on surfaces.
We observe that the round metrics on the sphere and the projective plane have
this property. We provide flat metrics on the torus and the Klein bottle which
also have this property.
Then we show that for the unit square flat metric on the Klein bottle there
exists a graph without shortest path embeddings. We show, moreover, that for
large g, there exist graphs G embeddable into the orientable surface of genus
g, such that with large probability a random hyperbolic metric does not admit a
shortest path embedding of G, where the probability measure is proportional to
the Weil-Petersson volume on moduli space.
Finally, we construct a hyperbolic metric on every orientable surface S of
genus g, such that every graph embeddable into S can be embedded so that every
edge is a concatenation of at most O(g) shortest paths.Comment: 22 pages, 11 figures: Version 3 is updated after comments of
reviewer
A fast algorithm to compute cohomology group generators of orientable 2-manifolds
In this paper a fast algorithm to compute cohomology group generators of cellular decomposition of any orientable closed 2-manifold is presented. The presented algorithm is a dual version of algorithm to compute homology generators presented by David Eppstein [12] and developed by Jeff Erickson and Kim Whittlesey [13]
Computational Geometric and Algebraic Topology
Computational topology is a young, emerging field of mathematics that seeks out practical algorithmic methods for solving complex and fundamental problems in geometry and topology. It draws on a wide variety of techniques from across pure mathematics (including topology, differential geometry, combinatorics, algebra, and discrete geometry), as well as applied mathematics and theoretical computer science. In turn, solutions to these problems have a wide-ranging impact: already they have enabled significant progress in the core area of geometric topology, introduced new methods in applied mathematics, and yielded new insights into the role that topology has to play in fundamental problems surrounding computational complexity.
At least three significant branches have emerged in computational topology: algorithmic 3-manifold and knot theory, persistent homology and surfaces and graph embeddings. These branches have emerged largely independently. However, it is clear that they have much to offer each other. The goal of this workshop was to be the first significant step to bring these three areas together, to share ideas in depth, and to pool our expertise in approaching some of the major open problems in the field