325 research outputs found
On the Enumeration of all Minimal Triangulations
We present an algorithm that enumerates all the minimal triangulations of a
graph in incremental polynomial time. Consequently, we get an algorithm for
enumerating all the proper tree decompositions, in incremental polynomial time,
where "proper" means that the tree decomposition cannot be improved by removing
or splitting a bag
Enumeration of non-orientable 3-manifolds using face pairing graphs and union-find
Drawing together techniques from combinatorics and computer science, we
improve the census algorithm for enumerating closed minimal P^2-irreducible
3-manifold triangulations. In particular, new constraints are proven for face
pairing graphs, and pruning techniques are improved using a modification of the
union-find algorithm. Using these results we catalogue all 136 closed
non-orientable P^2-irreducible 3-manifolds that can be formed from at most ten
tetrahedra.Comment: 37 pages, 34 figure
Admissible Colourings of 3-Manifold Triangulations for Turaev-Viro Type Invariants
Turaev-Viro invariants are amongst the most powerful tools to distinguish 3-manifolds. They are invaluable for mathematical software, but current algorithms to compute them rely on the enumeration of an extremely large set of combinatorial data defined on the triangulation, regardless of the underlying topology of the manifold.
In the article, we propose a finer study of these combinatorial data, called admissible colourings, in relation with the cohomology of the manifold. We prove that the set of admissible colourings to be considered is substantially smaller than previously known, by furnishing new upper bounds on its size that are aware of the topology of the manifold. Moreover, we deduce new topology-sensitive enumeration algorithms based on these bounds.
The paper provides a theoretical analysis, as well as a detailed experimental study of the approach. We give strong experimental evidence on large manifold censuses that our upper bounds are tighter than the previously known ones, and that our algorithms outperform significantly state of the art implementations to compute Turaev-Viro invariants
Counting and Enumerating Crossing-free Geometric Graphs
We describe a framework for counting and enumerating various types of
crossing-free geometric graphs on a planar point set. The framework generalizes
ideas of Alvarez and Seidel, who used them to count triangulations in time
where is the number of points. The main idea is to reduce the
problem of counting geometric graphs to counting source-sink paths in a
directed acyclic graph.
The following new results will emerge. The number of all crossing-free
geometric graphs can be computed in time for some .
The number of crossing-free convex partitions can be computed in time
. The number of crossing-free perfect matchings can be computed in
time . The number of convex subdivisions can be computed in time
. The number of crossing-free spanning trees can be computed in time
for some . The number of crossing-free spanning cycles
can be computed in time for some .
With the same bounds on the running time we can construct data structures
which allow fast enumeration of the respective classes. For example, after
time of preprocessing we can enumerate the set of all crossing-free
perfect matchings using polynomial time per enumerated object. For
crossing-free perfect matchings and convex partitions we further obtain
enumeration algorithms where the time delay for each (in particular, the first)
output is bounded by a polynomial in .
All described algorithms are comparatively simple, both in terms of their
analysis and implementation
Enumerating fundamental normal surfaces: Algorithms, experiments and invariants
Computational knot theory and 3-manifold topology have seen significant
breakthroughs in recent years, despite the fact that many key algorithms have
complexity bounds that are exponential or greater. In this setting,
experimentation is essential for understanding the limits of practicality, as
well as for gauging the relative merits of competing algorithms.
In this paper we focus on normal surface theory, a key tool that appears
throughout low-dimensional topology. Stepping beyond the well-studied problem
of computing vertex normal surfaces (essentially extreme rays of a polyhedral
cone), we turn our attention to the more complex task of computing fundamental
normal surfaces (essentially an integral basis for such a cone). We develop,
implement and experimentally compare a primal and a dual algorithm, both of
which combine domain-specific techniques with classical Hilbert basis
algorithms. Our experiments indicate that we can solve extremely large problems
that were once though intractable. As a practical application of our
techniques, we fill gaps from the KnotInfo database by computing 398
previously-unknown crosscap numbers of knots.Comment: 17 pages, 5 figures; v2: Stronger experimental focus, restrict
attention to primal & dual algorithms only, larger and more detailed
experiments, more new crosscap number
On Deletion in Delaunay Triangulation
This paper presents how the space of spheres and shelling may be used to
delete a point from a -dimensional triangulation efficiently. In dimension
two, if k is the degree of the deleted vertex, the complexity is O(k log k),
but we notice that this number only applies to low cost operations, while time
consuming computations are only done a linear number of times.
This algorithm may be viewed as a variation of Heller's algorithm, which is
popular in the geographic information system community. Unfortunately, Heller
algorithm is false, as explained in this paper.Comment: 15 pages 5 figures. in Proc. 15th Annu. ACM Sympos. Comput. Geom.,
181--188, 199
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