16 research outputs found
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
Nonorientable 3-manifolds admitting coloured triangulations with at most 30 tetrahedra
We present the census of all non-orientable, closed, connected 3-manifolds
admitting a rigid crystallization with at most 30 vertices. In order to obtain
the above result, we generate, manipulate and compare, by suitable computer
procedures, all rigid non-bipartite crystallizations up to 30 vertices.Comment: 18 pages, 3 figure
The complexity of the normal surface solution space
Normal surface theory is a central tool in algorithmic three-dimensional
topology, and the enumeration of vertex normal surfaces is the computational
bottleneck in many important algorithms. However, it is not well understood how
the number of such surfaces grows in relation to the size of the underlying
triangulation. Here we address this problem in both theory and practice. In
theory, we tighten the exponential upper bound substantially; furthermore, we
construct pathological triangulations that prove an exponential bound to be
unavoidable. In practice, we undertake a comprehensive analysis of millions of
triangulations and find that in general the number of vertex normal surfaces is
remarkably small, with strong evidence that our pathological triangulations may
in fact be the worst case scenarios. This analysis is the first of its kind,
and the striking behaviour that we observe has important implications for the
feasibility of topological algorithms in three dimensions.Comment: Extended abstract (i.e., conference-style), 14 pages, 8 figures, 2
tables; v2: added minor clarification
Fixed parameter tractable algorithms in combinatorial topology
To enumerate 3-manifold triangulations with a given property, one typically
begins with a set of potential face pairing graphs (also known as dual
1-skeletons), and then attempts to flesh each graph out into full
triangulations using an exponential-time enumeration. However, asymptotically
most graphs do not result in any 3-manifold triangulation, which leads to
significant "wasted time" in topological enumeration algorithms. Here we give a
new algorithm to determine whether a given face pairing graph supports any
3-manifold triangulation, and show this to be fixed parameter tractable in the
treewidth of the graph.
We extend this result to a "meta-theorem" by defining a broad class of
properties of triangulations, each with a corresponding fixed parameter
tractable existence algorithm. We explicitly implement this algorithm in the
most generic setting, and we identify heuristics that in practice are seen to
mitigate the large constants that so often occur in parameterised complexity,
highlighting the practicality of our techniques.Comment: 16 pages, 9 figure
Detecting genus in vertex links for the fast enumeration of 3-manifold triangulations
Enumerating all 3-manifold triangulations of a given size is a difficult but
increasingly important problem in computational topology. A key difficulty for
enumeration algorithms is that most combinatorial triangulations must be
discarded because they do not represent topological 3-manifolds. In this paper
we show how to preempt bad triangulations by detecting genus in
partially-constructed vertex links, allowing us to prune the enumeration tree
substantially.
The key idea is to manipulate the boundary edges surrounding partial vertex
links using expected logarithmic time operations. Practical testing shows the
resulting enumeration algorithm to be significantly faster, with up to 249x
speed-ups even for small problems where comparisons are feasible. We also
discuss parallelisation, and describe new data sets that have been obtained
using high-performance computing facilities.Comment: 16 pages, 7 figures, 3 tables; v2: minor revisions; to appear in
ISSAC 201
Optimizing the double description method for normal surface enumeration
Many key algorithms in 3-manifold topology involve the enumeration of normal
surfaces, which is based upon the double description method for finding the
vertices of a convex polytope. Typically we are only interested in a small
subset of these vertices, thus opening the way for substantial optimization.
Here we give an account of the vertex enumeration problem as it applies to
normal surfaces, and present new optimizations that yield strong improvements
in both running time and memory consumption. The resulting algorithms are
tested using the freely available software package Regina.Comment: 27 pages, 12 figures; v2: Removed the 3^n bound from Section 3.3,
fixed the projective equation in Lemma 4.4, clarified "most triangulations"
in the introduction to section 5; v3: replace -ise with -ize for Mathematics
of Computation (note that this changes the title of the paper
An edge-based framework for enumerating 3-manifold triangulations
A typical census of 3-manifolds contains all manifolds (under various
constraints) that can be triangulated with at most n tetrahedra. Al- though
censuses are useful resources for mathematicians, constructing them is
difficult: the best algorithms to date have not gone beyond n = 12. The
underlying algorithms essentially (i) enumerate all relevant 4-regular
multigraphs on n nodes, and then (ii) for each multigraph G they enumerate
possible 3-manifold triangulations with G as their dual 1-skeleton, of which
there could be exponentially many. In practice, a small number of multigraphs
often dominate the running times of census algorithms: for example, in a
typical census on 10 tetrahedra, almost half of the running time is spent on
just 0.3% of the graphs.
Here we present a new algorithm for stage (ii), which is the computational
bottleneck in this process. The key idea is to build triangulations by
recursively constructing neighbourhoods of edges, in contrast to traditional
algorithms which recursively glue together pairs of tetrahedron faces. We
implement this algorithm, and find experimentally that whilst the overall
performance is mixed, the new algorithm runs significantly faster on those
"pathological" multigraphs for which existing methods are extremely slow. In
this way the old and new algorithms complement one another, and together can
yield significant performance improvements over either method alone.Comment: 29 pages, 19 figure