597 research outputs found
The hardness of perfect phylogeny, feasible register assignment and other problems on thin colored graphs
AbstractIn this paper, we consider the complexity of a number of combinatorial problems; namely, Intervalizing Colored Graphs (DNA physical mapping), Triangulating Colored Graphs (perfect phylogeny), (Directed) (Modified) Colored Cutwidth, Feasible Register Assignment and Module Allocation for graphs of bounded pathwidth. Each of these problems has as a characteristic a uniform upper bound on the tree or path width of the graphs in “yes”-instances. For all of these problems with the exceptions of Feasible Register Assignment and Module Allocation, a vertex or edge coloring is given as part of the input. Our main results are that the parameterized variant of each of the considered problems is hard for the complexity classes W[t] for all t∈N. We also show that Intervalizing Colored Graphs, Triangulating Colored Graphs, and Colored Cutwidth are NP-Complete
Some Triangulated Surfaces without Balanced Splitting
Let G be the graph of a triangulated surface of genus . A
cycle of G is splitting if it cuts into two components, neither of
which is homeomorphic to a disk. A splitting cycle has type k if the
corresponding components have genera k and g-k. It was conjectured that G
contains a splitting cycle (Barnette '1982). We confirm this conjecture for an
infinite family of triangulations by complete graphs but give counter-examples
to a stronger conjecture (Mohar and Thomassen '2001) claiming that G should
contain splitting cycles of every possible type.Comment: 15 pages, 7 figure
Detecting Weakly Simple Polygons
A closed curve in the plane is weakly simple if it is the limit (in the
Fr\'echet metric) of a sequence of simple closed curves. We describe an
algorithm to determine whether a closed walk of length n in a simple plane
graph is weakly simple in O(n log n) time, improving an earlier O(n^3)-time
algorithm of Cortese et al. [Discrete Math. 2009]. As an immediate corollary,
we obtain the first efficient algorithm to determine whether an arbitrary
n-vertex polygon is weakly simple; our algorithm runs in O(n^2 log n) time. We
also describe algorithms that detect weak simplicity in O(n log n) time for two
interesting classes of polygons. Finally, we discuss subtle errors in several
previously published definitions of weak simplicity.Comment: 25 pages and 13 figures, submitted to SODA 201
Load-Balancing for Parallel Delaunay Triangulations
Computing the Delaunay triangulation (DT) of a given point set in
is one of the fundamental operations in computational geometry.
Recently, Funke and Sanders (2017) presented a divide-and-conquer DT algorithm
that merges two partial triangulations by re-triangulating a small subset of
their vertices - the border vertices - and combining the three triangulations
efficiently via parallel hash table lookups. The input point division should
therefore yield roughly equal-sized partitions for good load-balancing and also
result in a small number of border vertices for fast merging. In this paper, we
present a novel divide-step based on partitioning the triangulation of a small
sample of the input points. In experiments on synthetic and real-world data
sets, we achieve nearly perfectly balanced partitions and small border
triangulations. This almost cuts running time in half compared to
non-data-sensitive division schemes on inputs exhibiting an exploitable
underlying structure.Comment: Short version submitted to EuroPar 201
Computing trisections of 4-manifolds
Algorithms that decompose a manifold into simple pieces reveal the geometric
and topological structure of the manifold, showing how complicated structures
are constructed from simple building blocks. This note describes a way to
algorithmically construct a trisection, which describes a -dimensional
manifold as a union of three -dimensional handlebodies. The complexity of
the -manifold is captured in a collection of curves on a surface, which
guide the gluing of the handelbodies. The algorithm begins with a description
of a manifold as a union of pentachora, or -dimensional simplices. It
transforms this description into a trisection. This results in the first
explicit complexity bounds for the trisection genus of a -manifold in terms
of the number of pentachora (-simplices) in a triangulation.Comment: 15 pages, 9 figure
Implementing path coloring algorithms on planar graphs
Master's Project (M.S.) University of Alaska Fairbanks, 2017A path coloring of a graph partitions its vertex set into color classes such that each class induces a disjoint union of paths. In this project we implement several algorithms to compute path colorings of graphs embedded in the plane. We present two algorithms to path color plane graphs with 3 colors based on a proof by Poh in 1990. First we describe a naive algorithm that directly follows Poh's procedure, then we give a modified algorithm that runs in linear time. Independent results of Hartman and Skrekovski describe a procedure that takes a plane graph G and a list of 3 colors for each vertex, and computes a path coloring of G such that each vertex receives a color from its list. We present a linear time implementation based on Hartman and Skrekovski's proofs. A C++ implementation is provided for all three algorithms, utilizing the Boost Graph Library. Instructions are given on how to use the implementation to construct colorings for plane graphs represented by Boost data structures
Which point sets admit a k-angulation?
For k >= 3, a k-angulation is a 2-connected plane graph in which every
internal face is a k-gon. We say that a point set P admits a plane graph G if
there is a straight-line drawing of G that maps V(G) onto P and has the same
facial cycles and outer face as G. We investigate the conditions under which a
point set P admits a k-angulation and find that, for sets containing at least
2k^2 points, the only obstructions are those that follow from Euler's formula.Comment: 13 pages, 7 figure
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