1,856 research outputs found
Avoiding the Global Sort: A Faster Contour Tree Algorithm
We revisit the classical problem of computing the \emph{contour tree} of a
scalar field , where is a
triangulated simplicial mesh in . The contour tree is a
fundamental topological structure that tracks the evolution of level sets of
and has numerous applications in data analysis and visualization.
All existing algorithms begin with a global sort of at least all critical
values of , which can require (roughly) time. Existing
lower bounds show that there are pathological instances where this sort is
required. We present the first algorithm whose time complexity depends on the
contour tree structure, and avoids the global sort for non-pathological inputs.
If denotes the set of critical points in , the running time is
roughly , where is the depth of in
the contour tree. This matches all existing upper bounds, but is a significant
improvement when the contour tree is short and fat. Specifically, our approach
ensures that any comparison made is between nodes in the same descending path
in the contour tree, allowing us to argue strong optimality properties of our
algorithm.
Our algorithm requires several novel ideas: partitioning in
well-behaved portions, a local growing procedure to iteratively build contour
trees, and the use of heavy path decompositions for the time complexity
analysis
The Salesman's Improved Tours for Fundamental Classes
Finding the exact integrality gap for the LP relaxation of the
metric Travelling Salesman Problem (TSP) has been an open problem for over
thirty years, with little progress made. It is known that , and a famous conjecture states . For this problem,
essentially two "fundamental" classes of instances have been proposed. This
fundamental property means that in order to show that the integrality gap is at
most for all instances of metric TSP, it is sufficient to show it only
for the instances in the fundamental class. However, despite the importance and
the simplicity of such classes, no apparent effort has been deployed for
improving the integrality gap bounds for them. In this paper we take a natural
first step in this endeavour, and consider the -integer points of one such
class. We successfully improve the upper bound for the integrality gap from
to for a superclass of these points, as well as prove a lower
bound of for the superclass. Our methods involve innovative applications
of tools from combinatorial optimization which have the potential to be more
broadly applied
- …