2,016 research outputs found
Practical methods for approximating shortest paths on a convex polytope in R3
AbstractWe propose an extremely simple approximation scheme for computing shortest paths on the surface of a convex polytope in three dimensions. Given a convex polytope P with n vertices and two points p, q on its surface, let dP(p, q) denote the shortest path distance between p and q on the surface of P. Our algorithm produces a path of length at most 2dP(p, q) in time O(n). Extending this result, we can also compute an approximation of the shortest path tree rooted at an arbitrary point x ∈ P in time O(n log n). In the approximate tree, the distance between a vertex v ∈ P and x is at most cdP(x, v), where c = 2.38(1 + ε) for any fixed ε > 0. The best algorithms for computing an exact shortest path on a convex polytope take Ω(n2) time in the worst case; in addition, they are too complicated to be suitable in practice. We can also get a weak approximation result in the general case of k disjoint convex polyhedra: in O(n) time our algorithm gives a path of length at most 2k times the optimal
Geodesic Paths On 3D Surfaces: Survey and Open Problems
This survey gives a brief overview of theoretically and practically relevant
algorithms to compute geodesic paths and distances on three-dimensional
surfaces. The survey focuses on polyhedral three-dimensional surfaces
A Pseudopolynomial Algorithm for Alexandrov's Theorem
Alexandrov's Theorem states that every metric with the global topology and
local geometry required of a convex polyhedron is in fact the intrinsic metric
of a unique convex polyhedron. Recent work by Bobenko and Izmestiev describes a
differential equation whose solution leads to the polyhedron corresponding to a
given metric. We describe an algorithm based on this differential equation to
compute the polyhedron to arbitrary precision given the metric, and prove a
pseudopolynomial bound on its running time. Along the way, we develop
pseudopolynomial algorithms for computing shortest paths and weighted Delaunay
triangulations on a polyhedral surface, even when the surface edges are not
shortest paths.Comment: 25 pages; new Delaunay triangulation algorithm, minor other changes;
an abbreviated v2 was at WADS 200
Polyhedral computational geometry for averaging metric phylogenetic trees
This paper investigates the computational geometry relevant to calculations
of the Frechet mean and variance for probability distributions on the
phylogenetic tree space of Billera, Holmes and Vogtmann, using the theory of
probability measures on spaces of nonpositive curvature developed by Sturm. We
show that the combinatorics of geodesics with a specified fixed endpoint in
tree space are determined by the location of the varying endpoint in a certain
polyhedral subdivision of tree space. The variance function associated to a
finite subset of tree space has a fixed algebraic formula within
each cell of the corresponding subdivision, and is continuously differentiable
in the interior of each orthant of tree space. We use this subdivision to
establish two iterative methods for producing sequences that converge to the
Frechet mean: one based on Sturm's Law of Large Numbers, and another based on
descent algorithms for finding optima of smooth functions on convex polyhedra.
We present properties and biological applications of Frechet means and extend
our main results to more general globally nonpositively curved spaces composed
of Euclidean orthants.Comment: 43 pages, 6 figures; v2: fixed typos, shortened Sections 1 and 5,
added counter example for polyhedrality of vistal subdivision in general
CAT(0) cubical complexes; v1: 43 pages, 5 figure
Shapes of polyhedra and triangulations of the sphere
The space of shapes of a polyhedron with given total angles less than 2\pi at
each of its n vertices has a Kaehler metric, locally isometric to complex
hyperbolic space CH^{n-3}. The metric is not complete: collisions between
vertices take place a finite distance from a nonsingular point. The metric
completion is a complex hyperbolic cone-manifold. In some interesting special
cases, the metric completion is an orbifold. The concrete description of these
spaces of shapes gives information about the combinatorial classification of
triangulations of the sphere with no more than 6 triangles at a vertex.Comment: 39 pages. Published copy, also available at
http://www.maths.warwick.ac.uk/gt/GTMon1/paper25.abs.htm
Complexity of Bradley-Manna-Sipma Lexicographic Ranking Functions
In this paper we turn the spotlight on a class of lexicographic ranking
functions introduced by Bradley, Manna and Sipma in a seminal CAV 2005 paper,
and establish for the first time the complexity of some problems involving the
inference of such functions for linear-constraint loops (without precondition).
We show that finding such a function, if one exists, can be done in polynomial
time in a way which is sound and complete when the variables range over the
rationals (or reals). We show that when variables range over the integers, the
problem is harder -- deciding the existence of a ranking function is
coNP-complete. Next, we study the problem of minimizing the number of
components in the ranking function (a.k.a. the dimension). This number is
interesting in contexts like computing iteration bounds and loop
parallelization. Surprisingly, and unlike the situation for some other classes
of lexicographic ranking functions, we find that even deciding whether a
two-component ranking function exists is harder than the unrestricted problem:
NP-complete over the rationals and -complete over the integers.Comment: Technical report for a corresponding CAV'15 pape
Eight-Fifth Approximation for TSP Paths
We prove the approximation ratio 8/5 for the metric -path-TSP
problem, and more generally for shortest connected -joins.
The algorithm that achieves this ratio is the simple "Best of Many" version
of Christofides' algorithm (1976), suggested by An, Kleinberg and Shmoys
(2012), which consists in determining the best Christofides -tour out
of those constructed from a family \Fscr_{>0} of trees having a convex
combination dominated by an optimal solution of the fractional
relaxation. They give the approximation guarantee for
such an -tour, which is the first improvement after the 5/3 guarantee
of Hoogeveen's Christofides type algorithm (1991). Cheriyan, Friggstad and Gao
(2012) extended this result to a 13/8-approximation of shortest connected
-joins, for .
The ratio 8/5 is proved by simplifying and improving the approach of An,
Kleinberg and Shmoys that consists in completing in order to dominate
the cost of "parity correction" for spanning trees. We partition the edge-set
of each spanning tree in \Fscr_{>0} into an -path (or more
generally, into a -join) and its complement, which induces a decomposition
of . This decomposition can be refined and then efficiently used to
complete without using linear programming or particular properties of
, but by adding to each cut deficient for an individually tailored
explicitly given vector, inherent in .
A simple example shows that the Best of Many Christofides algorithm may not
find a shorter -tour than 3/2 times the incidentally common optima of
the problem and of its fractional relaxation.Comment: 15 pages, corrected typos in citations, minor change
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