729 research outputs found
Escaping a grid by edge-disjoint paths
We study the edge-disjoint escape problem in grids: Given a set of n sources in a two-dimensional grid, the problem is to connect all sources to the grid boundary using a set of n edge-disjoint paths. Different from the conventional approach that reduces the problem to network flow problem, we solve the problem by ensuring that no rectangles in the grid contain more sources than outlets, a necessary and sufficient condition for the existence of a solution. Based on this condition, we give a greedy algorithm which finds the paths in O(n2) time, which is faster than the previous approaches. This problem has applications in point-to-point delivery, VLSI reconfiguration and package routing.published_or_final_versio
The edge-disjoint path problem on random graphs by message-passing
We present a message-passing algorithm to solve the edge disjoint path
problem (EDP) on graphs incorporating under a unique framework both traffic
optimization and path length minimization. The min-sum equations for this
problem present an exponential computational cost in the number of paths. To
overcome this obstacle we propose an efficient implementation by mapping the
equations onto a weighted combinatorial matching problem over an auxiliary
graph. We perform extensive numerical simulations on random graphs of various
types to test the performance both in terms of path length minimization and
maximization of the number of accommodated paths. In addition, we test the
performance on benchmark instances on various graphs by comparison with
state-of-the-art algorithms and results found in the literature. Our
message-passing algorithm always outperforms the others in terms of the number
of accommodated paths when considering non trivial instances (otherwise it
gives the same trivial results). Remarkably, the largest improvement in
performance with respect to the other methods employed is found in the case of
benchmarks with meshes, where the validity hypothesis behind message-passing is
expected to worsen. In these cases, even though the exact message-passing
equations do not converge, by introducing a reinforcement parameter to force
convergence towards a sub optimal solution, we were able to always outperform
the other algorithms with a peak of 27% performance improvement in terms of
accommodated paths. On random graphs, we numerically observe two separated
regimes: one in which all paths can be accommodated and one in which this is
not possible. We also investigate the behaviour of both the number of paths to
be accommodated and their minimum total length.Comment: 14 pages, 8 figure
Endperiodic Automorphisms of Surfaces and Foliations
We extend the unpublished work of M. Handel and R. Miller on the
classification, up to isotopy, of endperiodic automorphisms of surfaces. We
give the Handel-Miller construction of the geodesic laminations, give an
axiomatic theory for pseudo-geodesic lamaniations, show the geodesic
laminations satisfy the axioms, and prove that paeudo-geodesic laminations
satisfying our axioms are ambiently isotopic to the geodesic laminations. The
axiomatic approach allows us to show that the given endperiodic automorphism is
isotopic to a smooth endperiodic automorphism preserving smooth laminations
ambiently isotopic to the original ones. Using the axioms, we also prove the
"transfer theorem" for foliations of 3-manifolds., namely that, if two depth
one foliations are transverse to a common one-dimensional foliation whose
monodromy on the noncompact leaves of the first foliation exhibits the nice
dynamics of Handel-Miller theory, then the transverse one-dimensional foliation
also induces monodromy on the noncompact leaves of the second foliation
exhibiting the same nice dynamics. Our theory also applies to surfaces with
infinitely many ends.Comment: Added Sergio Fenley as author. Moved material from Section 12.6 to a
new Section 6.7. Rewrote Section 7. Deleted material from Section 6.1 and
combined Sections 6.1 and 6.2 into new Section 6.1. Rewrote Section 4.6.
Corrected typos and errors and improved expositio
A Massively Parallel Dynamic Programming for Approximate Rectangle Escape Problem
Sublinear time complexity is required by the massively parallel computation
(MPC) model. Breaking dynamic programs into a set of sparse dynamic programs
that can be divided, solved, and merged in sublinear time.
The rectangle escape problem (REP) is defined as follows: For
axis-aligned rectangles inside an axis-aligned bounding box , extend each
rectangle in only one of the four directions: up, down, left, or right until it
reaches and the density is minimized, where is the maximum number
of extensions of rectangles to the boundary that pass through a point inside
bounding box . REP is NP-hard for . If the rectangles are points of a
grid (or unit squares of a grid), the problem is called the square escape
problem (SEP) and it is still NP-hard.
We give a -approximation algorithm for SEP with with time
complexity . This improves the time complexity of existing
algorithms which are at least quadratic. Also, the approximation ratio of our
algorithm for is which is tight. We also give a
-approximation algorithm for REP with time complexity and
give a MPC version of this algorithm for which is the first parallel
algorithm for this problem
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