729 research outputs found

    Escaping a grid by edge-disjoint paths

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    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

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    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

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    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

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    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 nn axis-aligned rectangles inside an axis-aligned bounding box BB, extend each rectangle in only one of the four directions: up, down, left, or right until it reaches BB and the density kk is minimized, where kk is the maximum number of extensions of rectangles to the boundary that pass through a point inside bounding box BB. REP is NP-hard for k>1k>1. 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 22-approximation algorithm for SEP with k2k\geq2 with time complexity O(n3/2k2)O(n^{3/2}k^2). This improves the time complexity of existing algorithms which are at least quadratic. Also, the approximation ratio of our algorithm for k3k\geq 3 is 3/23/2 which is tight. We also give a 88-approximation algorithm for REP with time complexity O(nlogn+nk)O(n\log n+nk) and give a MPC version of this algorithm for k=O(1)k=O(1) which is the first parallel algorithm for this problem
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