72,188 research outputs found

    Floer cohomology of torus fibers and real lagrangians in Fano toric manifolds

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    In this article, we consider the Floer cohomology (with Z2\Z_2 coefficients) between torus fibers and the real Lagrangian in Fano toric manifolds. We first investigate the conditions under which the Floer cohomology is defined, and then develop a combinatorial description of the Floer complex based on the polytope of the toric manifold. We show that if the Floer cohomology is defined, and the Floer cohomology of the torus fiber is non-zero, then the Floer cohomology of the pair is non-zero. We use this result to develop some applications to non-displaceability and the minimum number of intersection points under Hamiltonian isotopy.Comment: v2: Modified exposition and new corollary adde

    Two-Level Rectilinear Steiner Trees

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    Given a set PP of terminals in the plane and a partition of PP into kk subsets P1,...,PkP_1, ..., P_k, a two-level rectilinear Steiner tree consists of a rectilinear Steiner tree TiT_i connecting the terminals in each set PiP_i (i=1,...,ki=1,...,k) and a top-level tree TtopT_{top} connecting the trees T1,...,TkT_1, ..., T_k. The goal is to minimize the total length of all trees. This problem arises naturally in the design of low-power physical implementations of parity functions on a computer chip. For bounded kk we present a polynomial time approximation scheme (PTAS) that is based on Arora's PTAS for rectilinear Steiner trees after lifting each partition into an extra dimension. For the general case we propose an algorithm that predetermines a connection point for each TiT_i and TtopT_{top} (i=1,...,ki=1,...,k). Then, we apply any approximation algorithm for minimum rectilinear Steiner trees in the plane to compute each TiT_i and TtopT_{top} independently. This gives us a 2.372.37-factor approximation with a running time of O(PlogP)\mathcal{O}(|P|\log|P|) suitable for fast practical computations. The approximation factor reduces to 1.631.63 by applying Arora's approximation scheme in the plane

    Hyperorthogonal well-folded Hilbert curves

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    R-trees can be used to store and query sets of point data in two or more dimensions. An easy way to construct and maintain R-trees for two-dimensional points, due to Kamel and Faloutsos, is to keep the points in the order in which they appear along the Hilbert curve. The R-tree will then store bounding boxes of points along contiguous sections of the curve, and the efficiency of the R-tree depends on the size of the bounding boxes---smaller is better. Since there are many different ways to generalize the Hilbert curve to higher dimensions, this raises the question which generalization results in the smallest bounding boxes. Familiar methods, such as the one by Butz, can result in curve sections whose bounding boxes are a factor Ω(2d/2)\Omega(2^{d/2}) larger than the volume traversed by that section of the curve. Most of the volume bounded by such bounding boxes would not contain any data points. In this paper we present a new way of generalizing Hilbert's curve to higher dimensions, which results in much tighter bounding boxes: they have at most 4 times the volume of the part of the curve covered, independent of the number of dimensions. Moreover, we prove that a factor 4 is asymptotically optimal.Comment: Manuscript submitted to Journal of Computational Geometry. An abstract appeared in the 31st Int Symp on Computational Geometry (SoCG 2015), LIPIcs 34:812-82

    Automatic Markov Chain Monte Carlo Procedures for Sampling from Multivariate Distributions

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    Generating samples from multivariate distributions efficiently is an important task in Monte Carlo integration and many other stochastic simulation problems. Markov chain Monte Carlo has been shown to be very efficient compared to "conventional methods", especially when many dimensions are involved. In this article we propose a Hit-and-Run sampler in combination with the Ratio-of-Uniforms method. We show that it is well suited for an algorithm to generate points from quite arbitrary distributions, which include all log-concave distributions. The algorithm works automatically in the sense that only the mode (or an approximation of it) and an oracle is required, i.e., a subroutine that returns the value of the density function at any point x. We show that the number of evaluations of the density increases slowly with dimension. (author's abstract)Series: Preprint Series / Department of Applied Statistics and Data Processin

    Polychromatic Coloring for Half-Planes

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    We prove that for every integer kk, every finite set of points in the plane can be kk-colored so that every half-plane that contains at least 2k12k-1 points, also contains at least one point from every color class. We also show that the bound 2k12k-1 is best possible. This improves the best previously known lower and upper bounds of 43k\frac{4}{3}k and 4k14k-1 respectively. We also show that every finite set of half-planes can be kk colored so that if a point pp belongs to a subset HpH_p of at least 3k23k-2 of the half-planes then HpH_p contains a half-plane from every color class. This improves the best previously known upper bound of 8k38k-3. Another corollary of our first result is a new proof of the existence of small size \eps-nets for points in the plane with respect to half-planes.Comment: 11 pages, 5 figure

    Centroidal localization game

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    One important problem in a network is to locate an (invisible) moving entity by using distance-detectors placed at strategical locations. For instance, the metric dimension of a graph GG is the minimum number kk of detectors placed in some vertices {v1,,vk}\{v_1,\cdots,v_k\} such that the vector (d1,,dk)(d_1,\cdots,d_k) of the distances d(vi,r)d(v_i,r) between the detectors and the entity's location rr allows to uniquely determine rV(G)r \in V(G). In a more realistic setting, instead of getting the exact distance information, given devices placed in {v1,,vk}\{v_1,\cdots,v_k\}, we get only relative distances between the entity's location rr and the devices (for every 1i,jk1\leq i,j\leq k, it is provided whether d(vi,r)>d(v_i,r) >, <<, or == to d(vj,r)d(v_j,r)). The centroidal dimension of a graph GG is the minimum number of devices required to locate the entity in this setting. We consider the natural generalization of the latter problem, where vertices may be probed sequentially until the moving entity is located. At every turn, a set {v1,,vk}\{v_1,\cdots,v_k\} of vertices is probed and then the relative distances between the vertices viv_i and the current location rr of the entity are given. If not located, the moving entity may move along one edge. Let ζ(G)\zeta^* (G) be the minimum kk such that the entity is eventually located, whatever it does, in the graph GG. We prove that ζ(T)2\zeta^* (T)\leq 2 for every tree TT and give an upper bound on ζ(GH)\zeta^*(G\square H) in cartesian product of graphs GG and HH. Our main result is that ζ(G)3\zeta^* (G)\leq 3 for any outerplanar graph GG. We then prove that ζ(G)\zeta^* (G) is bounded by the pathwidth of GG plus 1 and that the optimization problem of determining ζ(G)\zeta^* (G) is NP-hard in general graphs. Finally, we show that approximating (up to any constant distance) the entity's location in the Euclidean plane requires at most two vertices per turn
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