6,154 research outputs found

    Packing and covering with balls on Busemann surfaces

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    In this note we prove that for any compact subset SS of a Busemann surface (S,d)({\mathcal S},d) (in particular, for any simple polygon with geodesic metric) and any positive number δ\delta, the minimum number of closed balls of radius δ\delta with centers at S\mathcal S and covering the set SS is at most 19 times the maximum number of disjoint closed balls of radius δ\delta centered at points of SS: ν(S)ρ(S)19ν(S)\nu(S) \le \rho(S) \le 19\nu(S), where ρ(S)\rho(S) and ν(S)\nu(S) are the covering and the packing numbers of SS by δ{\delta}-balls.Comment: 27 page

    Rectilinear Link Diameter and Radius in a Rectilinear Polygonal Domain

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    We study the computation of the diameter and radius under the rectilinear link distance within a rectilinear polygonal domain of nn vertices and hh holes. We introduce a \emph{graph of oriented distances} to encode the distance between pairs of points of the domain. This helps us transform the problem so that we can search through the candidates more efficiently. Our algorithm computes both the diameter and the radius in min{O(nω),O(n2+nhlogh+χ2)}\min \{\,O(n^\omega), O(n^2 + nh \log h + \chi^2)\,\} time, where ω<2.373\omega<2.373 denotes the matrix multiplication exponent and χΩ(n)O(n2)\chi\in \Omega(n)\cap O(n^2) is the number of edges of the graph of oriented distances. We also provide a faster algorithm for computing the diameter that runs in O(n2logn)O(n^2 \log n) time

    Constant-Factor Approximation for TSP with Disks

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    We revisit the traveling salesman problem with neighborhoods (TSPN) and present the first constant-ratio approximation for disks in the plane: Given a set of nn disks in the plane, a TSP tour whose length is at most O(1)O(1) times the optimal can be computed in time that is polynomial in nn. Our result is the first constant-ratio approximation for a class of planar convex bodies of arbitrary size and arbitrary intersections. In order to achieve a O(1)O(1)-approximation, we reduce the traveling salesman problem with disks, up to constant factors, to a minimum weight hitting set problem in a geometric hypergraph. The connection between TSPN and hitting sets in geometric hypergraphs, established here, is likely to have future applications.Comment: 14 pages, 3 figure

    Clustering with Neighborhoods

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    In the standard planar kk-center clustering problem, one is given a set PP of nn points in the plane, and the goal is to select kk center points, so as to minimize the maximum distance over points in PP to their nearest center. Here we initiate the systematic study of the clustering with neighborhoods problem, which generalizes the kk-center problem to allow the covered objects to be a set of general disjoint convex objects C\mathscr{C} rather than just a point set PP. For this problem we first show that there is a PTAS for approximating the number of centers. Specifically, if roptr_{opt} is the optimal radius for kk centers, then in nO(1/ε2)n^{O(1/\varepsilon^2)} time we can produce a set of (1+ε)k(1+\varepsilon)k centers with radius ropt\leq r_{opt}. If instead one considers the standard goal of approximating the optimal clustering radius, while keeping kk as a hard constraint, we show that the radius cannot be approximated within any factor in polynomial time unless P=NP\mathsf{P=NP}, even when C\mathscr{C} is a set of line segments. When C\mathscr{C} is a set of unit disks we show the problem is hard to approximate within a factor of 133236.99\frac{\sqrt{13}-\sqrt{3}}{2-\sqrt{3}}\approx 6.99. This hardness result complements our main result, where we show that when the objects are disks, of possibly differing radii, there is a (5+23)8.46(5+2\sqrt{3})\approx 8.46 approximation algorithm. Additionally, for unit disks we give an O(nlogk)+(k/ε)O(k)O(n\log k)+(k/\varepsilon)^{O(k)} time (1+ε)(1+\varepsilon)-approximation to the optimal radius, that is, an FPTAS for constant kk whose running time depends only linearly on nn. Finally, we show that the one dimensional version of the problem, even when intersections are allowed, can be solved exactly in O(nlogn)O(n\log n) time

    Magnetic-field dependence of transport in normal and Andreev billiards: a classical interpretation to the averaged quantum behavior

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    We perform a comparative study of the quantum and classical transport probabilities of low-energy quasiparticles ballistically traversing normal and Andreev two-dimensional open cavities with a Sinai-billiard shape. We focus on the dependence of the transport on the strength of an applied magnetic field BB. With increasing field strength the classical dynamics changes from mixed to regular phase space. Averaging out the quantum fluctuations, we find an excellent agreement between the quantum and classical transport coefficients in the complete range of field strengths. This allows an overall description of the non-monotonic behavior of the average magnetoconductance in terms of the corresponding classical trajectories, thus, establishing a basic tool useful in the design and analysis of experiments.Comment: 11 pages, 12 figures; minor revisions including updated inset of Fig. 4(b) and references; version as accepted for publication to Phys. Rev.

    Critical Percolation Exploration Path and SLE(6): a Proof of Convergence

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    It was argued by Schramm and Smirnov that the critical site percolation exploration path on the triangular lattice converges in distribution to the trace of chordal SLE(6). We provide here a detailed proof, which relies on Smirnov's theorem that crossing probabilities have a conformally invariant scaling limit (given by Cardy's formula). The version of convergence to SLE(6) that we prove suffices for the Smirnov-Werner derivation of certain critical percolation crossing exponents and for our analysis of the critical percolation full scaling limit as a process of continuum nonsimple loops.Comment: 45 pages, 14 figures; revised version following the comments of a refere
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