335 research outputs found

    Spanners for Geometric Intersection Graphs

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    Efficient algorithms are presented for constructing spanners in geometric intersection graphs. For a unit ball graph in R^k, a (1+\epsilon)-spanner is obtained using efficient partitioning of the space into hypercubes and solving bichromatic closest pair problems. The spanner construction has almost equivalent complexity to the construction of Euclidean minimum spanning trees. The results are extended to arbitrary ball graphs with a sub-quadratic running time. For unit ball graphs, the spanners have a small separator decomposition which can be used to obtain efficient algorithms for approximating proximity problems like diameter and distance queries. The results on compressed quadtrees, geometric graph separators, and diameter approximation might be of independent interest.Comment: 16 pages, 5 figures, Late

    Exact Computation of a Manifold Metric, via Lipschitz Embeddings and Shortest Paths on a Graph

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    Data-sensitive metrics adapt distances locally based the density of data points with the goal of aligning distances and some notion of similarity. In this paper, we give the first exact algorithm for computing a data-sensitive metric called the nearest neighbor metric. In fact, we prove the surprising result that a previously published 33-approximation is an exact algorithm. The nearest neighbor metric can be viewed as a special case of a density-based distance used in machine learning, or it can be seen as an example of a manifold metric. Previous computational research on such metrics despaired of computing exact distances on account of the apparent difficulty of minimizing over all continuous paths between a pair of points. We leverage the exact computation of the nearest neighbor metric to compute sparse spanners and persistent homology. We also explore the behavior of the metric built from point sets drawn from an underlying distribution and consider the more general case of inputs that are finite collections of path-connected compact sets. The main results connect several classical theories such as the conformal change of Riemannian metrics, the theory of positive definite functions of Schoenberg, and screw function theory of Schoenberg and Von Neumann. We develop novel proof techniques based on the combination of screw functions and Lipschitz extensions that may be of independent interest.Comment: 15 page

    Sparse geometric graphs with small dilation

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    Given a set S of n points in R^D, and an integer k such that 0 <= k < n, we show that a geometric graph with vertex set S, at most n - 1 + k edges, maximum degree five, and dilation O(n / (k+1)) can be computed in time O(n log n). For any k, we also construct planar n-point sets for which any geometric graph with n-1+k edges has dilation Omega(n/(k+1)); a slightly weaker statement holds if the points of S are required to be in convex position

    A Spanner for the Day After

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    We show how to construct (1+ε)(1+\varepsilon)-spanner over a set PP of nn points in Rd\mathbb{R}^d that is resilient to a catastrophic failure of nodes. Specifically, for prescribed parameters ϑ,ε(0,1)\vartheta,\varepsilon \in (0,1), the computed spanner GG has O(εcϑ6nlogn(loglogn)6) O\bigl(\varepsilon^{-c} \vartheta^{-6} n \log n (\log\log n)^6 \bigr) edges, where c=O(d)c= O(d). Furthermore, for any kk, and any deleted set BPB \subseteq P of kk points, the residual graph GBG \setminus B is (1+ε)(1+\varepsilon)-spanner for all the points of PP except for (1+ϑ)k(1+\vartheta)k of them. No previous constructions, beyond the trivial clique with O(n2)O(n^2) edges, were known such that only a tiny additional fraction (i.e., ϑ\vartheta) lose their distance preserving connectivity. Our construction works by first solving the exact problem in one dimension, and then showing a surprisingly simple and elegant construction in higher dimensions, that uses the one-dimensional construction in a black box fashion

    Robust Geometric Spanners

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    Highly connected and yet sparse graphs (such as expanders or graphs of high treewidth) are fundamental, widely applicable and extensively studied combinatorial objects. We initiate the study of such highly connected graphs that are, in addition, geometric spanners. We define a property of spanners called robustness. Informally, when one removes a few vertices from a robust spanner, this harms only a small number of other vertices. We show that robust spanners must have a superlinear number of edges, even in one dimension. On the positive side, we give constructions, for any dimension, of robust spanners with a near-linear number of edges.Comment: 18 pages, 8 figure
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