7,313 research outputs found

    Minimum Convex Partitions and Maximum Empty Polytopes

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    Let SS be a set of nn points in Rd\mathbb{R}^d. A Steiner convex partition is a tiling of conv(S){\rm conv}(S) with empty convex bodies. For every integer dd, we show that SS admits a Steiner convex partition with at most (n1)/d\lceil (n-1)/d\rceil tiles. This bound is the best possible for points in general position in the plane, and it is best possible apart from constant factors in every fixed dimension d3d\geq 3. We also give the first constant-factor approximation algorithm for computing a minimum Steiner convex partition of a planar point set in general position. Establishing a tight lower bound for the maximum volume of a tile in a Steiner convex partition of any nn points in the unit cube is equivalent to a famous problem of Danzer and Rogers. It is conjectured that the volume of the largest tile is ω(1/n)\omega(1/n). Here we give a (1ε)(1-\varepsilon)-approximation algorithm for computing the maximum volume of an empty convex body amidst nn given points in the dd-dimensional unit box [0,1]d[0,1]^d.Comment: 16 pages, 4 figures; revised write-up with some running times improve

    Approximating the Maximum Overlap of Polygons under Translation

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    Let PP and QQ be two simple polygons in the plane of total complexity nn, each of which can be decomposed into at most kk convex parts. We present an (1ε)(1-\varepsilon)-approximation algorithm, for finding the translation of QQ, which maximizes its area of overlap with PP. Our algorithm runs in O(cn)O(c n) time, where cc is a constant that depends only on kk and ε\varepsilon. This suggest that for polygons that are "close" to being convex, the problem can be solved (approximately), in near linear time

    Deconstructing Approximate Offsets

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    We consider the offset-deconstruction problem: Given a polygonal shape Q with n vertices, can it be expressed, up to a tolerance \eps in Hausdorff distance, as the Minkowski sum of another polygonal shape P with a disk of fixed radius? If it does, we also seek a preferably simple-looking solution P; then, P's offset constitutes an accurate, vertex-reduced, and smoothened approximation of Q. We give an O(n log n)-time exact decision algorithm that handles any polygonal shape, assuming the real-RAM model of computation. A variant of the algorithm, which we have implemented using CGAL, is based on rational arithmetic and answers the same deconstruction problem up to an uncertainty parameter \delta; its running time additionally depends on \delta. If the input shape is found to be approximable, this algorithm also computes an approximate solution for the problem. It also allows us to solve parameter-optimization problems induced by the offset-deconstruction problem. For convex shapes, the complexity of the exact decision algorithm drops to O(n), which is also the time required to compute a solution P with at most one more vertex than a vertex-minimal one.Comment: 18 pages, 11 figures, previous version accepted at SoCG 2011, submitted to DC

    Covering the Boundary of a Simple Polygon with Geodesic Unit Disks

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    We consider the problem of covering the boundary of a simple polygon on n vertices using the minimum number of geodesic unit disks. We present an O(n \log^2 n+k) time 2-approximation algorithm for finding the centers of the disks, with k denoting the number centers found by the algorithm

    Computational Geometry Column 42

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    A compendium of thirty previously published open problems in computational geometry is presented.Comment: 7 pages; 72 reference

    Separation-Sensitive Collision Detection for Convex Objects

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    We develop a class of new kinetic data structures for collision detection between moving convex polytopes; the performance of these structures is sensitive to the separation of the polytopes during their motion. For two convex polygons in the plane, let DD be the maximum diameter of the polygons, and let ss be the minimum distance between them during their motion. Our separation certificate changes O(log(D/s))O(\log(D/s)) times when the relative motion of the two polygons is a translation along a straight line or convex curve, O(D/s)O(\sqrt{D/s}) for translation along an algebraic trajectory, and O(D/s)O(D/s) for algebraic rigid motion (translation and rotation). Each certificate update is performed in O(log(D/s))O(\log(D/s)) time. Variants of these data structures are also shown that exhibit \emph{hysteresis}---after a separation certificate fails, the new certificate cannot fail again until the objects have moved by some constant fraction of their current separation. We can then bound the number of events by the combinatorial size of a certain cover of the motion path by balls.Comment: 10 pages, 8 figures; to appear in Proc. 10th Annual ACM-SIAM Symposium on Discrete Algorithms, 1999; see also http://www.uiuc.edu/ph/www/jeffe/pubs/kollide.html ; v2 replaces submission with camera-ready versio

    Fat Polygonal Partitions with Applications to Visualization and Embeddings

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    Let T\mathcal{T} be a rooted and weighted tree, where the weight of any node is equal to the sum of the weights of its children. The popular Treemap algorithm visualizes such a tree as a hierarchical partition of a square into rectangles, where the area of the rectangle corresponding to any node in T\mathcal{T} is equal to the weight of that node. The aspect ratio of the rectangles in such a rectangular partition necessarily depends on the weights and can become arbitrarily high. We introduce a new hierarchical partition scheme, called a polygonal partition, which uses convex polygons rather than just rectangles. We present two methods for constructing polygonal partitions, both having guarantees on the worst-case aspect ratio of the constructed polygons; in particular, both methods guarantee a bound on the aspect ratio that is independent of the weights of the nodes. We also consider rectangular partitions with slack, where the areas of the rectangles may differ slightly from the weights of the corresponding nodes. We show that this makes it possible to obtain partitions with constant aspect ratio. This result generalizes to hyper-rectangular partitions in Rd\mathbb{R}^d. We use these partitions with slack for embedding ultrametrics into dd-dimensional Euclidean space: we give a polylog(Δ)\mathop{\rm polylog}(\Delta)-approximation algorithm for embedding nn-point ultrametrics into Rd\mathbb{R}^d with minimum distortion, where Δ\Delta denotes the spread of the metric, i.e., the ratio between the largest and the smallest distance between two points. The previously best-known approximation ratio for this problem was polynomial in nn. This is the first algorithm for embedding a non-trivial family of weighted-graph metrics into a space of constant dimension that achieves polylogarithmic approximation ratio.Comment: 26 page
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