An Optimal-Time Algorithm for Slope Selection

Cole, Richard; Salowe, Jeffrey S.; Steiger, William L.; Szemerédi, Endre

doi:10.1137/0218055

Cited by 120 publications

(90 citation statements)

References 8 publications

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“…Previous works (such as cuttings [CF90,Mat90] or equipartitions [LS03]) have focused on identifying, and bounding the number of lines/hyperplanes intersecting a set of regions. Others [CSSS89] on partitioning the vertices of the arrangements rather than the lines themselves. Those results have found numerous applications.…”

Section: Introductionmentioning

confidence: 99%

A Center Transversal Theorem for Hyperplanes and Applications to Graph Drawing

Dujmović

Langerman

2012

Discrete Comput Geom

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Motivated by an open problem from graph drawing, we study several partitioning problems for line and hyperplane arrangements. We prove a ham-sandwich cut theorem: given two sets of n lines in R 2 , there is a line such that in both line sets, for both halfplanes delimited by , there are √ n lines which pairwise intersect in that halfplane, and this bound is tight; a centerpoint theorem: for any set of n lines there is a point such that for any halfplane containing that point there are n/3 of the lines which pairwise intersect in that halfplane. We generalize those results in higher dimension and obtain a center transversal theorem, a same-type lemma, and a positive portion Erdős-Szekeres theorem for hyperplane arrangements. This is done by formulating a generalization of the center transversal theorem which applies to set functions that are much more general than measures. Back to Graph Drawing (and in the plane), we completely solve the open problem that motivated our search: there is no set of n labelled lines that are universal for all n-vertex labelled planar graphs. As a side note, we prove that every set of n (unlabelled) lines is universal for all n-vertex (unlabelled) planar graphs.

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Section: Introductionmentioning

confidence: 99%

A Center Transversal Theorem for Hyperplanes and Applications to Graph Drawing

Dujmović

Langerman

2012

Discrete Comput Geom

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“…We run a binary search over these intersections, to shrink I further to a slab between two consecutive intersections (that contains a local minimum). To guide the binary search, we use the classical slope-selection procedure [10] that can compute, for a given slab I and a given parameter k, the k-th leftmost intersection point of the lines in S within I, in O(nm log(nm)) time. With this procedure at hand, the binary search performs O(log (nm)) steps, each taking O(nm log (nm)) time, both for finding the relevant intersection point, and for running Γ 1 at the corresponding vertical line.…”

Section: Minimum Hausdorff Rms Distance Under Translation In Two Dimementioning

confidence: 99%

Partial-Matching RMS Distance Under Translation: Combinatorics and Algorithms

et al. 2017

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We consider the RMS distance (sum of squared distances between pairs of points) under translation between two point sets in the plane, in two different setups. In the partial-matching setup, each point in the smaller set is matched to a distinct point in the bigger set. Although the problem is not known to be polynomial, we establish several structural properties of the underlying subdivision of the plane and derive improved bounds on its complexity. These results lead to the best known algorithm for finding a translation for which the partial-matching RMS distance between the point sets is minimized. In addition, we show how to compute a local minimum of the partial-matching RMS distance under translation, in polynomial time. In the Hausdorff setup, each point is paired to its nearest neighbor in the other set. We develop algorithms for finding a local minimum of the Hausdorff RMS distance in nearly linear time on the line, and in nearly quadratic time in the plane. These improve substantially the worst-case behavior of the popular ICP heuristics for solving this problem.

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“…All of these methods are based on intersecting higher-degree curves and/or surfaces, which are then searched (sometimes parametrically [1], [11], [12], [13], [30]) to find a global minimum. This reliance upon intersection computations leads to algorithms that are potentially numerically unstable, are conceptually complex, and have running times that are high for all but the most trivial motions.…”

Section: Previous Workmentioning

confidence: 99%