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Selfimproving algorithms
 in SODA ’06: Proceedings of the seventeenth annual ACMSIAM symposium on Discrete algorithm
"... We investigate ways in which an algorithm can improve its expected performance by finetuning itself automatically with respect to an arbitrary, unknown input distribution. We give such selfimproving algorithms for sorting and computing Delaunay triangulations. The highlights of this work: (i) an al ..."
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Cited by 26 (4 self)
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We investigate ways in which an algorithm can improve its expected performance by finetuning itself automatically with respect to an arbitrary, unknown input distribution. We give such selfimproving algorithms for sorting and computing Delaunay triangulations. The highlights of this work: (i) an algorithm to sort a list of numbers with optimal expected limiting complexity; and (ii) an algorithm to compute the Delaunay triangulation of a set of points with optimal expected limiting complexity. In both cases, the algorithm begins with a training phase during which it adjusts itself to the input distribution, followed by a stationary regime in which the algorithm settles to its optimized incarnation. 1
Transdichotomous Results in Computational Geometry, II: Offline Search
, 2010
"... We reexamine fundamental problems from computational geometry in the word RAM model, where input coordinates are integers that fit in a machine word. We develop a new algorithm for offline point location, a twodimensional analog of sorting where one needs to order points with respect to segments. T ..."
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Cited by 13 (3 self)
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We reexamine fundamental problems from computational geometry in the word RAM model, where input coordinates are integers that fit in a machine word. We develop a new algorithm for offline point location, a twodimensional analog of sorting where one needs to order points with respect to segments. This result implies, for example, that the convex hull of n points in three dimensions can be constructed in (randomized) time n·2 O( √ lglgn). Similar bounds hold for numerous other geometric problems, such as planar Voronoi diagrams, planar offline nearest neighbor search, line segment intersection, and triangulation of nonsimple polygons. In FOCS’06, we developed a data structure for online point location, which implied a bound of O(n lgn lglgn) for threedimensional convex hulls and the other problems. Our current bounds are dramatically better, and a convincing improvement over the classic O(nlgn) algorithms. As in the field of integer sorting, the main challenge is to find ways to manipulate information, while avoiding the online problem (in that case, predecessor search).
Computing Hereditary Convex Structures
 SCG'09
, 2009
"... Color red and blue the n vertices of a convex polytope P in R³. Can we compute the convex hull of each color class in o(n log n)? What if we have χ> 2 colors? What if the colors are random? Consider an arbitrary query halfspace and call the vertices of P inside it blue: can the convex hull of the bl ..."
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Cited by 8 (4 self)
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Color red and blue the n vertices of a convex polytope P in R³. Can we compute the convex hull of each color class in o(n log n)? What if we have χ> 2 colors? What if the colors are random? Consider an arbitrary query halfspace and call the vertices of P inside it blue: can the convex hull of the blue points be computed in time linear in their number? More generally, can we quickly compute the blue hull without looking at the whole polytope? This paper considers several instances of hereditary computation and provides new results for them. In particular, we resolve an eightyear old open problem by showing how to split a convex polytope in linear expected time.
Four soviets walk the dog  with an application to Alt’s conjecture
 CoRR
"... Given two polygonal curves in the plane, there are many ways to define a notion of similarity between them. One measure that is extremely popular is the Fréchet distance. Since it has been proposed by Alt and Godau in 1992, many variants and extensions have been studied. Nonetheless, even more than ..."
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Cited by 2 (1 self)
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Given two polygonal curves in the plane, there are many ways to define a notion of similarity between them. One measure that is extremely popular is the Fréchet distance. Since it has been proposed by Alt and Godau in 1992, many variants and extensions have been studied. Nonetheless, even more than 20 years later, the original O(n 2 log n) algorithm by Alt and Godau for computing the Fréchet distance remains the state of the art (here n denotes the number of vertices on each curve). This has led Helmut Alt to conjecture that the associated decision problem is 3SUMhard. In recent work, Agarwal et al. show how to break the quadratic barrier for the discrete version of the Fréchet distance, where one considers sequences of points instead of polygonal curves. Building on their work, we give a randomized algorithm to compute the Fréchet distance between two polygonal curves in time O(n 2 √ log n(log log n) 3/2) on a pointer machine and in time O(n 2 (log log n) 2) on a word RAM. Furthermore, we show that there exists an algebraic decision tree for the decision problem of depth O(n 2−ε), for some ε> 0. This provides evidence that the decision problem may not be 3SUMhard after all and reveals an intriguing new aspect of this wellstudied problem. 1
Bichromatic Line Segment Intersection Counting in O(n √ log n) Time
"... We give an algorithm for bichromatic line segment intersection counting that runs in O(n √ log n) time under the word RAM model via a reduction to dynamic predecessor search, offline point location, and offline dynamic ranking. This algorithm is the first to solve bichromatic line segment intersecti ..."
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We give an algorithm for bichromatic line segment intersection counting that runs in O(n √ log n) time under the word RAM model via a reduction to dynamic predecessor search, offline point location, and offline dynamic ranking. This algorithm is the first to solve bichromatic line segment intersection counting in o(n log n) time. 1
An Algorithm for Report Writing Research report for Geometric Algorithms (2IL55) – Spring semester 2010 Ruud Random Student number: 123456
, 2010
"... Introduce the topic of the report: state the problem, give motivation for studying the problem, refer to applications, and give an overview of your report. If you do not want to lose your reader right away, give her a smooth start! Get her ..."
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Introduce the topic of the report: state the problem, give motivation for studying the problem, refer to applications, and give an overview of your report. If you do not want to lose your reader right away, give her a smooth start! Get her
1 2 3 Triangulating the Square and Squaring the Triangle:
"... We show that Delaunay triangulations and compressed quadtrees are equivalent structures. More precisely, we give two algorithms: the first computes a compressed quadtree for a planar point set, given the Delaunay triangulation; the second finds the Delaunay triangulation, given a compressed quadtree ..."
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We show that Delaunay triangulations and compressed quadtrees are equivalent structures. More precisely, we give two algorithms: the first computes a compressed quadtree for a planar point set, given the Delaunay triangulation; the second finds the Delaunay triangulation, given a compressed quadtree. Both algorithms run in deterministic linear time on a pointer machine. Our work builds on and extends previous results by Krznaric and Levcopolous [38] and Buchin and Mulzer [9]. Our main tool for the second algorithm is the wellseparated pair decomposition (WSPD) [12], a structure that has been used previously to find Euclidean minimum spanning trees in higher dimensions [26]. We show that knowing the WSPD (and a quadtree) suffices to compute a planar Euclidean minimum spanning tree (EMST) in linear time. With the EMST at hand, we can find the Delaunay triangulation in linear time [20]. As a corollary, we obtain deterministic versions of many previous algorithms related to Delaunay triangulations, such as splitting planar Delaunay triangulations [18,19], preprocessing imprecise points for faster Delaunay computation [8, 40], and transdichotomous Delaunay triangulations [9, 14, 15].