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13
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 34 (6 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, I: Point Location in Sublogarithmic Time
, 2008
"... Given a planar subdivision whose coordinates are integers bounded by U ≤ 2 w, we present a linearspace data structure that can answer point location queries in O(min{lg n / lg lg n, √ lg U/lg lg U}) time on the unitcost RAM with word size w. Thisisthe first result to beat the standard Θ(lg n) bou ..."
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Cited by 23 (4 self)
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Given a planar subdivision whose coordinates are integers bounded by U ≤ 2 w, we present a linearspace data structure that can answer point location queries in O(min{lg n / lg lg n, √ lg U/lg lg U}) time on the unitcost RAM with word size w. Thisisthe first result to beat the standard Θ(lg n) bound for infinite precision models. As a consequence, we obtain the first o(n lg n) (randomized) algorithms for many fundamental problems in computational geometry for arbitrary integer input on the word RAM, including: constructing the convex hull of a threedimensional point set, computing the Voronoi diagram or the Euclidean minimum spanning tree of a planar point set, triangulating a polygon with holes, and finding intersections among a set of line segments. Higherdimensional extensions and applications are also discussed. Though computational geometry with bounded precision input has been investigated for a long time, improvements have been limited largely to problems of an orthogonal flavor. Our results surpass this longstanding limitation, answering, for example, a question of Willard (SODA’92).
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 20 (4 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.
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 ..."
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Cited by 9 (6 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.
Preprocessing Imprecise Points for Delaunay Triangulation: Simplified and Extended
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Triangulating the Square and Squaring the Triangle: Quadtrees and Delaunay Triangulations Are Equivalent
, 2011
"... 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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Cited by 4 (2 self)
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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 [40] and Buchin and Mulzer [10]. Our main tool for the second algorithm is the wellseparated pair decomposition (WSPD) [13], a structure that has been used previously to find Euclidean minimum spanning trees in higher dimensions [27]. We show that knowing the WSPD (and a quadtree) suffices to compute a planar EMST in linear time. With the EMST at hand, we can find the Delaunay triangulation in linear time [21]. As a corollary, we obtain deterministic versions of many previous algorithms related to Delaunay triangulations, such as splitting planar Delaunay triangulations [19, 20], preprocessing imprecise points for faster Delaunay computation [9, 42], and transdichotomous Delaunay triangulations [10, 15, 16].
A faster algorithm for computing motorcycle graphs
 PROC. 29TH SYMP. ON COMPUTATIONAL GEOMETRY, SOCG ’13, ACM
, 2013
"... We present a new algorithm for computing motorcycle graphs that runs in O(n4/3+ε) time for any ε> 0, improving on all previously known algorithms. The main application of this result is to computing the straight skeleton of a polygon. It allows us to compute the straight skeleton of a nondegener ..."
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Cited by 4 (1 self)
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We present a new algorithm for computing motorcycle graphs that runs in O(n4/3+ε) time for any ε> 0, improving on all previously known algorithms. The main application of this result is to computing the straight skeleton of a polygon. It allows us to compute the straight skeleton of a nondegenerate polygon with h holes in O(n h+ 1 log2 n + n4/3+ε) expected time. If all input coordinates are O(log n)bit rational numbers, we can compute the straight skeleton of a (possibly degenerate) polygon with h holes in O(n h+ 1 log3 n) expected time. In particular, it means that we can compute the straight skeleton of a simple polygon in O(n log3 n) expected time if all input coordinates are O(log n)bit rationals, while all previously known algorithms have worstcase running time ω(n3/2).
Nearlineartime deterministic plane Steiner spanners and TSP approximation for wellspaced point sets
 In Proceedings of the 24th Annual Canadian Conference on Computational Geometry (CCCG
, 2012
"... We describe an algorithm that takes as input n points in the plane and a parameter , and produces as output an embedded planar graph having the given points as a subset of its vertices in which the graph distances are a (1 + )approximation to the geometric distances between the given points. For po ..."
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Cited by 2 (0 self)
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We describe an algorithm that takes as input n points in the plane and a parameter , and produces as output an embedded planar graph having the given points as a subset of its vertices in which the graph distances are a (1 + )approximation to the geometric distances between the given points. For point sets in which the Delaunay triangulation has sharpest angle α, our algorithm’s output has O(β 2 n) vertices, its weight is O ( β α) times the minimum spanning tree weight where β = 1 α log 1 α. The algorithm’s running time, if a Delaunay triangulation is given, is linear in the size of the output. We use this result in a similarly fast deterministic approximation scheme for the traveling salesperson problem. 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
A Linear Time Euclidean Spanner on Imprecise Points
"... An sspanner on a set S of n points in Rd is a graph on S where for every two points p, q ∈ S, there exists a path between them in G whose length is less than or equal to s · pq  where pq  is the Euclidean distance between p and q. In this paper, we consider the construction of a Euclidean spann ..."
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An sspanner on a set S of n points in Rd is a graph on S where for every two points p, q ∈ S, there exists a path between them in G whose length is less than or equal to s · pq  where pq  is the Euclidean distance between p and q. In this paper, we consider the construction of a Euclidean spanner for imprecise points where we take advantage of prior, inexact knowledge of our input. In particular, in the first phase, we preprocess n ddimensional balls with radius r that are approximations of the input points with the guarantee that each input point lies within its respective ball. In the second phase, the specific points are revealed and we quickly compute a spanner using data from the preprocessing phase. We can compute (or update) the (1 + ε)spanner in time O(n(r+ 1ε) d log(r+ 1ε)) after O(n(r+