Dynamic planar convex hull

Brodal, Gerth Stølting; Jacob, Riko

doi:10.1109/sfcs.2002.1181985

Cited by 121 publications

(116 citation statements)

References 15 publications

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“…More precisely, they maintain an ε-coreset of size O(1/ √ ε) in O(log (1/ε)) amortized time per insertion. The width in a given direction can be efficiently computed from the coreset if we additionally maintain the convex hull of the coreset using the dynamic data structure by Brodal and Jacob [5]. This data structure uses linear space and can be updated in logarithmic time.…”

Section: Let W(i J ) Be the Width Of The Points In Subpath P (I J) mentioning

confidence: 99%

Streaming Algorithms for Line Simplification

Abam

Berg

Hachenberger

et al. 2009

Discrete Comput Geom

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We study the following variant of the well-known line-simplification problem: we are getting a (possibly infinite) sequence of points p 0 , p 1 , p 2 , . . . in the plane defining a polygonal path, and as we receive the points, we wish to maintain a simplification of the path seen so far. We study this problem in a streaming setting, where we only have a limited amount of storage, so that we cannot store all the points. We analyze the competitive ratio of our algorithms, allowing resource augmentation: we let our algorithm maintain a simplification with 2k (internal) points and compare the error of our simplification to the error of the optimal simplification with k Comput Geom (2010) 43: 497-515 points. We obtain the algorithms with O(1) competitive ratio for three cases: convex paths, where the error is measured using the Hausdorff distance (or Fréchet distance), xy-monotone paths, where the error is measured using the Hausdorff distance (or Fréchet distance), and general paths, where the error is measured using the Fréchet distance. In the first case the algorithm needs O(k) additional storage, and in the latter two cases the algorithm needs O(k 2 ) additional storage.

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Section: Let W(i J ) Be the Width Of The Points In Subpath P (I J) mentioning

confidence: 99%

Streaming Algorithms for Line Simplification

Abam

Berg

Hachenberger

et al. 2009

Discrete Comput Geom

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“…It may be possible to use a more efficient dynamic convex hull maintenance structure like the one of Brodal and Jacob [4], but it is unclear if all necessary operations that we need can be performed more efficiently using their data structure.…”

Section: Theorem 18mentioning

confidence: 99%

“…In Sect. 4, we continue the analysis of depth with algorithms to determine the deepest vertex in an arrangement, show a connection with k-sets, and deal with depth in degenerate arrangements. In Sect.…”

Section: Introductionmentioning

confidence: 99%

Efficient Algorithms for Maximum Regression Depth

Kreveld

Mitchell

Rousseeuw³

et al. 2007

Discrete Comput Geom

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We investigate algorithmic questions that arise in the statistical problem of computing lines or hyperplanes of maximum regression depth among a set of n points. We work primarily with a dual representation and find points of maximum undirected depth in an arrangement of lines or hyperplanes. An O(n d ) time and O(n d−1 ) space algorithm computes undirected depth of all points in d dimensions. Properties of undirected depth lead to an O(n log 2 n) time and O(n) space algorithm for computing a point of maximum depth in two dimensions, which has been improved to an O(n log n) time algorithm by Langerman and Steiger (Discrete Comput. Geom. 30(2):299-309, 2003). Furthermore, we describe the structure of depth in the plane and higher dimensions, leading to various other geometric and algorithmic results.

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“…Although it is possible to develop such applications using ad hoc algorithms and data structures to handle changing data, such algorithms can be challenging to design even for problems that are simple in the batch/static setting where changes to data are not allowed. A striking example is the two-dimensional convex hull problem, whose dynamic (incremental) version required decades of more research [30,7] than its static version (e.g., [17]). The field of incremental computation aims at deriving software that can respond automatically and efficiently to changing data.…”

Section: Introductionmentioning

confidence: 99%

Refinement Types for Incremental Computational Complexity

Çiçek

Garg

Acar

2015

Programming Languages and Systems

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Abstract. With recent advances, programs can be compiled to efficiently respond to incremental input changes. However, there is no language-level support for reasoning about the time complexity of incremental updates. Motivated by this gap, we present CostIt, a higher-order functional language with a lightweight refinement type system for proving asymptotic bounds on incremental computation time. Type refinements specify which parts of inputs and outputs may change, as well as dynamic stability, a measure of time required to propagate changes to a program's execution trace, given modified inputs. We prove our type system sound using a new step-indexed cost semantics for change propagation and demonstrate the precision and generality of our technique through examples.

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Dynamic planar convex hull

Cited by 121 publications

References 15 publications

Streaming Algorithms for Line Simplification

Streaming Algorithms for Line Simplification

Efficient Algorithms for Maximum Regression Depth

Refinement Types for Incremental Computational Complexity

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