A fast convex hull algorithm

Akl, Selim G.; Toussaint, Godfried T.

doi:10.1016/0020-0190(78)90003-0

Cited by 123 publications

(67 citation statements)

References 7 publications

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“…This is a kind of a Throw-Away Principle as introduced by Akl and Youssaint [1] and later by Devroye [10]. Finally we achieve the following main result.…”

Section: O(m + E(#v) 2 + E(#~) Log M) (16)supporting

confidence: 62%

“…This paper presents an algorithm and its probabilistic analysis for constructing the convex hull ofm given points in Rn, the n-dimensional Euclidean space. The algorithm under consideration combines the Gift-Wrapping concept with the so-called Throw-Away Principle (introduced by Aki and Toussaint [1 ] and later by Devroye [10]) for nonextremal points. The latter principle had been used for a convex-hull-construction algorithm in R 2 and for its probabilistic analysis in a recent paper by Borgwardt et al [5].…”

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confidence: 99%

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Average complexity of a gift-wrapping algorithm for determining the convex hull of randomly given points

Borgwardt

1997

Discrete Comput Geom

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Abstract. This paper presents an algorithm and its probabilistic analysis for constructing the convex hull ofm given points in Rn, the n-dimensional Euclidean space. The algorithm under consideration combines the Gift-Wrapping concept with the so-called Throw-Away Principle (introduced by Aki and Toussaint [1 ] and later by Devroye [10]) for nonextremal points. The latter principle had been used for a convex-hull-construction algorithm in R 2 and for its probabilistic analysis in a recent paper by Borgwardt et al. [5]. There, the considerations remained much simpler, because in R2 the construction of the convex hull essentially requires recognition of the extremal points and of their order only.In this paper the Simplex method is used to organize a walk over the surface of the convex hull. During this walk all facets are discovered. Under the condition of general position this information is sufficient, because the whole face lattice can simply be deduced when the set of facets is available.Exploiting the advantages of the revised Simplex method reduces the update effort to an n x n matrix and the number of calculated quotients for the pivot search to the points which are not thrown away.For this algorithm a probabilistic analysis can be carried out. We assume that our m random points are distributed identically, independently, and symmetrically under rotations in R n. Then the calculation of the expected effort becomes possible for a whole parametrical class of distributions over the unit ball. The results mean a progress in three directions:--a parametrization of the expected effort can be given; --the dependency on n--the dimension of the space---can be evaluated; --the additional work of preprocessing for detecting the vertices can be avoided without losing its advantages. 80K.H. Borgwardt

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“…This is a kind of a Throw-Away Principle as introduced by Akl and Youssaint [1] and later by Devroye [10]. Finally we achieve the following main result.…”

Section: O(m + E(#v) 2 + E(#~) Log M) (16)supporting

confidence: 62%

mentioning

confidence: 99%

Average complexity of a gift-wrapping algorithm for determining the convex hull of randomly given points

Borgwardt

1997

Discrete Comput Geom

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“…This style of data organization is familiar from databases, and has been utilized by several recent works [26,27,51]. 2 The key advantage of the relational form is in abstracting data from its representation, which permits a clear, highlevel description of what guards and methods do. This is also the rationale behind recent language features, such as the LINQ queries in C#.…”

Section: Preliminaries: Guards Methods and Relationsmentioning

confidence: 99%

Turning nondeterminism into parallelism

2013

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Nondeterminism is a useful and prevalent concept in the design and implementation of software systems. An important property of nondeterminism is its latent parallelism: A nondeterministic action can evaluate to multiple behaviors. If at least one of these behaviors does not conflict with concurrent tasks, then there is an admissible execution of the action in parallel with these tasks. Unfortunately, existing implementations of the atomic paradigm-optimistic as well as pessimistic-are unable to fully exhaust the parallelism potential of nondeterministic actions, lacking the means to guide concurrent tasks toward nondeterministic choices that minimize interference. This paper investigates the problem of utilizing parallelism due to nondeterminism. We observe that nondeterminism occurs in many real-world codes. We motivate the need for devising coordination mechanisms that can utilize available nondeterminism. We have developed a system featuring such mechanisms, which leverages nondeterminism in a wide class of query operations, allowing a task to look into the future of concurrent tasks that mutate the shared state during query evaluation and reduce conflict accordingly. We evaluate our system on a suite of 12 algorithmic benchmarks of wide applicability, as well as an industrial application. The results are encouraging.

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“…Since then numerous improvements [1,2,3,15,41] and extensions to polytopes [53,74] have been proposed. An overview can be found in [54,60].…”

Section: Convex Hullmentioning

confidence: 99%

“…Since the contents of s are ignored in our model, the new value of c only depends on the relationship between the index i and n which describes the position of the first nul character. The value of c is restricted to [1,255] if i < n, it is set to 0 if i = n and to [0,255] if i > n. Note that this model is valid for platforms where the C char type is unsigned. The last three equations that comprise the system are given by the following:…”

Section: String Buffer Analysismentioning

confidence: 99%

The two variable per inequality abstract domain

Simon

King

2010

Higher-Order Symb Comput

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Abstract. This article presents the Two Variable Per Inequality abstract domain (TVPI domain for short). This so-called weakly-relational domain is able to express systems of linear inequalities where each inequality has at most two variables. The domain represents a sweet-point in the performance-cost tradeoff between the faster Octagon domain and the more expressive domain of general convex polyhedra. In particular, we detail techniques to closely approximate integral TVPI systems, thereby finessing the problem of excessively growing coefficients, yielding -to our knowledge -the only relational domain that combines linear relations with arbitrary coefficients and strongly polynomial performance. Keywords: polyhedral analysis, integer programming, abstract interpretationStatic analysis methods have evolved from inferring prerequisite invariants for compiler optimisations to tools in their own right that are able to prove the absence of run-time errors of software. The abstract interpretation framework [21] provides a way of constructing and justifying program analyses. The key idea is to simulate each operation in a program with an abstract analog that operates on a description of the program state, rather than the state itself. The descriptions are chosen to trace a property of interest, for instance, an interval that encloses all possible values of a variable would be of value when reasoning about array bounds. These descriptions constitute what is known as an abstract domain. An abstract domain is usually presented as a lattice D, , , and the analysis itself is formulated as a set of recursive equations over D. The recursive equations are solved iteratively, until a fixpoint is reached which is detected using the ordering predicate . Each equation expresses how a program statement transforms the state at one program point to the state at another. Equations may be recursive because of loops in the program. The meet and join † This paper is a revised extract of the first author's PhD thesis [65], which, in turn, extends [72].

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A fast convex hull algorithm

Cited by 123 publications

References 7 publications

Average complexity of a gift-wrapping algorithm for determining the convex hull of randomly given points

Average complexity of a gift-wrapping algorithm for determining the convex hull of randomly given points

Turning nondeterminism into parallelism

The two variable per inequality abstract domain

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