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Probability Metrics and Recursive Algorithms
"... In this paper it is shown by several examples that probability metrics are a useful tool to study the asymptotic behaviour of (stochastic) recursive algorithms. The basic idea of this approach is to find a `suitable ' probability metric which yields contraction properties of the transformations des ..."
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Cited by 47 (9 self)
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In this paper it is shown by several examples that probability metrics are a useful tool to study the asymptotic behaviour of (stochastic) recursive algorithms. The basic idea of this approach is to find a `suitable ' probability metric which yields contraction properties of the transformations describing the limits of the algorithm. In order to demonstrate the wide range of applicability of this contraction method we investigate examples from various fields, some of them have been analyzed already in the literature.
Maintaining the Approximate Width of a Set of Points in the Plane (Extended Abstract)
, 1993
"... The width of a set of n points in the plane is the smallest distance between two parallel lines that enclose the set. We maintain the set of points under insertions and deletions of points and we are able to report an approximation of the width of this dynamic point set. Our data structure takes lin ..."
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Cited by 12 (1 self)
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The width of a set of n points in the plane is the smallest distance between two parallel lines that enclose the set. We maintain the set of points under insertions and deletions of points and we are able to report an approximation of the width of this dynamic point set. Our data structure takes linear space and allows for reporting the approximation with relative accuracy ffl in O( p 1=ffl log n) time; and the update time is O(log² n). The method uses the tentative pruneandsearch strategy of Kirkpatrick and Snoeyink.
Spacetime approximation with sparse grids
"... In this article we introduce approximation spaces for parabolic problems which are based on the tensor product construction of a multiscale basis in space and a multiscale basis in time. Proper truncation then leads to socalled spacetime sparse grid spaces. For a uniform discretization of the spa ..."
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Cited by 3 (1 self)
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In this article we introduce approximation spaces for parabolic problems which are based on the tensor product construction of a multiscale basis in space and a multiscale basis in time. Proper truncation then leads to socalled spacetime sparse grid spaces. For a uniform discretization of the spatial space of dimension d with O(N d) degrees of freedom, these spaces involve for d> 1 also only O(N d) degrees of freedom for the discretization of the whole spacetime problem. But they provide the same approximation rate as classical spacetime Finite Element spaces which need O(N d+1) degrees of freedoms. This makes these approximation spaces well suited for conventional parabolic and for timedependent optimization problems. We analyze the approximation properties and the dimension of these sparse grid spacetime spaces for general stable multiscale bases. We then restrict ourselves to an interpolatory multiscale basis, i.e. a hierarchical basis. Here, to be able to handle also complicated spatial domains Ω, we construct the hierarchical basis from a given spatial Finite Element basis as follows: First we determine coarse grid points recursively over the levels by the coarsening step of the algebraic multigrid method. Then, we derive interpolatory prolongation operators between the respective coarse and fine grid points by a least squares approach. This way we obtain an algebraic hierarchical basis for the spatial domain which we then use in our spacetime sparse grid approach. We give numerical results on the convergence rate of the interpolation error of these spaces for various spacetime problems with two spatial dimensions. Also implementational issues, data structures and questions of adaptivity are addressed to some extent.
Construction of Combinatorial Objects
, 1995
"... Isomorphism problems often can be solved by determining orbits of a group acting on the set of all objects to be classified. The paper centers around algorithms for this topic and shows how to base them on the same idea, the homomorphism principle. Especially it is shown that forming Sims chains, u ..."
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Cited by 1 (1 self)
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Isomorphism problems often can be solved by determining orbits of a group acting on the set of all objects to be classified. The paper centers around algorithms for this topic and shows how to base them on the same idea, the homomorphism principle. Especially it is shown that forming Sims chains, using an algorithmic version of Burnside's table of marks, computing double coset representatives, and computing Sylow subgroups of automorphism groups can be explained in this way. The exposition is based on graph theoretic concepts to give an easy explanation of data structures for group actions.
TETRISHASHING or Optimal Table Compression
"... In this paper, we present a new method for mapping a static set of n keys, each an integer between 0 and N \Gamma 1, into a hash table of size n without any collision. Our data structure requires only an additional array of n integers, each less than n, and achieves a worst case lookup time of O(1). ..."
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In this paper, we present a new method for mapping a static set of n keys, each an integer between 0 and N \Gamma 1, into a hash table of size n without any collision. Our data structure requires only an additional array of n integers, each less than n, and achieves a worst case lookup time of O(1). This method is based on a randomized compression scheme, and it finds a minimal perfect hash function in average time O(n). Our concept can be easily adapted for dynamic key sets. Then, the hash table has no longer minimal size but the storage location remains very small. Because of its simplicity our approach is particularly interesting for practical purposes.