Diskrepanz und Distanz von Ma�en bez�glich konvexer und Jordanscher Mengen

Abstract. Using number theoretical tools, we prove two main results for random r-regular circulant graphs with n vertices, when n is sufficiently large and r is fixed. First, for any fixed ε > 0, prime n and L ≥ n 1/r (log n) 1+1/r+ε , walks of length at most L terminate at every vertex with asymptotically the same probability. Second, for any n, there is a polynomial time algorithm to find a vertex bisector and an edge bisector, both of size less than n 1−1/r+o(1) . As circulant graphs are popular network topologies in distributed computing, we show that our results can be exploited for various information dissemination schemes. In particular, we provide lower bounds on the number of rounds required by any gossiping algorithms for any n.This settles an open question in an earlier work of the authors (2004) and shows that the generic gossiping algorithms of that work are nearly optimal.

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“…A relation between Δ(F ) and Γ (F , Ξ) for Ξ ∈ S h is given by the following inequality of [9] (see also [21]). Let e n (z) = exp(2πiz/n).…”

Section: Tools From the Theory Of Uniform Distributionmentioning

confidence: 99%

Random Walks and Bisections in Random Circulant Graphs

Mans

Shparlinski

2012

LATIN 2012: Theoretical Informatics

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“…However, in our case P is not absolutely continuous w.r.t. the Lebesgue measure on Ξ ⊂ R 2 , and, hence, the result in [17] does not apply.…”

Section: René Henrion Christian Küchler and Werner Römischmentioning

confidence: 99%

“…α B poly(W) , it is not surprising that α B poly(W) (P, P ε ) = 1 for every ε > 0. Note that it has been shown in [17], that, under certain conditions, α B poly(W) can be estimated against α B rect . However, in our case P is not absolutely continuous w.r.t.…”

Section: René Henrion Christian Küchler and Werner Römischmentioning

confidence: 99%

Discrepancy distances and scenario reduction in two-stage stochastic mixed-integer programming

Henrion¹,

Küchler²,

Römisch³

2008

Journal of Industrial &Amp; Management Optimization

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Abstract. Polyhedral discrepancies are relevant for the quantitative stability of mixed-integer two-stage and chance constrained stochastic programs. We study the problem of optimal scenario reduction for a discrete probability distribution with respect to certain polyhedral discrepancies and develop algorithms for determining the optimally reduced distribution approximately. Encouraging numerical experience for optimal scenario reduction is provided.

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“…Here we show that using some bounds from [1] in combination with some results of Laczkovich [14] and Niederreiter and Wills [16], from the theory of uniformly distributed sequences leads to a better error term in the asymptotic formula (6). Furthermore, we also consider a generalisation to the joint distribution of n 1 , .…”

Section: Introductionmentioning

confidence: 99%