On the Method of Typical Bounded Differences

Warnke, Lutz

doi:10.1017/s0963548315000103

Cited by 64 publications

(94 citation statements)

References 59 publications

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“…Roughly speaking, our contributions are as follows: We show that the power assignment functional has sufficiently nice properties in order to apply Yukich's general framework for Euclidean functionals to obtain concentration results (Section 3). Combining these insights with a recent generalization of the Azuma‐Hoeffding bound by Warnke , we obtain concentration of measure and complete convergence of the power assignment functional for all combinations of d and

p \geq 1

, even for the case

p \geq d

(Section 4). In addition, we obtain complete convergence for

p \geq d

for minimum‐weight spanning tree functional.…”

Section: Introductionmentioning

confidence: 67%

“…The following theorem is a simplified version of a result by Lutz Warnke , Theorem 1.2, Remark 1]. Theorem (Warnke , Theorem 1.2, Remark 1]). Let

F : {({[0, 1]}^{d})}^{n} \to ℝ

.…”

Section: Convergencementioning

confidence: 98%

“…Unfortunately, this situation is not covered by McDiarmid's or Azuma‐Hoeffding's inequality. Fortunately, Warnke proved a generalization specifically for the case that the influence of single variables is typically bounded and fulfills a weaker bound in the worst case.…”

Section: Convergencementioning

confidence: 99%

“…As examples, we have obtained complete convergence for the

M S T

(minimum‐spanning tree) and the

P A

(power assignment) functional. To prove this, we have used a recent concentration of measure result by Warnke . His concentration inequality might be of independent interest to the algorithms community.…”

Section: Conclusion and Open Problemsmentioning

confidence: 99%

“…Second, we apply a recent generalization of Azuma‐Hoeffding's inequality to prove complete convergence for the case

p \geq d

for both power assignments and minimum spanning trees. We introduce the notion of typically smooth Euclidean functionals, prove convergence of such functionals, and show that minimum spanning trees and power assignments are typically smooth.…”

Section: Introductionmentioning

confidence: 99%

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Probabilistic analysis of power assignments

Graaf

Manthey

2017

Random Struct Algorithms

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ABSTRACT:A fundamental problem for wireless ad hoc networks is the assignment of suitable transmission powers to the wireless devices such that the resulting communication graph is connected. The goal is to minimize the total transmit power in order to maximize the life-time of the network. Our aim is a probabilistic analysis of this power assignment problem. We prove complete convergence for arbitrary combinations of the dimension d and the distance-power gradient p. Furthermore, we prove that the expected approximation ratio of the simple spanning tree heuristic is strictly less than its worst-case ratio of 2.Our main technical novelties are two-fold: First, we find a way to deal with the unbounded degree that the communication network induced by the optimal power assignment can have. Minimum spanning trees and traveling salesman tours, for which strong concentration results are known in Euclidean space, have bounded degree, which is heavily exploited in their analysis. Second, we apply a recent generalization of Azuma-Hoeffding's inequality to prove complete convergence for the case p ≥ d for both power assignments and minimum spanning trees (MSTs). As far as we are aware, complete convergence for p > d has not been proved yet for any Euclidean functional.

show abstract

p \geq 1

, even for the case

p \geq d

(Section 4). In addition, we obtain complete convergence for

p \geq d

for minimum‐weight spanning tree functional.…”

Section: Introductionmentioning

confidence: 67%

“…The following theorem is a simplified version of a result by Lutz Warnke , Theorem 1.2, Remark 1]. Theorem (Warnke , Theorem 1.2, Remark 1]). Let

F : {({[0, 1]}^{d})}^{n} \to ℝ

.…”

Section: Convergencementioning

confidence: 98%

Section: Convergencementioning

confidence: 99%

“…As examples, we have obtained complete convergence for the

M S T

(minimum‐spanning tree) and the

P A

Section: Conclusion and Open Problemsmentioning

confidence: 99%

“…Second, we apply a recent generalization of Azuma‐Hoeffding's inequality to prove complete convergence for the case

p \geq d

Section: Introductionmentioning

confidence: 99%

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Probabilistic analysis of power assignments

Graaf

Manthey

2017

Random Struct Algorithms

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Large monochromatic components in 3‐edge‐colored Steiner triple systems

DeBiasio

Tait

2020

J of Combinatorial Designs

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It is known that in any r‐coloring of the edges of a complete r‐uniform hypergraph, there exists a spanning monochromatic component. Given a Steiner triple system on n vertices, what is the largest monochromatic component one can guarantee in an arbitrary 3‐coloring of the edges? Gyárfás proved that (2n+3)/3 is an absolute lower bound and that this lower bound is best possible for infinitely many n. On the other hand, we prove that for almost all Steiner triple systems the lower bound is actually (1−o(1))n. We obtain this result as a consequence of a more general theorem which shows that the lower bound depends on the size of a largest 3‐partite hole (ie, disjoint sets X1,X2,X3 with false|X1false|=false|X2false|=false|X3false| such that no edge intersects all of X1,X2,X3) in the Steiner triple system (Gyárfás previously observed that the upper bound depends on this parameter). Furthermore, we show that this lower bound is tight unless the structure of the Steiner triple system and the coloring of its edges are restricted in a certain way. We also suggest a variety of other Ramsey problems in the setting of Steiner triple systems.

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The Reverse H‐free Process for Strictly 2‐Balanced Graphs

Makai

2014

Journal of Graph Theory

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Abstract:Consider the random graph process that starts from the complete graph on n vertices. In every step, the process selects an edge uniformly at random from the set of edges that are in a copy of a fixed graph H and removes it from the graph. The process stops when no more copies of H exist. When H is a strictly 2-balanced graph we give the exact asymptotics on the number of edges remaining in the graph when the process terminates and investigate some basic properties namely the size of the maximal independent set and the presence of subgraphs. C 2014 Wiley Periodicals, Inc. J. Graph Theory 79: [125][126][127][128][129][130][131][132][133][134][135][136][137][138][139][140][141][142][143][144] 2015

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On the Method of Typical Bounded Differences

Cited by 64 publications

References 59 publications

Probabilistic analysis of power assignments

Probabilistic analysis of power assignments

Large monochromatic components in 3‐edge‐colored Steiner triple systems

The Reverse H‐free Process for Strictly 2‐Balanced Graphs

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