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### Five Proofs of Chernoff's Bound with Applications [article]

Wolfgang Mulzer
2019 arXiv   pre-print
We discuss five ways of proving Chernoff's bound and show how they lead to different extensions of the basic bound.  ...  Chernoff's bound is one of the most basic and versatile tools in the life of a theoretical computer scientist, with a seemingly endless amount of applications.  ...  A Proof via Differential Privacy The fifth proof of Chernoff's bound is due to Steinke and Ullman [22] , and it uses methods from the theory of differential privacy [11] .  ...

### The Moment Bound Is Tighter Than Chernoff's Bound for Positive Tail Probabilities

Thomas K. Philips, Randolph Nelson
1995 American Statistician
In this article we show that for all positive t and for all distributions, the moment bound is tighter than Chemnoff's bound.  ...  By way of example, we demonstrate that the improvement is often substantial.  ...  Perhaps the most famous of these bounds is Chernoff's bound (Chernoff 1952) .  ...

### Dynamic Monopolies for Degree Proportional Thresholds in Connected Graphs of Girth at least Five and Trees [article]

Michael Gentner, Dieter Rautenbach
2016 arXiv   pre-print
For a set D of vertices of G, let the set H_ρ(D) arise by starting with the set D, and iteratively adding further vertices u to the current set if they have at least ρ d_G(u) neighbors in it.  ...  If H_ρ(D) contains all vertices of G, then D is known as an irreversible dynamic monopoly or a perfect target set associated with the threshold function uρ d_G(u).  ...  In fact, especially for small values of ρ, and graphs with many vertices of small degrees, the bound (3) can be much better than the bound (1).  ...

### 1-factorizations of pseudorandom graphs [article]

Asaf Ferber, Vishesh Jain
2018 arXiv   pre-print
Moreover, we also obtain a lower bound for the number of distinct 1-factorizations of such graphs G which is off by a factor of 2 in the base of the exponent from the known upper bound.  ...  This lower bound is better by a factor of 2^nd/2 than the previously best known lower bounds, even in the simplest case where G is the complete graph.  ...  The underlying idea is similar in both cases, but the proof in the sparse case is technically more involved as a standard use of Chernoff's bounds and the union bound does not work (and therefore, we will  ...

### Randomly coloring graphs of girth at least five

Thomas P. Hayes
2003 Proceedings of the thirty-fifth ACM symposium on Theory of computing - STOC '03
We improve rapid mixing results for the simple Glauber dynamics designed to generate a random k-coloring of a bounded-degree graph.  ...  Thanks also to my advisor Laci Babai, and to all the theory group at the University of Chicago, for much encouragement and patient listening.  ...  Section 6 contains a fairly detailed outline of the proof of Theorem 2, including some of the proofs. Our improved version of Chernoff's bound is proved in Section 6.1.  ...

### Randomly coloring graphs of girth at least five

Thomas P. Hayes
2003 Proceedings of the thirty-fifth ACM symposium on Theory of computing - STOC '03
We improve rapid mixing results for the simple Glauber dynamics designed to generate a random k-coloring of a bounded-degree graph.  ...  Thanks also to my advisor Laci Babai, and to all the theory group at the University of Chicago, for much encouragement and patient listening.  ...  Section 6 contains a fairly detailed outline of the proof of Theorem 2, including some of the proofs. Our improved version of Chernoff's bound is proved in Section 6.1.  ...

### Towards the linear arboricity conjecture [article]

Asaf Ferber, Jacob Fox, Vishesh Jain
2018 arXiv   pre-print
Our proofs of these results further give probabilistic polynomial time algorithms for finding such decompositions into linear forests.  ...  of G.  ...  The existence of such a partitioning is guaranteed by Lemma 2.8, which is proved by a standard application of Chernoff's bounds (Lemma 2.1) followed by the Lovász Local Lemma (Lemma 2.4).  ...

### The Total Acquisition Number of Random Geometric Graphs [article]

Ewa Infeld, Dieter Mitsche, Pawel Pralat
2016 arXiv   pre-print
The total acquisition number of G, denoted by a_t(G), is the minimum cardinality of the set of vertices with positive weight at the end of the process.  ...  In this paper, we investigate random geometric graphs G(n,r) with n vertices distributed u.a.r. in [0,√(n)]^2 and two vertices being adjacent if and only if their distance is at most r.  ...  The remaining part is a simple consequence of Chernoff's bound and the union bound over all vertices.  ...

### On a theorem of Chernoff on rank one Riemannian symmetric spaces [article]

Pritam Ganguly, Ramesh Manna, Sundaram Thangavelu
2021 arXiv   pre-print
In this paper, we prove an exact analogue of Chernoff's theorem for all rank one Riemannian symmetric spaces (of noncompact and compact types) using iterates of the associated Laplace-Beltrami operators  ...  Chernoff used iterates of the Laplacian on ℝ^n to prove an L^2 version of the Denjoy-Carleman theorem which provides a sufficient condition for a smooth function on ℝ^n to be quasi-analytic.  ...  Ph.D. scholarship from Indian Institute of Science.  ...

### Number of 1-factorizations of regular high-degree graphs [article]

Asaf Ferber, Vishesh Jain, Benny Sudakov
2019 arXiv   pre-print
A 1-factor in an n-vertex graph G is a collection of n/2 vertex-disjoint edges and a 1-factorization of G is a partition of its edges into edge-disjoint 1-factors.  ...  In this paper we address the natural question of estimating F(n,d), the number of 1-factorizations in d-regular graphs on an even number of vertices, provided that d≥n/2+ε n.  ...  Acknowledgment: The first author is grateful to Kyle Luh and Rajko Nenadov for helpful discussions at the first step of this project.  ...

### The Total Acquisition Number of Random Geometric Graphs

Ewa Infeld, Dieter Mitsche, Paweł Prałat
2017 Electronic Journal of Combinatorics
The total acquisition number of $G$, denoted by $a_t(G)$, is the minimum cardinality of the set of vertices with positive weight at the end of the process.  ...  In this paper, we investigate random geometric graphs $\mathcal{G}(n,r)$ with $n$ vertices distributed uniformally at random in $[0,\sqrt{n}]^2$ and two vertices being adjacent if and only if their distance  ...  The remaining part is a simple consequence of Chernoff's bound and the union bound over all vertices.  ...

### Mixing in Random Digraphs with Application to the Forward-Secure Key Evolution in Wireless Sensor Networks

Marek Klonowski, Mirosław Kutyłowski, Michał Ren, Katarzyna Rybarczyk
2014 ACM transactions on sensor networks
The article provides a rigorous mathematical analysis of the distribution of keys and supplements it with experimental results.  ...  It is shown that with probability close to 1 the mixing time is of small order and the fluctuations of the distribution are limited.  ...  Acknowledgments Katarzyna Rybarczyk acknowledge a partial support of the National Science Center, under grant DEC-2011/01/B/ST1/03943.  ...

### A Simple Distribution-Free Approach to the Max k-Armed Bandit Problem [chapter]

Matthew J. Streeter, Stephen F. Smith
2006 Lecture Notes in Computer Science
The max k-armed bandit problem is a recently-introduced online optimization problem with practical applications to heuristic search.  ...  We demonstrate the effectiveness of our approach by applying it to the task of selecting among priority dispatching rules for the resource-constrained project scheduling problem with maximal time lags  ...  sampled by any of the five heuristics (on any of the 10, 000 stored runs of each of the five heuristics).  ...

### Optimal Best Arm Identification with Fixed Confidence [article]

Aurélien Garivier
2016 arXiv   pre-print
We give a complete characterization of the complexity of best-arm identification in one-parameter bandit problems. We prove a new, tight lower bound on the sample complexity.  ...  It consists in a new sampling rule (which tracks the optimal proportions of arm draws highlighted by the lower bound) and in a stopping rule named after Chernoff, for which we give a new analysis.  ...  The authors acknowledge the support of the French Agence Nationale de la Recherche (ANR), under grants ANR-13-BS01-0005 (project SPADRO) and ANR-13-CORD-0020 (project ALICIA).  ...

### Edge and Pair Queries – Random Graphs and Complexity [article]

Dariusz Dereniowski, Przemysław Gordinowicz, Paweł Prałat
2022 arXiv   pre-print
We concentrate on investigating the two associated graph parameters for binomial random graphs, and showing that determining any of the two parameters is NP-hard for bounded degree graphs.  ...  We investigate two types of query games played on a graph, pair queries and edge queries.  ...  The conclusion follows from Chernoff's bound (1) (applied with ε = ω−1/2 ), since P |Q − E[Q]| ≥ ε E[Q] ≤ 2 exp −Ω(ε 2 E[Q]) = o(1/n 3 ). The proof of part (g) is exactly the same.  ...
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