Randomized Distributed Edge Coloring via an Extension of the Chernoff--Hoeffding Bounds
TLDR
Fast and simple randomized algorithms for edge coloring a graph in the synchronous distributed point-to-point model of computation and new techniques for proving upper bounds on the tail probabilities of certain random variables which are not stochastically independent are introduced.Abstract:
Certain types of routing, scheduling, and resource-allocation problems in a distributed setting can be modeled as edge-coloring problems We present fast and simple randomized algorithms for edge coloring a graph in the synchronous distributed point-to-point model of computation Our algorithms compute an edge coloring of a graph $G$ with $n$ nodes and maximum degree $\Delta$ with at most $16 \Delta + O(\log^{1+ \delta} n)$ colors with high probability (arbitrarily close to 1) for any fixed $\delta > 0$; they run in polylogarithmic time The upper bound on the number of colors improves upon the $(2 \Delta - 1)$-coloring achievable by a simple reduction to vertex coloring
To analyze the performance of our algorithms, we introduce new techniques for proving upper bounds on the tail probabilities of certain random variables The Chernoff--Hoeffding bounds are fundamental tools that are used very frequently in estimating tail probabilities However, they assume stochastic independence among certain random variables, which may not always hold Our results extend the Chernoff--Hoeffding bounds to certain types of random variables which are not stochastically independent We believe that these results are of independent interest and merit further studyread more
Citations
More filters
Posted Content
Optimal Gossip-Based Aggregate Computation
Jen-Yeu Chen,Gopal Pandurangan +1 more
TL;DR: In this paper, the authors present the first provably almost-optimal gossip-based algorithms for aggregate computation that are both time optimal and message-optimality. But they do not consider the non-address oblivious model.
Posted Content
A structure theorem for almost low-degree functions on the slice
Nathan Keller,Ohad Klein +1 more
TL;DR: In this article, it was shown that for Boolean functions whose domain is the ''slice'' of the Fourier-Walsh expansion, the total weight on coefficients beyond degree $k$ is at most
Book
The cheat sheet
TL;DR: In this article, it was shown that a function Φ : [n]m 7→R satisfies the Lipschitz property with constant d if for every ® x, ® x ∈ [n]-m which differ in a single coordinate, it holds that |Φ(® x) − Φ (® x ′)| ≤ d.
Proceedings ArticleDOI
Distributed scheduling of parallel I/O in the presence of data replication
Jan-Jan Wu,Pangfeng Liu +1 more
TL;DR: A distributed scheduling algorithm, highest degree lowest workload first (HDLWF), which approximates the augmenting path algorithm in distributed environments, and results indicate that HDLWF yields schedules close to the centralized optimal solution, and in some cases within 3% of the optimal solution.
Journal ArticleDOI
A structure theorem for almost low-degree functions on the slice
Nathan Keller,Ohad Klein +1 more
TL;DR: It is shown that if in the representation of f, the total weight beyond degree $k$ is at most $\epsilon, then f can be approximated by a Boolean-valued function depending on at most $O(2^k)$ variables, and the approximation rate can be improved.
References
More filters
Book ChapterDOI
Probability Inequalities for sums of Bounded Random Variables
TL;DR: In this article, upper bounds for the probability that the sum S of n independent random variables exceeds its mean ES by a positive number nt are derived for certain sums of dependent random variables such as U statistics.
Book
The Probabilistic Method
TL;DR: A particular set of problems - all dealing with “good” colorings of an underlying set of points relative to a given family of sets - is explored.
Journal ArticleDOI
A Measure of Asymptotic Efficiency for Tests of a Hypothesis Based on the sum of Observations
TL;DR: In this paper, it was shown that the likelihood ratio test for fixed sample size can be reduced to this form, and that for large samples, a sample of size $n$ with the first test will give about the same probabilities of error as a sample with the second test.
Journal ArticleDOI
Locality in distributed graph algorithms
TL;DR: This model focuses on the issue of locality in distributed processing, namely, to what extent a global solution to a computational problem can be obtained from locally available data.