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
Proceedings ArticleDOI
Distributed (∆+1)-coloring in sublogarithmic rounds
TL;DR: A new randomized coloring algorithm for (∆+1)-coloring running in O(√log ∆)+ 2^O( ∼log log n) rounds with probability 1-1/n^Ω(1) in a graph with n nodes and maximum degree ∆.
Book ChapterDOI
Constructive proofs of concentration bounds
TL;DR: In this article, the authors give a combinatorial proof of the Chernoff-Hoeffding concentration bound, which says that the sum of independent {0, 1}-valued random variables is highly concentrated around the expected value.
Journal ArticleDOI
Link scheduling in wireless sensor networks: Distributed edge-coloring revisited
TL;DR: This work considers the problem of link scheduling in a sensor network employing a TDMA MAC protocol and develops a distributed edge-coloring algorithm that is the first distributed algorithm that can edge-color a graph using at most (@D+1) colors.
Proceedings ArticleDOI
Simultaneous approximations for adversarial and stochastic online budgeted allocation
TL;DR: This paper designs algorithms that achieve a competitive ratio better than 1 -- 1/e on average, while preserving a nearly optimal worst case competitive ratio, and designs an algorithm with the optimal competitive ratio in both the adversarial and random arrival models.
Book ChapterDOI
The Graph Coloring Problem: A Bibliographic Survey
TL;DR: In this article, an arbitrary undirected graph without loops is defined as a graph where V = {v 1, v 2,…, v n } is its vertex set and E = {e 1,e 2, e m } ⊂ (E ×E) is its edge set.
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.