The Ramsey number R(3, t) has order of magnitude t2/log t
Random Structures & Algorithms
Nearly optimal distributed edge coloring in O(log log n) rounds
Random Structures & Algorithms
Nearly optimal distributed edge colouring in O(log log n) rounds
SODA '97 Proceedings of the eighth annual ACM-SIAM symposium on Discrete algorithms
Fast distributed algorithms for Brooks-Vizing colourings
Proceedings of the ninth annual ACM-SIAM symposium on Discrete algorithms
Divide and conquer martingales and the number of triangles in a random graph
Random Structures & Algorithms
Distributed coloring in Õ (√log n) Bit Rounds
IPDPS'06 Proceedings of the 20th international conference on Parallel and distributed processing
Distributed graph coloring in a few rounds
Proceedings of the 30th annual ACM SIGACT-SIGOPS symposium on Principles of distributed computing
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Often when analysing randomized algorithms, especially parallel or distributed algorithms, one is called upon to show that some function of many independent choices is tightly concentrated about its expected value. For example, the algorithm might colour the vertices of a given graph with two colours and one would wish to show that, with high probability, very nearly half of all edges are monochromatic.The classic result of Chernoff [3] gives such a large deviation result when the function is a sum of independent indicator random variables. The results of Hoeffding [5] and Azuma [2] give similar results for functions which can be expressed as martingales with a bounded difference property. Roughly speaking, this means that each individual choice has a bounded effect on the value of the function. McDiarmid [9] nicely summarized these results and gave a host of applications. Expressed a little differently, his main result is as follows.