CS2800-Probability_part_e_v.2

CS2800-Probability_part_e_v.2 - Discrete Math CS 2800 Prof....

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1 Discrete Math CS 2800 Prof. Bart Selman [email protected] Module Probability --- Part e) 1) The Probabilistic Method 2) Randomized Algorithms
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2 2 The Probabilistic Method
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3 The Probabilistic Method Method for providing non-constructive existence proofs: Thm. If the probability that a randomly selected element of the set S does not have a particular property is less than 1, then there exists an element in S with this property . Alternatively: If the probability that a random element of S has a particular property is larger than 0 , then there exists at least one element with that property in S. Note: We saw an earlier example of the probabilistic method when discussing the 7/8 alg. for 3-CNF.
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4 Example: Lower bound for Ramsey numbers Recall the definition of Ramsey number R(k,k): Let R(k,k) be the minimal n such that if the edges of the complete graph on n nodes are colored Red and Blue, then there either is a complete subgraph of k nodes with all edges Red or a complete subgraph of k nodes with all edges Blue . R(3,3) = 6. So, any complete 6 node graphs has either a Red or a Blue triangle. (Proof: see “party problem”.)
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Reminder: “The party problem” Dinner party of six: Either there is a group of 3 who all know each other, or there is a group of 3 who are all strangers. Consider one person. She either knows or doesn’t know each other person. But there are 5 other people! So, she knows, or doesn’t know, at least 3 others. (GPH) Let’s say she knows 3 others. If any of those 3 know each other, we have a blue , which means 3 people know each other. Contradicts assumption. So they all must be strangers. But then we have three strangers. Contradicts assumption. The case where she doesn’t know 3 others is similar. Also, leads to constradiction. So, such a party does not exist! QED By contradiction. Assume we have a party of six where no three people all know each other and no three people are all strangers.
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How do we get a lower bound on R(k,k)? E.g.,
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CS2800-Probability_part_e_v.2 - Discrete Math CS 2800 Prof....

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