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Unformatted text preview: 1 Discrete Math CS 2800 Prof. Bart Selman [email protected] Module Probability  Part e) 1) The Probabilistic Method 2) Randomized Algorithms 2 2 The Probabilistic Method 3 The Probabilistic Method Method for providing nonconstructive 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 3CNF. 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”.) 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. How do we get a lower bound on R(k,k)? E.g., lower bound for R(3,3)? We need to find an n such that the complete graph on n nodes with a Red/Blue edge coloring does not contain a Red or a Blue triangle. So, R(3,3) > 5. I.e., we have a lower bound on the Ramsey Number for k =3. E.g. Can we do this for R(k,k) in general? Very difficult to construct the colorings, but… we can prove they exist for nontrivial k . Thm. For k ≥ 4, R(k,k) ≥ 2 k/2 So, e.g., k = 20 , then there exists a Red / Blue coloring of the complete graph with 1023 nodes that does not have any complete monochromatic sub graph of size 20 . (But we have no idea of how to find such a coloring!) Proof: Consider a sample space where each possible coloring of the nnode complete graph is equally likely. A sample coloring can be obtained by randomly coloring each edge. I.e., with probability ½ set edge to...
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This note was uploaded on 10/25/2009 for the course CS 2800 at Cornell.
 '07
 SELMAN
 Algorithms

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