review1sol - Computer Science 340 Reasoning about...

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Computer Science 340 Reasoning about Computation Review Session 1 Thursday, January 10, 2008 Problem 1 Scientists at the Princeton Genomics Institute have discovered the following process: Given a test tube filled with DNA strands, it is possible to insert an enzyme that will connect two end points of stands (possibly of the same strand), and then dissolve. Experiments show every pair of endpoints is equally likely to be joined. If two endpoints of the same strand connect, then a ”DNA cycle” is formed. Suppose that n enzymes are inserted into a test tube with n strands. What is the expected number of DNA cycles that will be formed? Solution: Assume that the n enzymes act one at a time. Suppose there are i strands currently. The action of a single enzyme reduces the number of strands by 1 and results in a cycle with probability 1 2 i - 1 . (Once you fix one end point that the enzyme acts on, there are 2 i - 1 choices for the other end point. Of these, exactly one will result in the formation of a cycle.) Let X i be an indicator random variable for the event that a cycle is formed when i strands remain in the test tube. Then E[ X i ] = 1 2 i - 1 . The expected number of cycles formed is n i =1 E[ X i ] = n i =1 1 2 i - 1 = Θ(log n ). Problem 2 A model recently proposed in social networks to model the effect of distance on social relationships is the following: Consider a graph where the vertices are integer points ( x, y ) such that x ∈ { 0 , . . . , n - 1 } and y ∈ { 0 , . . . , n - 1 } . Place an edge between two points ( x 1 , y 1 ) and ( x 2 , y 2 ) with probability p ( x 1 , y 1 , x 2 , y 2 ) = 1 max( | x 1 - x 2 | , | y 1 - y 2 | ) α , where α > 0 is a constant. Let X n be the degree of the vertex (0 , 0). Give an asymptotic expression for the expectation E[ X n ], i.e. find a function f
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