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**Unformatted text preview: **Comprehensive Exam: Algorithms and Concrete Mathematics Autumn 2002 This is a one hour closed-book exam and the point total for all questions is 100. 1. (10 points) Let f (n) = 2Jl0gn and g(n) = ne, for some constant E . Which one of the following claims is true: Prove your answer. 2. (15 points) Prove a tight asymptotic bound on the behavior of T(n). T(n) = T(r0.4nl) + T(r0.5nl) + n, where for k 5 10 we are given that T ( k ) = @(I). Do not disregard the "ceiling" operations, i.e. provide a derivation that takes them into account or expIicitIy shows that they do not change the result. 3. (20 points) Prove that the following algorithm generates a random permutation of the numbers 1,2,. . . , n: Set A[i] <- i; For i := 1,2, ..., n : j <- Random uniformly distributed integer i n range [i,n]; Swap(ACi1 , A C j l l ; endf or In particular, prove for all i, j that Pr(A[i] = j) = l/n. 4. (20 points) You are given a graph G(V, E) with n nodes and two length functions: Zl:E-+{l,2 ,..., n } a n d 1 2 : E - + { l , 2...

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