Again, we skip the proof of correctness of this algorithm. (d) What is the worst-case running time of Find-Index-2(A[1 : n)? What about its worst-case expected running time? Remember to prove your answer formally.

Prove f(n) = n3 - (n2 log2 n) + 2n f(n) ∈ O(n3) by giving the constants (c, n0) and arguing that your constants hold as n goes to infinity by either "chaining up", "chaining down", carefully treating the inequality as an equality and doing some algebra, or even weak induction. Also, what is the minimum val4ue for c (in the definition of O() ) forj this problem and is it an exclusive or inclusive minimum/bound.

Formally prove or disprove the following claims, using any method

We mentioned that if we want to prove P ≠ NP, we only need to pick up any one NPC problem and prove that polynomial-time algorithm does not exist for the problem. If you want to prove P ≠ NP, select one NPC problem based on your preference and describe your idea of why polynomial-time algorithm does not exist for the problem. It does not have to be a formal proof, a description of your idea would be fine.

Solving recurrences using the Substitution method. Give asymptotic upper and lower bounds for T(n) in each of the following recurrences. Solve using the substitution method. Assume that T(n) is constant n ≤ 2. Make your bounds as tight as possible and justify your answers. Hint: You may use the recursion trees or Master method to make an initial guess and prove it through induction a. T(n) = 2T(n-1) + 1 b. T(n) = 8T(n/2) + n^3

Use the Transform-and-Conquer algorithm design technique with Instance Simplification variant to design an O(nlogn) algorithm for the problem below. Show the pseudocode. Given a set S of n integers and another integer x, determine whether or not there exist two elements in S whose sum is exactly x.

The runtime complexity, T(n), of the three following recurrence relation solved by Master's Theorem) are T(n) = 6T(n/3) + n2 logn T(n) = 64T(n/8)- n2 log n T(n) = 4T(n/2) + n/logn (The solution for the three relations are respectively given by) A: (nlogn), 9(n2), (n2 logn) B: 9(n2), (n2 logn), (n log n) C: (n2logn), Master's Theorem does not apply, 9(n2) D: 0(n2), Master's Theorem does not apply, 9(n2 log n) E: 0(n2 logn), 9(n2), Master's Theorem does not apply F: 9(n2), (n2 logn), Master's Theorem does not apply

Give asymptotically tight upper and lower bounds for T (n) in each of the followingalgorithmic recurrences. Justify your answers.E. ?(?) = 3?((?/3) - 2) + ?/2 (Hint: think about how you can use an assumptionabout the importance of the -2 to apply the Masters Theorem)

Implement Algorithm to Knuth Version of Schreier-Sims procedure A j( g ); Input: ( S, T, F ) up to date of order j; g E -i-fJ]; Output~ ( S, T, F ) up to date of order j and g ~ S;