Consider the following algorithm segment. For each positive integer n, let b be the number of iterations of the following while loop. while (n > 0) n := n div(3) end while Find a recurrence relation for b

A problem called S reduces to a problem called T if a T solver can be used as a subroutine to solve S. In pseudocode: Solves(...): ... SolveT(...) ... Assuming that this reduction is correct, answer the following questions regarding what the reduction tells us. If we know that an algorithm exists for solving Problem S, what does that tell us about Problem T? [ Select] If we know that an algorithm cannot exist for solving Problem S, what does that tell us about Problem T? [ Select] If we know that an algorithm exists for solving Problem T, what does that tell us about Problem S? [ Select ] [ Select ] An algorithm cannot exist for solving Problem S,r solving Problem T, what does that tell us about Nothing An algorithm exists for solving Problem S

Build a recursion tree for the following recurrence equation and then solve for T(n)

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1. Determine the running time of the following algorithm. Write summations to represent loops and solve using bounding. Be sure to show work on both the upper bound and lower bound, justify the split, and check that the bounds differ by only a constant factor. Use asymptotic notation to write the answer. Func1(n) 1 2 3 4 5 6 7 8 9 10 11 S← 0; for i ←n to n² do for j← 1 to i do for k9n to 10n² do for mi to k end end end return (s); end s+s + i- j + k −m;

Please help me solve Part A of this problem

4. Practice with the iteration method. We have already had a recurrence relation ofan algorithm, which is T(n) = 4T(n/2) + n log n. We know T(1) ≤ c.(a) Solve this recurrence relation, i.e., express it as T(n) = O(f(n)), by using the iteration method.Answer:(b)  Prove, by using mathematical induction, that the iteration rule you have observed in 4(a) is correct and you have solved the recurrence relation correctly. [Hint: You can write out the general form of T(n) at the iteration step t, and prove that this form is correct for any iteration step t by using mathematical induction.Then by finding out the eventual number of t and substituting it into your generalform of T(n), you get the O(·) notation of T(n).]

Time comp.

Sort the following functions in terms of asymptotic growth from smallest to largest. In particular, the resulting order should be such that f₁ =0(f₂), f₂=0 (f 3), and so on. 1 2 I I I - ✓ 3" ✓ log (23) 14 1 1010 n 11/10 ✓ nln(12n) ✓ 52! ✓ log_²(n) ✓20200.1

1. Let S = {1,2...n} be a set activities, where activity i € S has a start time si, a finish time f; and a profit p, asSociated with it. Design an dynamic programming algorithm for computing a set of compatible activities of maximum profit. Argue that the time complexity of you algorithm is O(n log(n)). (You must start with the correct definitions, and then write a recurrence relation.)