Another recursive algorithm is applied to some data A = (1, ..., am) where m = 2* (i.e. 2, 4, 8,16 ...) where x is an integer ≥ 1. The running time T is characterised using the following recurrence equations: T(1) = c when the size of A is 1 T(m) = 2T ()+c otherwise Determine the running time complexity of this algorithm.

Let m be a matrix with n rows and n columns whose entries are either 1 or 0. recall that the element of m on row i and column j is denoted by mij . the diagonal entries of m are {mii} for 1 ≤ i ≤ n. We call M permutable if it is possible to swap some of the rows and some of columns so that all diagonal entries are 1. Design a polynomial time algorithm that decides whether a binary matrix M is permutable or not. Note that we can swap any two rows or two columns. Also, the order in which these swaps are done is not important.

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

Find the running time for each of the following algorithms. Show work by finding a table of values for each while loop, writing the summations, then solving. 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. c) Func4(n) 1 2 3 4 567∞∞ s = 0; for i 1 to 5n do 8 j← 3i; while (j < i3) do s+ s + i - j; j+5 x j; end end 8 return (s);

Please solve using iterative method: Solve the following recurrences and compute the asymptotic upper bounds. Assume that T(n) is a constant for sufficiently small n. Make your bounds as tight as possible. a. T(n) = T(n − 2) + √n b.T(n) = 2T(n − 1) + c

Determine the expected case running time of the algorithm. Fully justify your solution. Determine the best case running time of the algorithm and state what condition must be true forit to occur. No justification is needed! Determine the worst case running time of the algorithm and state what condition must be true forit to occur. No justification is needed!

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;

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).]

PLEASE explain how to solve this STEP BY STEP

Time comp.

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A recursive algorithm is applied to some data A = (a1,..., am) where m≥ 2. The running time T is characterised using the following recurrence equations: T(2) = c when the size of A is 2 T(m) = T(m-1) + 2c otherwise Determine the running time complexity of this algorithm.

Explain the complete work !