T(n)=3T(n/2) +n², where T(1) = 0(1). ou must state which case of the Master Theorem holds, and prove that it does apply.
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- Set up a recurrence relation for the BinarySearch_v1 and solve it step by step. Write the time efficiency class that this algorithm belongs to?(a) For each of the following recurrences, give an expression for theruntime T (n) if the recurrence can be solved with the Master Theorem.Otherwise, indicate that the Master Theorem does not apply.(i) T (n) = T (n/2) + T(n/2) + T(n/2) + n2(ii) T (n) = 0.5T (n/2)+ 1/n(iii) T (n) = 3T (n/3) + n(iv) T (n) = 4T (n/2) + nlognCalculate an upper-bound runtime for each of the following recurrence relations. Use any method we’ve seen in class—Master Theorem, Substitution Method, algebra. I. T(n) = 18T(n/9) + O(n √ n), where T(1) = 1. II. T(n) = T(n − 1) + 1, where T(1) = 1.
- Given the following code snippet, find the following:1) Recurrence relation of func (x,y) when x>0 and length of y is n.T(x,n) = T(_) + O(_) (Just fill in the blanks)For 2-3, consider worst-case scenario and initial value of y as [1]2) Time complexity of func (x,y)3) Auxiliary Space Complexity of func (x,y)a)Write a recursive definition for the set of odd positive integers. b)Use master theorem to find the solution to the recurrence relation f(n) = 4f(n/2) + 2? ! ,when n = 2" , where k is a positive integer and f(1) = 1.Question 3 Please solve the recurrence and show its proof by induction of: T(1) = 3 T(n) = T(n/3) + 2n, n > 1
- Using the code in the picture (Phyton 3): Find the Recurrence relation for foo(a, b) when b > 0 (Follow the format) T(n) = ___ T ( ___ / ___ ) + O ( ___ ) What is the worst-case time complexity of foo(a, b)? What is the worst-case auxiliary space complexity of foo(a,b)?(b) Prove, by using mathematical induction, that the iteration rule you haveobserved 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 provethat 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). See image for reference to part a, the answer to part a came out to be O(n^2). Need help with b?Give asymptotic upper and lower bounds for T(n) in each of the following recurrences. Assume that T(n) is constant for n ≤ 2. Make your bounds as tight as possible, and justify your answers. T(n)=9 T(n/5)+n^2=θ(n^2) T(n)=T(√n)+4
- Find a bound for each recurrence below using Master Theorem:(a) T(n) = 3T(n=2) + n2(b) T(n) = 2T(n=4) + n0:49(c) T(n) = 3T(n=3) + n=23. Solving RecurrencesGive asymptotic upper and lower bounds for T(n) in each of the following recurrences.Assume that T(n) is constant for n ≤ 2. Make your bounds as tight as possible, andjustify your answers. (1) T(n) = 4T(n/4) + 5n(2) T(n) = 5T(n/4) + 4n(3) T(n) = 25T(n/5) + n2Please explain Give asymptotic upper and lower bounds for each of the following recurrences. Justify your answer. T(n)=√nT(√n)+n