question Use substitution method: Show that the solution of T(n) = T(n-1) + n is O(n²) Use master method to find tight asymptotic bounds: T(n) = 2*T(n/4) + n T(n) = 2*T(n/4) + n²
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please answer with proper explanation and step by step solution.
answer for 2 is theta(n)
answer for 3 is Theta (n²)
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- 3. 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) + n2recurrence relation obtain L(1) = 0; L(N) = n+4 * L(n/4) T(1) = 1; T(n)= n+2*T(n/2)Use the master method to give tight asymptotic bounds for the following recurrence T(n) = 2T(n/4) + 1 Group of answer choices 1. ϴ(n0.5lgn) 2. ϴ(n0.5) 3. ϴ(n2) 4. ϴ(n)
- Solve the following recurrences using recursion tree method and write the asymptotic time-complexity. 1. T(n) = 3T (n/4) + n^22. T(n) = T (n/5) + T(4n/5) + n3. T(n) = 3T(n − 1) + n^4 4. T(n) = T (n/2) + n^2Lets say an algorithm has 2 base cases. PsuedoCode: if n == 1 return 0 if n == 2 return 1 return A[n..............n-1]solve the recurrence relation for the above simple recursive algorithm?(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) + nlogn
- Solve the recurrence relations shown below. a. T(n)={4T(n−1)+1if n>11if n=1T(n)={4T(n−1)+1if n>11if n=1 -- b. T(n)={3T(n/3)+4nif n>11if n=1Arrange the functions √n (square root of n), 1000log(n), nlog(n), 2n!, 2n, 3n, n2/100000 in increasing order for their Big-O estimate.Solve the recurrence relation by substitution: T(n) = T(n-1) + 2^n