1. Theorem 4.1 (Master Theorem). Let a 2 1 and b >1 be constants, let f(n)...
Using the Master Method give asymptotic bounds for T(n) in each of the following recurrences. Assume that T(n) is constant for n ≤ 4. (a) T(n) = 4 T(n/4) + n lg2 n (b) T(n) = 3 T(n/4) + n lg n c) T(n) = 4 T(n/5) + √? (d) T(n) = 4 T(n/2) + n2 lg n
Problem 1 Use the master method to give tight asymptotic bounds for the following recurrences. a) T(n) = T(2n/3) +1 b) T(n) = 2T("/2) +n4 c) T(n) = T(71/10) +n d) T(n) = 57(n/2) + n2 e) T(n) = 7T(1/2) + 12 f) T(n) = 27(1/4) + Vn g) T(n) = T(n − 2) +n h) T(n) = 27T(n/3) + n° lgn
Let f(n) = 5n^2. Prove that f(n) = O(n^3). Let f(n) = 7n^2. Prove that f(n) = Ω(n). Let f(n) = 3n. Prove that f(n) =ꙍ (√n). Let f(n) = 3n+2. Prove that f(n) = Θ (n). Let k > 0 and c > 0 be any positive constants. Prove that (n + k)c = O(nc). Prove that lg(n!) = O(n lg n). Let g(n) = log10(n). Prove that g(n) = Θ(lg n). (hint: ???? ? = ???? ?)???? ?...
Question 6 (20 points) Solve the following recurrences using the Master Theorem. T(n) = 2T (3/4)+1 T(n) = 2T (n/4) + va 7(n) = 2T (n/4) +n T(n) = 2T (3/4) + n
Give asymptotic upper bounds (in terms of O) for T(n) in each of the following recurrences. Assume that T(n) is constant for n < 2. Make your bounds as tight as posible. a) T(n)=T(H) +1; b) T(n) = T(n-1) + 1/n;
Problem 2 Solve the following recurrences. You only need to obtain the asymptotic solution (in e) notation). If you use the master theorem, you must specify all parameters and briefly verify all conditions. 1. (5%) T(n) = 25T(F) + n2 +n, T(1) = 5.
1. [12 marks] For each of the following recurrences, use the “master theorem” and give the solution using big-O notation. Explain your reasoning. If the “master theorem” does not apply to a recurrence, show your reasoning, but you need not give a solution. (a) T(n) = 3T(n/2) + n lg n; (b) T(n) = 9T(3/3) + (n? / 1g n); (c) T(n) = T([n/41) +T([n/4])+ Vn; (d) T(n) = 4T([n/7])+ n.
r the recurrence relation o. Consider T(n) = Vn T(Vn) + n a. Why can't you solve this with the master theorem? b. S t involves a constant C, tell me what it is in terms of T(O), T(1), or whatever your inequality by induction. Show the base case. Then show the how that T( n)= 0(n lg ig n). First, clearly indicate the inequality that you wish to hen proceed to prove the inductive hypothesis inductive case, and clearly...
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 answer. *Hint : You can use Master method to obtain Θ(.). (a) T(n) = 4T(n/4) + 5n (b) T(n) = 4T(n/5) + 5n (c) T(n) = 5T(n/4) + 4n (d) T(n) = 25T(n/5) + n^2 (e) T(n) = 4T(n/5) + lg n (f) T(n) = 4T(n/5) + lg^5 n...
Let n be a nonnegative integer and let F 22 + 1 be a Fermat number. Prove that if is a prime number, then either n=0 or 3--1mod F. [Hint: If n 2 1, use the law of quadratic reciprocity to evaluate the Legendre symbol (3/F). Now use Euler's Criterion (Theorem 4.4).] Let n be a nonnegative integer and let F 22 + 1 be a Fermat number. Prove that if is a prime number, then either n=0 or 3--1mod...