4. Suppose T (n) satisfies the recurrence equations T(n) = 2 * T( n/2 ) +...
Recurrence equations using the Master Theorem: Characterize each of the following recurrence equations using the master method (assuming that T(n) = c for n < d, for constants c > 0 and d > = 1). T(n) = c for n < d, for constants c > 0 and d greaterthanorequalto 1). a. T(n) = 2T(n/2) + log n b. T(n) = 8T(n/2) + n^2 c. T(n)=16T(n/2) + (n log n)^4 d. T(n) = 7T(n/3) + n
1. Solve the recurrence relation T(n) = 2T(n/2) + n, T(1) = 1 and prove your result is correct by induction. What is the order of growth? 2. I will give you a shortcut for solving recurrence relations like the previous problem called the Master Theorem. Suppose T(n) = aT(n/b) + f(n) where f(n) = Θ(n d ) with d≥0. Then T(n) is: • Θ(n d ) if a < bd • Θ(n d lg n) if a = b...
Consider the recurrence T (n) = 3 · T (n/2) + n. • Use the recursion tree method to guess an asymptotic upper bound for T(n). Show your work. • Prove the correctness of your guess by induction. Assume that values of n are powers of 2.
Find the best big O bound you can on T(n) if it satisfies the recurrence T(n) ≤ T(n/4) + T(n/2) + n, with T(n) = 1 if n < 4.
Prove the proposition given in a) with induction or by picture. a) If D(N) satisfies D(N) = 2 D(N/2)+N for N > 1, with D(1) = 0, then D(N)=Nlog N. Where D(N) is the number of operations required to solve the problem of size N. b) Also, assume an example where N = 1000000 (i.e. 10^6) is problem size. If given CPU is capable of performing 100 (i.e. 10^2) operations per second. How much time it will take (in seconds)...
Suppose that the function f satisfies the recurrence rela- tion f (n)2f(Vn)+1 whenever n is a perfect square greater than 1 and f (2) 1. a) Find f(16). . b) Give a big-O estimate for f(n). [Hint: Make the sub- stitution m log n
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: ???? ? = ???? ?)???? ?...
Prove that the solution of the recurrence T(n) = T(n/2) +6(logk n) with T(1-6(1), for any integer k 2 0, is T(n) = Θ(logk+1 n) (Hint: the upper bound T(n) = O(logk+1 n) is easy; the lower bound T(n) = Ω(logk +1 n) is harder.) Prove that the solution of the recurrence T(n) = T(n/2) +6(logk n) with T(1-6(1), for any integer k 2 0, is T(n) = Θ(logk+1 n) (Hint: the upper bound T(n) = O(logk+1 n) is easy;...
use master's theorem to solve the following recurrence relation T(n) = 8T(n/2) + nlog(n)
6. Consider the recurrence relation T(n) = 2T(n-1) + 5 for integers n 1 and T(O) = 0. Find a closed-form solution Using induction, prove your solution correct for all integers n 20.