What is the asymptotic behavior of T(n) = 4T(n/2) + O(n3)? Select tightest bound that holds.

Question

BlackTom AI · Accepted Answer

Apply the Master Theorem for recurrences of the form T(n) = a T(n/b) + f(n). Here a = 4, b = 2, and f(n) = O(n^3) (we take f(n) proportional to n^3 for the asymptotic comparison). 1. Relevant concepts: - Master Theorem compares f(n) to n^{log_b a}. Compute n^{log_b a} = n^{log_2 4} = n^2. - If f(n) = Theta(n^{log_b a + epsilon}) for some epsilon > 0 and a regularity condition holds, then T(n) = Theta(f(n)) (case 3). 2. Step-by-step reasoning and intermediate calculations: - Compute exponent: log_b a = log_2 4 = 2, so n^{log_b a} = n^2. - Compare f(n) ~ n^3 with n^2: n^3 = n^{2+1} so epsilon = 1 > 0. Thus f(n) grows polynomially faster than n^{log_b a}. - Verify the regularity condition: check a f(n/b) ≤ c f(n) for some c < 1. If f(n) = C n^3, then a f(n/2) = 4 * C * (n/2)^3 = 4 * C * (n^3 / 8) = (1/2) C n^3 = (1/2) f(n), so we can take c = 1/2 < 1. Regularity condition holds. 3. Conclusion from Master Theorem: - By case 3, T(n) = Theta(f(n)) = Theta(n^3). Therefore the tight asymptotic bound is Theta(n^3), which implies O(n^3). 4. Why the correct answer is correct: - Because the work per level (f(n) = n^3) dominates the total cost compared to the tree-structure contribution n^2, the recurrence's growth is driven by f(n), yielding Theta(n^3). 5. Why other plausible bounds are incorrect: - O(n^2) or O(n^2 log n) are too small because they ignore the n^3 non-recursive work which dominates. - O(n^4) is correct as an upper bound but not tight; Theta(n^3) is the tight growth, so O(n^4) is weaker. Final confirmation: The asymptotic behavior is O(n^3), matching the provided answer.

What is the asymptotic behavior of T(n) = 4T(n/2) + O(n3)? Select tightest bound that holds.单项选择题

类似问题

In case 2 of the master theorem, If f(n) = O(n^k) where k = logb(a), then---------------.

In the following recurrence what are a and b? T(n)=3T(n/2)+n

In the context of the Master Theorem 𝑇 ( 𝑛 ) = 𝑎 𝑇 ( 𝑛 / 𝑏 ) + 𝑓 ( 𝑛 ) , the total work done by all internal nodes in the recursion tree is primarily determined by:

What is the asymptotic behavior of T(n) = 2T(n/2) + O(1)? Select the tightest bound that holds.

What is the asymptotic behavior of T(n) = 4T(n/2) + O(n2)? Select tightest bound that holds.

What is the asymptotic behavior of T(n) = 4T(n/2) + O(n)? Select tightest bound that holds.

What is the asymptotic behavior of T(n) = 4T(n/2) + O(1)? Select tightest bound that holds.

What is the asymptotic behaviour of T(n) = T(n/2) + O(n)? Select the tightest bound that holds.

更多留学生实用工具

智能学习助手

风格化写作助手

论文查重助手

文献引用助手

课堂转译助手

课堂笔记助手

Quiz搜索助手

学校历年真题

智能刷题助手

智能匹配练习

希望你的学习变得更简单