[Solved] What is the time complexity of the bottom-up dynamic programming solution for... What is the time complexity of the bottom-up dynamic programming solution for the integer break problem shown below? | Dynamic Programming: Knapsack

Dynamic Programming: Knapsack - Integer Break

What is the time complexity of the bottom-up dynamic programming solution for the integer break problem shown below?

def integer_break(n):
    dp = [0] * (n + 1)
    dp[1] = 1
    for i in range(2, n + 1):
        max_product = 0
        for j in range(1, i):
            max_product = max(max_product, max(j, dp[j]) * max(i - j, dp[i - j]))
        dp[i] = max_product
    return dp[n]

AO(n log n) time

BO(n) time

CO(2^n) time

DO(n^2) time

Step-by-Step Solution

Step 1: Identify loops
Outer loop runs from 2 to n (≈ n iterations). Inner loop runs from 1 to i (up to n iterations).
Step 2: Calculate total operations
Nested loops yield roughly 1 + 2 + ... + n ≈ n(n+1)/2 = O(n^2) operations.
Final Answer:
Option D -> Option D
Quick Check:
Double nested loops with linear bounds -> quadratic time [OK]

Quick Trick: Nested loops with linear bounds usually mean O(n²) [OK]

Common Mistakes:

MISTAKES

Confusing with O(n) by ignoring inner loop
Thinking recursion stack adds exponential time here

Trap Explanation:

PITFALL

Some candidates mistake the problem for linear due to single dp array or confuse with brute force exponential time.

Interviewer Note:

CONTEXT

Checks understanding of DP time complexity vs brute force recursion

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More Dynamic Programming: Knapsack Quizzes

What is the time complexity of the bottom-up dynamic programming solution for the integer break problem shown below?

Step 1: Identify loops

Step 2: Calculate total operations

Final Answer:

Quick Check:

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