[Solved] Consider the following code for integer break. What is the value of dp[3] after the outer loop iteration i = 3 completes?... What is the value of dp[3] after the outer loop iteration i = 3 completes? | Dynamic Programming: Knapsack

Dynamic Programming: Knapsack - Integer Break

Consider the following code for integer break. What is the value of dp[3] after the outer loop iteration i = 3 completes?

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]

print(integer_break(4))  # Output not asked here

A2

B4

C3

D1

Step-by-Step Solution

Step 1: Trace dp for i=2
dp[1] = 1 (given). For i=2, j=1: max_product = max(0, max(1,1)*max(1,1)) = 1, so dp[2] = 1.
Step 2: Trace dp for i=3
j=1: max_product = max(0, max(1,1)*max(2,dp[2])) = max(0,1*2) = 2 j=2: max_product = max(2, max(2,dp[2])*max(1,dp[1])) = max(2,2*1) = 2 So dp[3] = 2.
Final Answer:
Option A -> Option A
Quick Check:
dp[3] = 2 matches manual calculation [OK]

Quick Trick: Trace inner loop carefully for each j [OK]

Common Mistakes:

MISTAKES

Off-by-one in loop bounds
Confusing dp[i-j] with dp[j]

Trap Explanation:

PITFALL

Candidates often miscalculate max_product by mixing indices or skipping max(j, dp[j]) logic.

Interviewer Note:

CONTEXT

Tests ability to mentally execute DP code and track intermediate states

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More Dynamic Programming: Knapsack Quizzes

Consider the following code for integer break. What is the value of dp[3] after the outer loop iteration i = 3 completes?

Step 1: Trace dp for i=2

Step 2: Trace dp for i=3

Final Answer:

Quick Check:

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