Dynamic Programming: Knapsack - Integer Break
Given the following bottom-up DP code for integer break, what is the value of dp[4] after all iterations?
Given the following bottom-up DP code for integer break, what is the value of dp[4] after all iterations?
easy🧾 Code Trace Q3 of Q15
Step-by-Step Solution
Solution:
Step 1: Trace dp[2] and dp[3]
dp[2] = max(1*1) = 1, dp[3] = max(1*2, 2*1) = 2Step 2: Calculate dp[4]
Check splits: j=1 -> max(1,dp[1]) * max(3,dp[3]) = 1*3=3; j=2 -> max(2,dp[2]) * max(2,dp[2])=2*2=4; j=3 -> max(3,dp[3]) * max(1,dp[1])=3*1=3; max is 4, so dp[4]=4Final Answer:
Option A -> Option AQuick Check:
dp array matches [0,1,1,2,4] [OK]
Quick Trick: Trace dp array stepwise for small n [OK]
Common Mistakes:
MISTAKES- Off-by-one in loops
- Confusing dp[i] with i
Trap Explanation:
PITFALL- Candidates often miscalculate dp[4] by missing max(j, dp[j]) usage.
Interviewer Note:
CONTEXT- Tests ability to mentally execute DP code and understand state transitions
Master "Integer Break" in Dynamic Programming: Knapsack
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