Which of the following changes correctly adapts the algorithm?
hard🎤 Interviewer Follow-up Q15 of Q15
Dynamic Programming: Knapsack - Perfect Squares
Suppose the problem is modified so that you want to find the minimum number of perfect squares summing to n, but now each perfect square can only be used at most once. Which of the following changes correctly adapts the algorithm?
AKeep the same bottom-up DP but iterate squares in increasing order and update dp[i] from dp[i - square] without reuse
BUse a top-down memoized recursion with a visited set to avoid reusing squares
CUse a 2D DP array where dp[i][j] represents minimum squares using first j squares to sum to i
DChange the inner loop to iterate over squares in decreasing order and update dp[i] accordingly
Step-by-Step Solution
Solution:
Step 1: Understand the constraint change
Limiting each perfect square to be used at most once converts the problem from unbounded to 0/1 knapsack variant, requiring tracking which squares are used.
Step 2: Identify correct DP adaptation
A 2D DP array dp[i][j] that tracks minimum squares to sum to i using first j squares correctly models the usage constraint. Other options either do not prevent reuse or do not track usage properly.
Step 3: Confirm why other options fail
Keep the same bottom-up DP but iterate squares in increasing order and update dp[i] from dp[i - square] without reuse still allows reuse implicitly. Use a top-down memoized recursion with a visited set to avoid reusing squares is inefficient and complex. Change the inner loop to iterate over squares in decreasing order and update dp[i] accordingly's order change alone doesn't enforce usage limits.
