Dynamic Programming: Knapsack - Number of Ways to Make Change
What is the space complexity of the top-down memoized solution for counting ways to make change with n coins and amount W?
What is the space complexity of the top-down memoized solution for counting ways to make change with n coins and amount W?
medium🪤 Complexity Trap Q6 of Q15
Step-by-Step Solution
Solution:
Step 1: Analyze memo dictionary size
Memo stores results for states (index, amount), up to n * W entries.Step 2: Analyze recursion stack depth
Recursion depth can be up to n + W in worst case.Step 3: Total space complexity
Space = memo + recursion stack = O(n * W) + O(n + W) -> dominated by O(n * W).Final Answer:
Option B -> Option BQuick Check:
Memo size dominates space, recursion stack is smaller order [OK]
Quick Trick: Memo size plus recursion stack depth matter [OK]
Common Mistakes:
MISTAKES- Ignoring recursion stack space
- Assuming constant space
- Confusing memo size with stack
Trap Explanation:
PITFALL- Candidates often forget recursion stack space, underestimating total space complexity.
Interviewer Note:
CONTEXT- Tests understanding of auxiliary space in recursive DP solutions.
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