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What is the time complexity of the optimal bottom-up dynamic programming solution for the problem, given strs has length l, and the constraints are m zeros and n ones?

medium🪤 Complexity Trap Q13 of Q15

Dynamic Programming: Knapsack - Ones and Zeroes (2D Knapsack)

What is the time complexity of the optimal bottom-up dynamic programming solution for the problem, given strs has length l, and the constraints are m zeros and n ones?

AO(l * (m + n))

BO(l * m * n)

CO(l * m * n * max_length_of_string)

DO(2^l)

Step-by-Step Solution

Solution:

Step 1: Identify loops in the code
Outer loop iterates over all strings (l), inner loops iterate over m and n capacities.
Step 2: Calculate total operations
Each string causes updates over a 2D dp array of size (m+1)*(n+1), so total complexity is O(l * m * n).
Final Answer:
Option B -> Option B
Quick Check:
Nested loops over l, m, n -> O(l * m * n) [OK]

Quick Trick: Nested loops over strings and both capacities multiply complexities [OK]

Common Mistakes:

MISTAKES

Confusing sum with product of m and n
Including string length in complexity unnecessarily

Trap Explanation:

PITFALL

Option A looks simpler but ignores that both m and n loops multiply, not add.

Interviewer Note:

CONTEXT

Checks if candidate understands DP complexity with multiple resource constraints.

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