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What is the time complexity of the bottom-up dynamic programming solution for Last Stone Weight II, given n stones and total sum W?

medium📊 Complexity Q5 of Q15
Dynamic Programming: Knapsack - Last Stone Weight II
What is the time complexity of the bottom-up dynamic programming solution for Last Stone Weight II, given n stones and total sum W?
AO(n * W)
BO(n^2)
CO(W^2)
DO(n + W)
Step-by-Step Solution
Solution:

  1. Step 1: Understand DP dimensions

    The DP table size is n by W/2 (half sum), so roughly O(n * W).

  2. Step 2: Analyze nested loops

    Outer loop runs n times, inner loop runs up to W/2 times, so total O(n * W).

  3. Final Answer:

    Option B -> Option B

  4. Quick Check:

    DP iterates over all stones and sums [OK]
Quick Trick: DP loops over stones and sums -> O(n*W) [OK]
Common Mistakes:
MISTAKES
  • Assuming quadratic in n only
  • Ignoring sum dimension in complexity
  • Confusing with space complexity
Trap Explanation:
PITFALL
  • Ignoring the sum dimension leads to underestimating complexity.
Interviewer Note:
CONTEXT
  • Tests understanding of DP time complexity analysis.
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