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Consider two approaches to count bits from 0 to n: (1) Dynamic Programming using right shift, and (2) Brian Kernighan's bit counting trick. When is the DP approach preferable over Brian Kernighan's?

hard⚖️ Approach Comparison Q8 of Q15
Bit Manipulation - Counting Bits (0 to N)
Consider two approaches to count bits from 0 to n: (1) Dynamic Programming using right shift, and (2) Brian Kernighan's bit counting trick. When is the DP approach preferable over Brian Kernighan's?
AWhen memory is limited, Brian Kernighan's uses less space than DP
BWhen n is very large and average set bits per number is high, DP is faster due to O(n) time
CWhen n is small, Brian Kernighan's is always faster due to fewer operations
DWhen numbers have very few set bits, DP is more efficient than Brian Kernighan's
Step-by-Step Solution
Solution:

  1. Step 1: DP approach runs in O(n) time regardless of bit distribution

    Brian Kernighan's depends on total set bits, which can be costly if many bits set.

  2. Step 2: For large n and dense bits, DP is preferable

    DP avoids overhead of clearing bits repeatedly.

  3. Step 3: For small n, Brian Kernighan's may be faster due to low overhead

    But question asks when DP is preferable, so large n with many set bits is the answer.

  4. Final Answer:

    Option B -> Option B

  5. Quick Check:

    DP is O(n), Brian Kernighan's depends on set bits count [OK]
Quick Trick: DP is O(n), Brian Kernighan's depends on set bits count [OK]
Common Mistakes:
MISTAKES
  • Assuming DP always better
  • Ignoring overhead for small inputs
  • Confusing space trade-offs
Trap Explanation:
PITFALL
  • Candidates often think DP is always better, ignoring constant overhead for small inputs.
Interviewer Note:
CONTEXT
  • Tests tradeoff reasoning between two common bit counting approaches
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