Which approach guarantees the optimal time complexity for this problem?
easy🔍 Pattern Recognition Q11 of Q15
Bit Manipulation - Counting Bits (0 to N)
You need to compute the number of set bits (1s) in the binary representation of every integer from 0 up to a given number N efficiently. Which approach guarantees the optimal time complexity for this problem?
AUse Brian Kernighan's bit counting trick to count set bits for each number by repeatedly clearing the least significant set bit.
BUse a dynamic programming approach that counts bits by building results from smaller subproblems using right shifts.
CUse a greedy approach that counts bits by checking each bit position independently for all numbers.
DConvert each number to a binary string and count the '1' characters using built-in string functions.
Step-by-Step Solution
Solution:
Step 1: Understand the problem requirements
The problem requires counting set bits for all numbers from 0 to N efficiently.
Step 2: Evaluate approaches for optimality
Brian Kernighan's trick counts set bits per number in O(k) time, where k is the number of set bits, but doing this for all numbers results in O(n log n) in worst case. The dynamic programming approach uses previously computed results and right shifts to achieve O(n) time complexity, which is optimal.
Final Answer:
Option B -> Option B
Quick Check:
DP approach is optimal for counting bits from 0 to N [OK]
Quick Trick:DP approach builds on smaller subproblems for O(n) time [OK]
Common Mistakes:
MISTAKES
Assuming Brian Kernighan's trick is optimal for all numbers
Using string conversion for speed
Greedy bit checking is optimal
Trap Explanation:
PITFALL
While Brian Kernighan's trick is efficient per number, DP approach is optimal overall for counting bits from 0 to N.
Interviewer Note:
CONTEXT
Tests if candidate can identify the most efficient bit counting method beyond naive approaches.
Master "Counting Bits (0 to N)" in Bit Manipulation
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