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Suppose you want to count set bits for all numbers from 0 to n, but now n can be extremely large (up to 10^12). Which approach is best suited and why?

hard🎤 Interviewer Follow-up Q10 of Q15

Bit Manipulation - Counting Bits (0 to N)

Suppose you want to count set bits for all numbers from 0 to n, but now n can be extremely large (up to 10^12). Which approach is best suited and why?

ABrute force counting bits individually for each number

BMathematical formula or divide-and-conquer approach without enumerating all numbers

CBrian Kernighan's bit counting trick for each number

DDynamic programming with an array of size n+1

Step-by-Step Solution

Solution:

Step 1: Recognize that enumerating all numbers up to 10^12 is infeasible
Both brute force and DP require O(n) space/time, impossible for large n.
Step 2: Brian Kernighan's trick also requires iterating all numbers, so not feasible
Need a mathematical or divide-and-conquer method to count bits without enumeration.
Step 3: Such methods use bit patterns and recursion on bit positions to count set bits efficiently
This approach scales to very large n.
Final Answer:
Option B -> Option B
Quick Check:
Only mathematical approach handles huge n efficiently [OK]

Quick Trick: Large n requires formula-based counting, not enumeration [OK]

Common Mistakes:

MISTAKES

Trying DP or brute force for huge n
Ignoring space/time limits
Not considering mathematical solutions

Trap Explanation:

PITFALL

Candidates often default to known DP or bit tricks without considering scalability to huge inputs.

Interviewer Note:

CONTEXT

Tests candidate's ability to adapt approach for large input constraints

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