Bit Manipulation Algorithms

Core Operators

Bit Masks

To operate on bit position k, create a mask 1 << k.

set:    n |  (1 << k)
clear:  n & ~(1 << k)
toggle: n ^  (1 << k)
test:  (n &  (1 << k)) != 0

These operations are constant-time under the usual fixed-width machine-integer model.

Count Set Bits

A simple method examines every bit. Brian Kernighan's method repeatedly clears the lowest set bit:

count = 0
while n != 0:
    n = n & (n - 1)
    count++

The loop runs once per set bit, so its time is O(number of 1-bits).

Power of Two Test

A positive power of two has exactly one set bit.

isPowerOfTwo(n):
    return n > 0 and (n & (n - 1)) == 0

For example, 8 = 1000₂ and 7 = 0111₂, so 8 & 7 = 0.

Subset Representation

An n-bit integer can represent a subset of n items: bit i is 1 if item i is included.

for mask = 0 to (1 << n) - 1:
    process subset represented by mask

This is common in exhaustive search and bitmask dynamic programming. Enumerating all subsets takes Θ(2ⁿ).

XOR Patterns

XOR has useful identities: x ^ x = 0, x ^ 0 = x, and it is associative and commutative.

This can find a unique value when every other value appears exactly twice.

result = 0
for x in array:
    result ^= x

The classic XOR-swap trick exists, but normal temporary-variable swapping is clearer and usually preferable in real code.