What are common bit-manipulation tricks?

Question

Accepted Answer

**Bit manipulation** operates directly on the binary representation of integers using AND, OR, XOR, NOT, and shifts. It enables compact, branch-free, and very fast operations. ## Core operations ```python x & 1 # is x odd? (lowest bit) x << 1 # multiply by 2 x >> 1 # integer divide by 2 x & (1 << k) # is bit k set? x | (1 << k) # set bit k x & ~(1 << k) # clear bit k x ^ (1 << k) # toggle bit k ``` ## Useful tricks ```python x & (x - 1) # clears the lowest set bit x & (x - 1) == 0 # is x a power of two? (x>0) x & (-x) # isolates the lowest set bit bin(x).count('1') # popcount (number of set bits) ``` ## Example: find the single number (XOR trick) ```python def single_number(nums): result = 0 for n in nums: result ^= n # pairs cancel: a ^ a == 0 return result # the lone unpaired value remains single_number([4, 1, 2, 1, 2]) # -> 4 ``` ## Complexity Bit ops are O(1); the XOR scan above is **O(n) time, O(1) space**. ## When to use / pitfalls Use for **bitmasks** (subset enumeration, DP over states), flags, fast math, and low-level/embedded code. Beware sign bits, language-specific shift behavior, and readability — comment non-obvious tricks. ## Why it matters Bit manipulation yields elegant O(1)-space solutions (like the XOR single-number trick) that astonish in interviews. Bitmask DP makes many combinatorial state-space problems tractable. It is essential in systems, graphics, compression, and performance-critical code where every cycle counts.