Bit manipulation involves operations on binary representations, useful for optimizing space, performing quick arithmetic, and solving subset problems efficiently.

Bit manipulation is an advanced technique in competitive programming that leverages the binary representation of integers to perform operations quickly and efficiently. Common bit manipulation operations include AND, OR, XOR, and bit shifts, which can be used to optimize space and perform fast calculations. For example, bit manipulation allows you to toggle, set, or clear specific bits in an integer, which can be useful in problems involving binary flags or permissions. In subset or combinatorial problems, bitwise operations can efficiently represent subsets of elements using bit masks, where each bit in an integer corresponds to the presence or absence of an element in a subset. Bit manipulation is also used in tasks like counting bits, finding the parity of numbers, and performing fast arithmetic in situations where speed is crucial. Mastering bit manipulation techniques opens up new possibilities for solving problems that might be inefficient with traditional methods.