XORsummias

XORsummias is a term used in competitive programming and computer science to describe a specific type of problem involving the bitwise XOR operation. These problems typically revolve around finding a subset of numbers from a given set whose XOR sum equals a particular target value. Alternatively, they might ask for the maximum possible XOR sum achievable from a subset, or properties related to the XOR sum of all possible subsets.

The XOR operation, denoted by the symbol '^', has unique properties that make these problems interesting. For instance, XORing a number with itself results in zero (a ^ a = 0), and XOR is associative and commutative. These properties are often exploited in algorithms designed to solve XORsummia problems.

Common techniques for tackling XORsummia problems include Gaussian elimination over GF(2) (the finite field with two elements), basis construction using techniques like the XOR basis or linear basis, and dynamic programming. The XOR basis allows for efficient representation and manipulation of the XOR sums of subsets. The complexity of solving these problems often depends on the size of the input set and the magnitude of the numbers involved. These problems appear frequently in programming contests and algorithmic challenges, testing a participant's understanding of bitwise operations and efficient data structures.