sparse data structures

Sparse data structures are ways to store data efficiently when most of the values are zeros or empty. Instead of saving every position, these structures only record the locations and values of the non-zero entries. This saves memory and speeds up processing, especially for large datasets like matrices or graphs where only a few elements contain meaningful information. Common examples include sparse matrices and sparse vectors, which are widely used in scientific computing, machine learning, and data analysis to handle large, mostly empty datasets effectively.