sparse matrix format

see Spare Matrix wikipedia item,
and scipy's documentation on different choices of sparse matrix type

sparse matrix storage, only store non-zero entries. there're multiple possible data structures for this, and can be divided into 2 groups

• support efficient modification
  • DOK (dictory of keys)
  • LIL (list of lists)
  • COO (coordiate list)
• support efficient access
  • CSR/CSC (compressed sparse row/column)

Dictionary of Keys (DOK)

• a dictionary that maps (row, col)-pair to the value;
• good for incremental build;
• poor for iterating;
• often used for building matrix, and convert to another format

List of Lists (LIL)

• matrix is a list of lists, one list for each row;
• each row list stores the (col, val) pair list;
• efficient for creation/insertion

Coordinate List (COO) aka IJV format

• sotre a list of (row, col, value) triplets, and ideally sorted by row then col;
• also known as IJV or Triplet format.

Compressed Sparse Row (CSR)

• an m*n matrix is represented as 3 vectors: vals, row_ptr, col_idx;
• vals: all values in row-major; length is number of non-zero matrix elements;
• col_idx: all values' column index in row-major order; same length with vals;
• row_ptr: row_ptr[0] = 0, row_ptr[k] = number-of-vals in first k rows; i.e. row_ptr[k+1]-row_ptr[k] is number of elements at row k;
• this is extremely optimized for row-by-row iteration: only access current portion of vals and col_idx, and 2 elements of row_ptr to determine the portion - super cache friendly;
• thus very suitable for cases like matrix-multiplication, matrix-vector-multiplication;