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Sparse Matrices
sparse … many elements are zero dense … few elements are zero
Example Of Sparse Matrices
diagonal tridiagonal lower triangular (?)
These are structured sparse matrices. May be mapped into a 1D array so that a mapping function can be used to locate an element.
Unstructured Sparse Matrices
Airline flight matrix. airports are numbered 1 through n flight(i,j) = list of nonstop flights from airport i to airport j n = 1000 (say) n x n array of list references => 4 million bytes total number of flights = 20,000 (say) need at most 20,000 list references => at most 80,000 bytes
Unstructured Sparse Matrices
Web page matrix. web pages are numbered 1 through n web(i,j) = number of links from page i to page j
Web analysis. authority page … page that has many links to it hub page … links to many authority pages
Web Page Matrix
n = 2 billion (and growing by 1 million a day) n x n array of ints => 16 * 10^18 bytes (16 * 10^9 GB) each page links to 10 (say) other pages on average on average there are 10 nonzero entries per row space needed for nonzero elements is approximately 20 billion x 4 bytes = 80 billion bytes (80 GB)
Representation Of Unstructured
Sparse Matrices
Single linear list in row-major order. scan the nonzero elements of the sparse matrix in row- major order each nonzero element is represented by a triple (row, column, value) the list of triples may be an array list or a linked list (chain)
Single Linear List Example
list = row 1 1 2 2 4 4 column 3 5 3 4 2 3 value 3 4 5 7 2 6
Array Linear List Representation
row 1 1 2 2 4 4 list = column 3 5 3 4 2 3 value 3 4 5 7 2 6
element 0 1 2 3 4 5 row 1 1 2 2 4 4 column 3 5 3 4 2 3 value 3 4 5 7 2 6
Array Of Row Chains
row[]
3 3
null 5 4
3 5
null 4 7
2 2
null 3 6
null
Orthogonal List Representation Both row and column lists.
Node structure. row col down next
value
Row Lists
0 2 6 0 0 (^) null
(^1 3 3 1 5 )
(^2 3 5 2 4 )
(^4 2 2 4 3 )
n
n
n
Column Lists
(^1 3 3 1 5 )
(^2 3 5 2 4 )
(^4 2 2 4 3 )
n
n n
Orthogonal Lists
0 2 6 0 0 (^) null
row[]
(^1 3 3 1 5 )
(^2 3 5 2 4 )
(^4 2 2 4 3 )
n n
n
n nn
Variations
May use circular lists instead of chains.
Approximate Memory Requirements
500 x 500 matrix with 1994 nonzero elements
2D array 500 x 500 x 4 = 1million bytes Single Array List 3 x 1994 x 4 = 23,928 bytes One Chain Per Row 23928 + 500 x 4 = 25,
Runtime Performance
Matrix Transpose 500 x 500 matrix with 1994 nonzero elements
2D array 210 ms Single Array List 6 ms One Chain Per Row 12 ms
