Sparse Matrices. sparse many elements are zero dense few elements are zero

Transcription

1 Sparse Matrices sparse many elements are zero dense few elements are zero

2 Special Matrices A square matrix has the same number of rows and columns. Some special forms of square matrices are Diagonal: M(i,j) = 0 for i j Tridiagonal: M(i,j) = 0 for i-j < 1 Lower triangular: M(i,j) = 0 for i < j Upper triangular: M(i,j) = 0 for i > j Symmetric M(i,j) = M(j,i) for all i and j See Figure 2

3 Special Matrices 3

4 Example Of Sparse Matrices Diagonal Tri-diagonal lower triangular (?) Two possible representations of sparse matrix: 1. array 2. linked list

5 For example consider a matrix having 100 row and 100 column that have only 2000 non zero elements. Space required by above matrix is 1000,000. So we can store all the non-zero elements of above matrix in a 1D matrix of size or Create a linked list of non-zero elements.

6 Array Representation of Sparse Matrix The nonzero entries may be mapped into a 1D array in row-major order To reconstruct the matrix structure, need to record the row and column each nonzero comes from 6

7 Linked Representation of Sparse Matrix A shortcoming of the 1-D array of a sparse matrix is that we need to know the number of nonzero terms in each of the sparse matrices when the array is created. A linked representation can overcome this shortcoming. 7

8 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

9 Representation Of Unstructured Sparse Matrices Single linear list in row-major order. scan the nonzero elements of the sparse matrix in rowmajor 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)

10 Single Linear List Example list = row column value

11 Array Linear List Representation row list = column value element row column value

12 Node structure. Chain Representation row value col next

13 Single Chain row list = column value null firstnode

14 Multi-Linked Lists A multilinked list is a more general linked list with multiple links from nodes. In multi-linked list, each node can have any number of pointers to other nodes, and there may or may not be inverses for each pointer. Multi-lists are essentially the technique of embedding multiple lists into a single data structure. A multi-list has more than one next pointer, like a doubly linked list, but the pointers create separate lists

15 head Multi-lists

16 head Multi-lists

17 Linked Structures A doubly-linked list or multi-list is a data structure with multiple pointers in each node. In a doubly-linked list the two pointers create bidirectional links. In a multi-list the pointers used to make multiple link routes through the data.

18 Example : List of List Suppose you want to make list of all Customers in particular bank. Bank will have multiple branches. Each branch will have multiple customers.

19 Multi-list linked representation of Sparse Matrix

20 3-Tuple Representation Of Sparse Matrix sparse matrix 3-Tuple Representation Of Sparse Matrix Using Array: no of row no of col no of non zero value

21 Elements in the first row represents the number of rows, columns and non-zero values in sparse matrix. First Row rows 3 - columns 2 - non- zero values Elements in the other rows gives information about the location and value of non-zero elements ( Second Row) - represents value 1 at 0th Row, 0th column (Third Row) - represents value 4 at 2nd Row, 1st column

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23 Multi-list Representation Both row and column lists. Node structure. row col value down next

24 Row Lists null n n n

25 Column Lists n n n

26 Muti linked Lists null n n n row[] n n n

27 * structure for col headnode */ struct cheadnode { int colno ; struct node *down ; struct cheadnode *next ; } ; /* structure for row headnode */ struct rheadnode { int rowno ; struct node * right ; struct rheadnode *next ; } ;

28 /* structure for node to store element */ struct node { int row ; int col ; int val ; struct node *right ; struct node *down ; } ; /* structure for special headnode */ struct spmat { struct rheadnode *firstrow ; struct cheadnode *firstcol ; int noofrows ; int noofcols ; } ;

29 struct sparse { int *sp ; int row ; struct spmat *smat ; struct cheadnode *chead[max2] ; struct rheadnode *rhead[max1] ; struct node *nd ; } ;

30 One Linear List Per Row row1 = [(3, 3), (5,4)] row2 = [(3,5), (4,7)] row3 = [] row4 = [(2,2), (3,6)]

31 Node structure. Array Of Row Chains col next value

32 Array Of Row Chains null null null null row[]

33 Orthogonal List Representation Both row and column lists. Node structure. row down col next value

34 Row Lists null n n n

35 Column Lists n n n

36 Orthogonal Lists null n n n row[] n n n

37 Variations May use circular lists instead of chains.

38 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 x 4 = 25,928

39 Runtime Performance Matrix Transpose 500 x 500 matrix with 1994 nonzero elements 2D array Single Array List One Chain Per Row 210 ms 6 ms 12 ms

40 Performance Matrix Addition. 500 x 500 matrices with 1994 and 999 nonzero elements 2D array Single Array List One Chain Per Row 880 ms 18 ms 29 ms