DATA STRUCTURES AND ALGORITHMSModule 2SPARSE MATRIX

SPARSE ARRAYA special array that contains more number of zero values than the non-zero values for their elements.

Eg: No of zero elements =6No. of non zero elements = 3Therefore, its a sparse matrix

106000002

SPARSE MATRIXA sparse matrix =2D sparse arrayA matrix is said to be a sparse matrix if most of its elements are zero.dense matrix =A matrix that is not sparseThe density of a matrix is the percentage of entries that are non-zero

Alternative Representations

If most of the elements are zero then the occurrence of zero elements in a large array is both a computational and storage inconvenience

Alternative Representations Array representation Dynamic representation

Array Representation (Tuple matrix)All non-zero elements are stored in another array of tripletNo of rows in the new array = No. of non zero elements + 1No. of columns in the new array = 3Triplet contains row number of the non-zero element column number of the non-zero element Value of non-zero elementTriplet can be represented by

Tuple matrix(0,0) No of rows in sparse matrix(0,1) No of columns in sparse matrix(0,2) No of non-zero elements in sparse matrixArray Representation (Tuple matrix)

100000500000

342001125

QuestionConvert sparse matrix to tuple matrix0 0 0 0 7 0 9 00 0 0 5

Sparse matrix to tuple matrixI/p: sparse matrix A[][]O/p: tuple matrix TUPLE[][3]i=0, j=0, k=1,count=0While(i

Sparse Matrix additionConvert sparse matrix to tuple matrix

040000706000

000310002000

343033101202

343014127206

+Sparse Matrix addition

343014127206

343033101202

345014033101127208

Sparse Matrix addition case 1If ((TUPLE1 [i][0] < TUPLE2 [j][0] )) SUM [ptr][0] =TUPLE1 [i][0] SUM [ptr][1] =TUPLE1 [ i][1] SUM [ptr][2] =TUPLE1 [ i][2] i=i+1, ptr=ptr+1, elem=elem+1

Sparse Matrix addition case 2If ((TUPLE1 [i][0] > TUPLE2 [j][0] )) SUM [ptr][0] =TUPLE2 [j][0] SUM [ptr][1] =TUPLE2 [ j][1] SUM [ptr][2] =TUPLE2 [ j][2] j=j+1, ptr=ptr+1, elem=elem+1

Sparse Matrix addition case 3If((TUPLE1 [i][0] = TUPLE2 [j][0] ) AND (TUPLE1 [i][1] =TUPLE2 [j][1] )) SUM [ptr][0] =TUPLE1 [i][0] SUM [ptr][1] =TUPLE1 [i][1] SUM [ptr][2] =TUPLE1 [i][2] +TUPLE2 [j][2] Ptr=ptr+1, i=i+1, j=j+1, elem=elem+1

Sparse Matrix addition case 4If ((TUPLE1 [i][0] = TUPLE2 [j][0] ) && (TUPLE1 [i][1]

Sparse Matrix addition case 5If ((TUPLE1 [i][0] = TUPLE2 [j][0] ) && (TUPLE1 [i][1] >TUPLE2 [j][1] ) ) SUM [ptr][0] =TUPLE2 [j][0] SUM [ptr][1] =TUPLE2[ j][1] SUM [ptr][2] =TUPLE2[ j][2] j=j+1, ptr=ptr+1, elem=elem+1

Sparse Matrix addition case 6aWhile(i

Sparse Matrix addition case 6bWhile(j

Algorithm SPARSE MATRIX ADDITION

Input: two sparse Matrices MAT1 and MAT2Output: Resultant Matrix SUM is the sum of two matricesData Structure: Sparse Matrix implemented by using array.

Steps:I=1,j=1,SUM[100][3],ptr=1,elem=0TUPLE1=SPARSE_TO_TUPLE(MAT1)TUPLE2=SPARSE_TO_TUPLE(MAT2)r1=row size,c1=column size of MAT1r2=row size, c2=column size of MAT2n1=ROWSIZE(TUPLE1)n2=ROWSIZE(TUPLE2)If(COMPARE(r1,r2) =FALSE) OR(COMPARE(c1,c2)=FALSE)Print Addition is not possibleExitElse SUM[0][0]=r1SUM[0][1]=c1SUM[0][2]=elem Stop

Limitations of array representation

We do not know the number of non-zero elements in advanceInsertion and deletion is not an easy task in an array

Solution:Use dynamic representation i.e Use linked lists

Special types of matricesSquare matrixA matrix in which no. of rows = no. of columns

Diagonal matrixA matrix in which only diagonal elements are non-zero

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147562938

Upper triangular matrixA matrix in which all the non-zero elements occur only on or above the diagonal

Lower Triangular matrixA matrix in which all the non-zero elements occur only on or below the diagonalSpecial types of matrices

147062008

100260358

Scalar matrixA diagonal matrix in which all diagonal elements are same

Identity or Unit matrixA diagonal matrix in which all diagonal elements are 1Special types of matrices

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100010001