1. Sparse Matrices
Steve Paks

2. Sparse Matrices
•
Definition
–
head node
•
•
•
•
using
the down field to link into a column list
using the right field to link into a row list
the next field links the head nodes
the total number of head nods
= max{number of rows, number of columns}
entry
–
entry node
•
•
the down field links to the next nonzero term in the same column
the right field links to the next nonzero term in the same row

3. Sparse Matrices(Cont’d)
•
Declarations
class SparseMatrices{
int MAX_SIZE = 50;
MatrixNode hdnode[MAX_SIZE];
enum TagField{
head, entry
}
class EntryNode{
int row;
int col;
int value;
}
class MatrixNode{
MatrixNode down;
MatrixNode right;
TagField tag;
MatrixNode next;
EntryNode entry;
}
}

4. Sparse Matrices(Cont’d)
•
mread()
Matrix
Sparse Matrix
Linked Representation Of The Sparse Matrix

5. Sparse Matrices(Cont’d)
entry
•
Algorithm
node
temp
e
h
hdnode[0]
last
temp
…
h
temp
h
hdnode[4]
currentRow = 0
h
e
h
e
last
temp
2
0
2
11
temp
h
0
11
temp
hdnode[0]
hdnode[0]
4
hdnode[1]
hdnode[0]
last
4
e
0
11
2

6. Sparse Matrices(Cont’d)
entry
•
Algorithm(Cont’d)
h
h
hdnode[0]
currentRow = 0
hdnode[2]
e
last
0
2
11
h
h
hdnode[0]
hdnode[2]
last
temp
e
0
11
2
currentRow = 1

7. Sparse Matrices(Cont’d)
entry
•
Algorithm(Cont’d)
h
h
hdnode[0]
last
hdnode[1]
hdnode[2]
h
e
0
11
h
h
hdnode[0]
last
hdnode[1]
2
hdnode[2]
h
e
0
2
11
e
temp
1
12
0
currentRow = 1

8. Sparse Matrices(Cont’d)
entry
•
Algorithm(Cont’d)
currentRow = 1
h
h
hdnode[0]
hdnode[2]
h
e
0
2
11
hdnode[1]
e
last
temp
1
12
0

9. Sparse Matrices(Cont’d)
entry
•
Algorithm(Cont’d)
currentRow = 1
h
h
hdnode[0]
hdnode[2]
h
e
0
2
11
hdnode[1]
e
last
temp
1
12
0

10. Sparse Matrices(Cont’d)
entry
•
Algorithm(Cont’d)
currentRow = 1
h
h
hdnode[0]
hdnode[2]
h
e
0
2
11
hdnode[1]
e
last
temp
1
12
0

11. Sparse Matrices(Cont’d)
entry
•
Algorithm(Cont’d)
currentRow = 2
h
h
last
hdnode[2]
hdnode[0]
h
e
0
2
11
hdnode[1]
e
temp
1
12
0

12. Sparse Matrices(Cont’d)
entry
•
Algorithm(Cont’d)
currentRow = 2
h
h
hdnode[0]
hdnode[2]
h
e
0
2
11
hdnode[1]
e
1
12
temp
last
e
2
-4
1
0

13. Sparse Matrices(Cont’d)
entry
•
Algorithm(Cont’d)
currentRow = 2
h
h
hdnode[0]
hdnode[2]
h
e
0
2
11
hdnode[1]
e
1
12
temp
e
last
2
-4
1
0

14. Sparse Matrices(Cont’d)
entry
•
Algorithm(Cont’d)
currentRow = 2
h
h
hdnode[0]
hdnode[2]
h
e
0
2
11
hdnode[1]
e
1
12
temp
e
last
2
-4
1
0

15. Sparse Matrices(Cont’d)
entry
•
Algorithm(Cont’d)
currentRow = 3
h
last
h
hdnode[0]
hdnode[2]
h
h
hdnode[3]
e
0
2
11
hdnode[1]
e
1
0
12
e
2
1
-4
e
temp
3
-15
3

16. Sparse Matrices(Cont’d)
entry
•
Algorithm(Cont’d)
currentRow = 3
h
last
h
hdnode[0]
hdnode[2]
h
h
hdnode[3]
e
0
2
11
hdnode[1]
e
1
0
12
e
2
1
-4
e
temp
3
-15
3

17. Sparse Matrices(Cont’d)
entry
•
Algorithm(Cont’d)
currentRow = 3
h
h
hdnode[0]
h
hdnode[2]
h
hdnode[3]
e
0
2
11
hdnode[1]
e
1
0
12
e
2
1
-4
temp
last
e
3
-15
3

18. Sparse Matrices(Cont’d)
entry
•
Algorithm(Cont’d)
currentRow = 3
h
h
hdnode[0]
h
hdnode[2]
h
hdnode[3]
e
0
2
11
hdnode[1]
e
1
0
12
e
2
1
-4
temp
last
e
3
-15
3

19. Sparse Matrices(Cont’d)
•
e
Algorithm(Cont’d)
4
4
node
h
h
hdnode[0]
h
hdnode[2]
h
hdnode[3]
e
0
2
11
hdnode[1]
e
1
0
12
e
2
1
-4
temp
last
e
3
-15
3

20. Sparse Matrices(Cont’d)
•
Analysis of mread
–
–
–
–
–
O(max{numRows, numCols}) = new keyword works in a constant amount of time.
O(numTerms) = in a constant amount of time to set up each none zero
O(max{numRows, numCols}, numTerms) = O(numRows + numCols + numTerms)
Sparse Matrix using two-dimensional array
• O(numRows∙numCols)
linked list is better, but it is slightly worse than sequential method

21. Sparse Matrices(Cont’d)
•
mwrite()
–
–
–
–
•
using two for loop
outer for loop = the number of rows
inner for loop = the number of terms
O(numRows + numTerms)
merase()
–
–
–
–
–
–
resembles the structure found in mwrite()
erasing the entry nodes resembles the structure found in mwrite()
O(numRows + numTerms)
requiring extra time to erase the head nodes
O(numRows + numCols)
finally, O(numRows + numCols + numTerms)

22. Sparse Matrices(Cont’d)
•
Union in c vs Inheritance in java
AbstractNode
down : AbstractNode
right : AbstractNode
tag : TagField
getEntry()
getNext()
setNext(AbstractNode)
VS
HeadNode
next : AbstractNode
getEntry()
getNext()
setNext(AbstractNode)
EntryNode
entry : Entry
entry
EntryNode()
getEntry()