Software Project: Multiplication of Sparse Matrices

1
Software ProjectMultiplication of Sparse
Matrices

• November 26-27, 2006

2
Sparse Matrices

• A sparse matrix is a matrix populated primarily
  with zeros.
• Manipulating huge sparse matrices with the
  standard algorithms may be impossible due to
  their large size.
• Sparse data is by its nature easily compressed,
  which can yield enormous savings in memory usage.

3
Exercise 3

• Goals
• Implement several data structures for compressed
  representation of sparse matrices.
• Analyze the advantages and the disadvantages of
  each implementation.
• Measure the running times and select the most
  efficient implementation.

4
Exercise 3.1

• Use the linked list representation.
• The naïve implementation
• Replace each row by a linked list.

5
Disadvantages

• Accessing the columns elements is inconvenient.
  We need to go over all the rows to search for a
  certain column.
• It is more convenient to view matrix
  multiplication as multiplication of a row by a
  column.

(,j)
(i,j)

(i,)
A
B
C
Row-wise
6
Doubly Linked List Representation
7
Data Type Definition

• typedef struct cell_t
• struct cell_t row_next / pointer to the next
  element in the row. /
• struct cell_t col_next / pointer to the next
  element in the column. /
• int rowind / index in row /
• int colind / index in column /
• int value / value of the elem
  /
• cell_t / matrix cell data type /
• typedef struct
• int n / size /
• cell_t rows / array of row lists /
• cell_t cols / array of col lists /
• sparse_matrix_lst / sparse matrix
  representation /

8
Question 3.1 sparse_mlpl_lst

• Implement the multiplication of sparse matrices
  represented as doubly linked lists.
• Create sparse matrices of different size and fill
  them with random integers.
• Perform multiplication of matrices of different
  sizes.
• Measure the performance using the clock()
  function.
• Prepare 2 graphs which plot the running times in
  CPU ticks as the function of matrix size and the
  number of its non zero elements.

9
User Interface

• Input
• Case 1 0
• Case 2 An integer number followed by a matrix
  values
• Output
• Case 1 The running times
• Case 2 A matrix, which is the square of the
  input one.

10
Disadvantages of Linked Lists

• Memory allocation is performed for each non zero
  element of the matrix.
• Not a cache friendly code, i.e. we can not
  optimize it for a better cache memory utilization.

11
Exercise 3.2

• Use a compressed array representation.

values
rowind
colptr
12
Compressed data structure
22
84
61
values
2
1
0
rowind
colptr
13
Data Type Definition

• typedef struct
• int n / size /
• int colptr / pointers to where
  columns begin in rowind and values /
• int rowind / row indexes /
• elem values
• sparse_matrix_arr

14
Question 3.2 sparse_mlpl_arr

• Implement the multiplication of sparse matrices
  represented by the compressed array data
  structure.
• Perform multiplication of matrices of different
  sizes.
• Measure the performance using the clock()
  function.
• Prepare 2 graphs which plot the running times in
  CPU ticks as the function of matrix size and the
  number of its non zero elements.
• Compare the performance of the two
  representations of sparse matrices.
• Discuss the advantages and the disadvantages of
  each.

15
Compressed Array Implementation

• Observations
• The described data type is compressed by
  column.
• Accessing row elements is inconvenient.

22
84
61
values
2
1
0
rowind
colptr
16
Multiplication by column
The inner loop can be viewed as
Remember the jki loop ordering from exercise 2.1
(,j)
(,k)
(k,j)
/ jki / for (j0 jltn j) for (k0 kltn
k) r bkj for (i0 iltn i)
cij aik r
A
B
C
17
Multiplication by column
The second loop can be viewed as
Remember the jki loop ordering from exercise 2.1
(,j)
(,k)
(k,j)
/ jki / for (j0 jltn j) for (k0 kltn
k) r bkj for (i0 iltn i)
cij aik r
A
B
C
The column j in C is obtained by the
multiplication of the matrix A by the column
vector of B.
18
Matrix by vector multiplication
b
A

• Sometimes a better way to look at it
• Ab is a linear combination of As columns!

19
Files and locations

• All of your files should be located under your
  home directory /soft-proj07/assign3/sparse_mlpl_l
  st
• The source files and the executable should match
  the exercise name (e.g. sparse_mlpl_lst.c)
• Strictly follow the provided prototypes and the
  file framework.

20
Exercise 3 Notes

• Avoid unnecessary memory allocations.
• Read the input directly into the sparse matrix.
• Free all the allocated memory.
• Consult the website for recent updates and
  announcements.
• Use your own judgment!!!

21
Good Luck in the Exercise!!!