CIS 330 C++ and UnixLecture 8

Memory and Pointers II (with Graphs and BFS)

Live Coding � Pointers and memory

Matrix Multiplication

Matrix Multiplication ci,n =∑"#$% &', ) ∗ +), ,

M

M

NN

c3,3 = a3,1 * b1,3 + a3,2 * b2,3

Example

� Sparse matrix – matrix with only a small number of values that are not zeros

� SpMV – Sparse Matrix Vector Multiply� Systems of equations � Graph processing

SpMV0 1 2 0 0

3 0 1 0 0

0 0 0 0 0

1 1 0 0 6

0 0 1 9 0

1

3

2

5

1

7

5

0

10

47

SpMV0 1 2 0 0

3 0 1 0 0

0 0 0 0 0

1 1 0 0 6

0 0 1 9 0

1

3

2

5

1

7

5

0

10

47

SpMV in COO

0 1 2 0 0

3 0 1 0 0

0 0 0 0 0

1 1 0 0 6

0 0 1 9 0

row col val

1 2 1

1 3 2

2 1 3

2 3 1

4 1 1

4 2 1

4 5 6

5 3 1

5 4 9

Example Sparse Matrices

sparsity = 0.33%

Compressed Sparse Row

row col val

1 2 1

1 3 2

2 1 3

2 3 1

4 1 1

4 2 1

4 5 6

5 3 1

5 4 9

2 3 1 3 1 2 5 3 4

1 2 3 1 1 1 6 1 9

col

val

1 3 5 5 8 10 row_ptr

Algorithm

� Input: A ∈ℝI×J, x ∈ℝJ

� Output: Ax = y ∈ℝI

� for i = 1 to I /* number of rows */

� get begin and end index into col and val for row I

� for j = begin to end

� y[i] += val[j] * x[ col[j] ]

� end

� end

SpMV� The concept is simple when you understand it, but initially difficult

to grasp the indexing

� Questions?