1. Lecture 21: Matrix Operations and All-pair Shortest Paths
Shang-Hua Teng

2. Matrix Basic Vector: array of numbers; unit vector
Inner product, outer product, norm
Matrix: rectangular table of numbers, square matrix; Matrix transpose
All zero matrix and all one matrix
Identity matrix
0-1 matrix, Boolean matrix, matrix of graphs

3. Matrix of Graphs Else A(i, j) = 0. Adjacency Matrix:
If A(i, j) = 1: edge exists
Else A(i, j) = 0. 1 1 2 3 4 2 -3 4 3

4. Matrix of Graphs Else A(i, j) = infty. Weighted Matrix:
If A(i, j) = w(i,j): edge exists
Else A(i, j) = infty. 1 1 2 3 4 2 -3 4 3

5. Matrix Operations Matrix-vector operation Matrix operations
System of linear equations
Eigenvalues and Eigenvectors
Matrix operations

6. Matrix Addition:

7. 2. Scalar Multiplication:

8. 3. Matrix Multiplication

9. Add and Multiply Rings: Commutative, Associative Distributive
Other rings

10. Matrix Multiplication Can be Defined on any Ring

11. Two Graph Problems
Transitive closure: whether there exists a path between every pair of vertices
generate a matrix closure showing all transitive closures
for instance, if a path exists from i to j, then closure[i, j] =1
All-pair shortest paths: shortest paths between every pair of vertices
Doing better than Bellman-Ford O(|V|2|E|)
They are very similar

12. Transitive Closure D E
Given a digraph G, the transitive closure of G is the digraph G* such that
G* has the same vertices as G
if G has a directed path from u to v (u  v), G* has a directed edge from u to v
The transitive closure provides reachability information about a digraph B G C A D E B C A G*

13. Transitive Closure and Matrix Multiplication
Let A be the adjacency matrix of a graph G 1 2 3 4 -3 A

14. Floyd-Warshall, Iteration 2
BOS v
ORD 4
JFK v 2 v 6
SFO
DFW
LAX v 3 v 1
MIA v 5

15. Transitive Closure and Matrix Multiplication

16. A Better Idea

17. Even Better idea: Dynamic Programming; Floyd-Warshall
Number the vertices 1, 2, …, n.
Consider paths that use only vertices numbered 1, 2, …, k, as intermediate vertices:
Uses only vertices numbered 1,…,k
(add this edge if it’s not already in) i j
Uses only vertices
numbered 1,…,k-1
Uses only vertices
numbered 1,…,k-1 k

18. Floyd-Warshall’s Algorithm
A is the original matrix, T is the transitive matrix
T  A
for(k=1:n)
for(j=1:n)
for(i=1:n)
T[i, j] = T[i, j] OR
(T[i, k] AND T[k, j])
It should be obvious that the complexity is (n3) because of the 3 nested for-loops
T[i, j] =1 if there is a path from vertex i to vertex j

19. Floyd-Warshall Example
BOS v
ORD 4
JFK v 2 v 6
SFO
DFW
LAX v 3 v 1
MIA v 5

20. Floyd-Warshall, Iteration 1
BOS v
ORD 4
JFK v 2 v 6
SFO
DFW
LAX v 3 v 1
MIA v 5

21. Floyd-Warshall, Iteration 3
BOS v
ORD 4
JFK v 2 v 6
SFO
DFW
LAX v 3 v 1
MIA v 5

22. Floyd-Warshall, Iteration 4
BOS v
ORD 4
JFK v 2 v 6
SFO
DFW
LAX v 3 v 1
MIA v 5

23. Floyd-Warshall, Iteration 5
BOS v
ORD 4
JFK v 2 v 6
SFO
DFW
LAX v 3 v 1
MIA v 5

24. Floyd-Warshall, Iteration 6
BOS v
ORD 4
JFK v 2 v 6
SFO
DFW
LAX v 3 v 1
MIA v 5

25. Floyd-Warshall, Conclusion
BOS v
ORD 4
JFK v 2 v 6
SFO
DFW
LAX v 3 v 1
MIA v 5