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 Adjacency Matrix: If A(i, j) = 1: edge exists Else A(i, j) = 0. 12 34 1 -3 3 2 4

4. Matrix of Graphs Weighted Matrix: If A(i, j) = w(i,j): edge exists Else A(i, j) = infty. 12 34 1 -3 3 2 4

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

6. 1. 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 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 A D C E B A D C E G G*

13. Transitive Closure and Matrix Multiplication Let A be the adjacency matrix of a graph G 12 34 1 -3 3 2 4 A

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

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: k j i Uses only vertices numbered 1,…,k-1 Uses only vertices numbered 1,…,k-1 Uses only vertices numbered 1,…,k (add this edge if it’s not already in)

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

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

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

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

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

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

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

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