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

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

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

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

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

Matrix Addition:

2. Scalar Multiplication:

3. Matrix Multiplication

Add and Multiply Rings: Commutative, Associative Distributive Other rings

Matrix Multiplication Can be Defined on any Ring

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

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*

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

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

Transitive Closure and Matrix Multiplication

A Better Idea

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

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

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

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

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

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

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

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

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