Lecture 21: Matrix Operations and All-pair Shortest Paths

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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

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### 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

1

2

4

3

Matrix of Graphs

• If A(i, j) = 1: edge exists

Else A(i, j) = 0.

1

2

-3

4

3

1

2

4

3

Matrix of Graphs

Weighted Matrix:

• If A(i, j) = w(i,j): edge exists

Else A(i, j) = infty.

1

2

-3

4

3

Matrix Operations
• Matrix-vector operation
• System of linear equations
• Eigenvalues and Eigenvectors
• Matrix operations
• Rings:
• Commutative, Associative
• Distributive
• Other rings
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*

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Transitive Closure and Matrix Multiplication
• Let A be the adjacency matrix of a graph G

A

Floyd-Warshall, Iteration 2

BOS

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JFK

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SFO

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MIA

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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

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

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Floyd-Warshall, Iteration 1

BOS

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Floyd-Warshall, Iteration 3

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Floyd-Warshall, Iteration 4

BOS

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BOS

Floyd-Warshall, Iteration 5

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BOS

Floyd-Warshall, Iteration 6

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BOS

Floyd-Warshall, Conclusion

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