All Pairs Shortest Path Examples While the illustrations which follow only show solutions from vertex A (or 1) for simplicity, students should note that.

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Presentation transcript:

All Pairs Shortest Path Examples While the illustrations which follow only show solutions from vertex A (or 1) for simplicity, students should note that as an All-Pairs problem the algorithm is computing similar information for each vertex.

Matrix Multiply Method for APSP

The graph for the Slow “Matrix Multiply” Method 2 3 4 1 3 8 2 -4 -5 1 7 4 5 6

π: 1 d: 3 Slow “Matrix Multiply” Method Initial conditions, e.g., L(1) 2 3 4 π: 1 d: 8 1 3 8 2 -4 -5 1 7 4 5 6 π: nil d: ∞ π: 1 d: -4

π: 1 d: 3 Slow “Matrix Multiply” Method L(2) – bolded values indicate change 2 3 4 π: 1 d: 8 1 3 8 2 -4 -5 1 7 4 5 6 π: 5 d: 2 π: 1 d: -4

π: 1 d: 3 Slow “Matrix Multiply” Method L(3) – bolded values indicate change 2 3 4 π: 4 d: -3 1 3 8 2 -4 -5 1 7 4 5 6 π: 5 d: 2 π: 1 d: -4

π: 3 d: 1 Slow “Matrix Multiply” Method L(4) – bolded values indicate change 2 3 4 π: 4 d: -3 1 3 8 2 -4 -5 1 7 4 5 6 π: 5 d: 2 π: 1 d: -4

The graph for the Fast “Matrix Multiply” Method 2 3 4 1 3 8 2 -4 -5 1 7 4 5 6

π: 1 d: 3 Fast “Matrix Multiply” Method Initial conditions, e.g., L(1) 2 3 4 π: 1 d: 8 1 3 8 2 -4 -5 1 7 4 5 6 π: nil d: ∞ π: 1 d: -4

π: 1 d: 3 Fast “Matrix Multiply” Method L(2) – bolded values indicate change 2 3 4 π: 1 d: 8 1 3 8 2 -4 -5 1 7 4 5 6 π: 5 d: 2 π: 1 d: -4

π: 3 d: 1 Fast “Matrix Multiply” Method L(4) – bolded values indicate change 2 3 4 π: 4 d: -3 1 3 8 2 -4 -5 1 7 4 5 6 π: 5 d: 2 π: 1 d: -4

Floyd-Warshall Method for APSP

The graph for the Floyd- Warshall Method 2 3 4 1 3 8 2 -4 -5 1 7 4 5 6

π: 1 d: 3 Floyd-Warshall Method Initial conditions, e.g., D(0) 2 3 4 π: 1 d: 8 1 3 8 2 -4 -5 1 7 4 5 6 π: nil d: ∞ π: 1 d: -4

π: 1 d: 3 Floyd-Warshall Method D(1) – No Changes 2 3 4 π: 1 d: 8 1 3 8 2 -4 -5 1 7 4 5 6 π: nil d: ∞ π: 1 d: -4

π: 1 d: 3 Floyd Warshall Method D(2) – bolded values indicate change 2 3 4 π: 1 d: 8 1 3 8 2 -4 -5 1 7 4 5 6 π: 2 d: 4 π: 1 d: -4

π: 1 d: 3 Floyd Warshall Method D(3) – No changes 2 3 4 π: 1 d: 8 1 3 8 2 -4 -5 1 7 4 5 6 π: 2 d: 4 π: 1 d: -4

π: 1 d: 3 Floyd Warshall Method D(4) – bolded values indicate change 2 3 4 π: 4 d: -1 1 3 8 2 -4 -5 1 7 4 5 6 π: 2 d: 4 π: 1 d: -4

π: 3 d: 1 Floyd Warshall Method D(5) – bolded values indicate change 2 3 4 π: 4 d: -3 1 3 8 2 -4 -5 1 7 4 5 6 π: 5 d: 2 π: 1 d: -4