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18-Shortest-Paths-II

18-Shortest-Paths-II - Algorithms LECTURE 18 Shortest Paths...

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Algorithms L18.1 Professor Ashok Subramanian L ECTURE 18 Shortest Paths II Bellman-Ford algorithm Linear programming and difference constraints VLSI layout compaction Algorithms
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Algorithms L18.2 Negative-weight cycles Recall: If a graph G = ( V , E ) contains a negative- weight cycle, then some shortest paths may not exist. Example: u v < 0
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Algorithms L18.3 Negative-weight cycles Recall: If a graph G = ( V , E ) contains a negative- weight cycle, then some shortest paths may not exist. Example: u v < 0 Bellman-Ford algorithm: Finds all shortest-path lengths from a source s V to all v V or determines that a negative-weight cycle exists.
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Algorithms L18.4 Bellman-Ford algorithm d [ s ] 0 for each v V – { s } do d [ v ] for i 1 to | V | – 1 do for each edge ( u , v ) E do if d [ v ] > d [ u ] + w ( u , v ) then d [ v ] d [ u ] + w ( u , v ) for each edge ( u , v ) E do if d [ v ] > d [ u ] + w ( u , v ) then report that a negative-weight cycle exists initialization At the end, d [ v ] = δ ( s , v ) , if no negative-weight cycles. Time = O ( V E ) . relaxation step
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Algorithms L18.5 Example of Bellman-Ford A B E C D –1 4 1 2 –3 2 5 3
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Algorithms L18.6 Example of Bellman-Ford A B E C D –1 4 1 2 –3 2 5 3 0 Initialization.
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Algorithms L18.7 Example of Bellman-Ford A B E C D –1 4 1 2 –3 2 5 3 0 1 2 3 4 5 7 8 Order of edge relaxation. 6
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Algorithms L18.8 Example of Bellman-Ford A B E C D –1 4 1 2 –3 2 5 3 0 1 2 3 4 5 7 8 6
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Algorithms L18.9 Example of Bellman-Ford A B E C D –1 4 1 2 –3 2 5 3 0 1 2 3 4 5 7 8 6
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Algorithms L18.10 Example of Bellman-Ford A B E C D –1 4 1 2 –3 2 5 3 0 1 2 3 4 5 7 8 6
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Algorithms L18.11 -1 Example of Bellman-Ford A B E C D –1 4 1 2 –3 2 5 3 0 1 2 3 4 5 7 8 6
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Algorithms L18.12 4 -1 Example of Bellman-Ford A B E C D –1 4 1 2 –3 2 5 3 0 1 2 3 4 5 7 8 6
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Algorithms L18.13 4 -1 Example of Bellman-Ford A B E C D –1 4 1 2 –3 2 5 3 0 1 2 3 4 5 7 8 6
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Algorithms L18.14 4 2 -1 Example of Bellman-Ford A B E C D –1 4 1 2 –3 2 5 3 0 1 2 3 4 5 7 8 6
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Algorithms L18.15 2 -1 Example of Bellman-Ford A B E C D –1 4 1 2 –3 2 5 3 0 1 2 3 4 5 7 8 6
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