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Lecture 6 Shortest Path Problem

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s t

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Dynamic Programming

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Dijkstra’s Algorithm is motivated from a way to implement of this dynamic programming.

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Dynamic Programming

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Lemma Proof

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Theorem 2 -1 2 1

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Counterexample

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Smart Implementation

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An Example 1 2 3 4 5 6 2 4 2 1 3 4 2 3 2 Initialize 1 0 Select the node with the minimum temporary distance label.

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Update Step 2 3 4 5 6 2 4 2 1 3 4 2 3 2 2 4 0 1

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Choose u such that N_(u) S 1 3 4 5 6 2 4 2 1 3 4 2 3 2 2 4 0 2

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Update Step 1 2 3 4 5 6 2 4 2 1 3 4 2 3 2 2 4 6 4 3 0 The predecessor of node 3 is now node 2

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Choose u Such That N_(u) S 1 24 5 6 2 4 2 1 3 4 2 3 2 2 3 6 4 0 3

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Update 1 24 5 6 2 4 2 1 3 4 2 3 2 0 d(5) is not changed. 3 2 3 6 4

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Choose u s.t. N_(u) S 1 24 6 2 4 2 1 3 4 2 3 2 0 3 2 3 6 4 5

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Update 1 24 6 2 4 2 1 3 4 2 3 2 0 3 2 3 6 4 5 d(4) is not changed 6

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Choose u s.t. N_(u) S 1 2 6 2 4 2 1 3 4 2 3 2 0 3 2 3 6 4 5 6 4

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Update 1 2 6 2 4 2 1 3 4 2 3 2 0 3 2 3 6 4 5 6 4 d(6) is not updated

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Choose u s.t. N_(u) S 1 2 2 4 2 1 3 4 2 3 2 0 3 2 3 6 4 5 6 4 6 There is nothing to update

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End of Algorithm 1 2 2 4 2 1 3 4 2 3 2 0 3 2 3 6 4 5 6 4 6 All nodes are now permanent The predecessors form a tree The shortest path from node 1 to node 6 can be found by tracing back predecessors

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Dijkstra’s Algorithm

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Lemma

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Proof of Lemma s u w S T

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Theorem

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Counterexample 3 -1 -2 2 1

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Dijkstra’s Algorithm

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An Example 1 2 3 4 5 6 2 4 2 1 3 4 2 3 2 Initialize 1 0 Select the node with the minimum temporary distance label.

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Update Step 2 3 4 5 6 2 4 2 1 3 4 2 3 2 2 4 0 1

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Choose Minimum Temporary Label 1 3 4 5 6 2 4 2 1 3 4 2 3 2 2 4 0 2

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Update Step 1 2 3 4 5 6 2 4 2 1 3 4 2 3 2 2 4 6 4 3 0 The predecessor of node 3 is now node 2

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Choose Minimum Temporary Label 1 24 5 6 2 4 2 1 3 4 2 3 2 2 3 6 4 0 3

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Update 1 24 5 6 2 4 2 1 3 4 2 3 2 0 d(5) is not changed. 3 2 3 6 4

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Choose Minimum Temporary Label 1 24 6 2 4 2 1 3 4 2 3 2 0 3 2 3 6 4 5

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Update 1 24 6 2 4 2 1 3 4 2 3 2 0 3 2 3 6 4 5 d(4) is not changed 6

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Choose Minimum Temporary Label 1 2 6 2 4 2 1 3 4 2 3 2 0 3 2 3 6 4 5 6 4

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Update 1 2 6 2 4 2 1 3 4 2 3 2 0 3 2 3 6 4 5 6 4 d(6) is not updated

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Choose Minimum Temporary Label 1 2 2 4 2 1 3 4 2 3 2 0 3 2 3 6 4 5 6 4 6 There is nothing to update

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End of Algorithm 1 2 2 4 2 1 3 4 2 3 2 0 3 2 3 6 4 5 6 4 6 All nodes are now permanent The predecessors form a tree The shortest path from node 1 to node 6 can be found by tracing back predecessors

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Dijkstra’s Algorithm with simple buckets (also known as Dial’s algorithm)

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An Example 1 2 3 4 5 6 2 4 2 1 3 4 2 3 2 Initialize distance labels 1 0 Select the node with the minimum temporary distance label. 01234567 1 2 3 4 5 6 Initialize buckets.

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Update Step 01234567 1 2 3 4 5 6 23 2 3 4 5 6 2 4 2 1 3 4 2 3 2 2 4 0 1

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Choose Minimum Temporary Label 01234567 4 5 6 23 Find Min by starting at the leftmost bucket and scanning right till there is a non-empty bucket. 1 3 4 5 6 2 4 2 1 3 4 2 3 2 2 4 0 2

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Update Step 1 2 3 4 5 6 2 4 2 1 3 4 2 3 2 2 4 6 4 3 0 01234567 4 5 6 23 3 4 5

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Choose Minimum Temporary Label 1 24 5 6 2 4 2 1 3 4 2 3 2 2 3 6 4 0 3 01234567 6 3 45 Find Min by starting at the leftmost bucket and scanning right till there is a non-empty bucket.

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Update 1 24 5 6 2 4 2 1 3 4 2 3 2 0 3 2 3 6 4 01234567 6 3 45

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Choose Minimum Temporary Label 1 24 6 2 4 2 1 3 4 2 3 2 0 3 2 3 6 4 5 01234567 6 45

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Update 1 24 6 2 4 2 1 3 4 2 3 2 0 3 2 3 6 4 5 6 01234567 6 45 6

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Choose Minimum Temporary Label 1 2 6 2 4 2 1 3 4 2 3 2 0 3 2 3 6 4 5 6 4 01234567 4 6

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Update 1 2 6 2 4 2 1 3 4 2 3 2 0 3 2 3 6 4 5 6 4 01234567 4 6

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Choose Minimum Temporary Label 1 2 2 4 2 1 3 4 2 3 2 0 3 2 3 6 4 5 6 4 6 01234567 6 There is nothing to update

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End of Algorithm 1 2 2 4 2 1 3 4 2 3 2 0 3 2 3 6 4 5 6 4 6 All nodes are now permanent The predecessors form a tree The shortest path from node 1 to node 6 can be found by tracing back predecessors

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Implementations With min-priority queue, Dijkstra algorithm can be implemented in time With Fibonacci heap, Dijkstra algorithm can be implemented in time With Radix heap, Dijkstra algorithm can be implemented in time

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