Lecture 6 Shortest Path Problem

s t

Dynamic Programming

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

Dynamic Programming

Lemma Proof

Theorem

Counterexample

Smart Implementation

An Example Initialize 1 0      Select the node with the minimum temporary distance label.

Update Step      1

Choose u such that N_(u) S   

Update Step    The predecessor of node 3 is now node 2

Choose u Such That N_(u) S  3

Update d(5) is not changed 

Choose u s.t. N_(u) S  5

Update  5 d(4) is not changed 6

Choose u s.t. N_(u) S

Update d(6) is not updated

Choose u s.t. N_(u) S There is nothing to update

End of Algorithm 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

Dijkstra’s Algorithm

Lemma

Proof of Lemma s u w S T

Theorem

Counterexample

Dijkstra’s Algorithm

An Example Initialize 1 0      Select the node with the minimum temporary distance label.

Update Step      1

Choose Minimum Temporary Label   

Update Step    The predecessor of node 3 is now node 2

Choose Minimum Temporary Label  3

Update d(5) is not changed 

Choose Minimum Temporary Label  5

Update  5 d(4) is not changed 6

Choose Minimum Temporary Label

Update d(6) is not updated

Choose Minimum Temporary Label There is nothing to update

End of Algorithm 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

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

An Example Initialize distance labels 1 0      Select the node with the minimum temporary distance label  Initialize buckets.

Update Step       1

Choose Minimum Temporary Label  Find Min by starting at the leftmost bucket and scanning right till there is a non-empty bucket   

Update Step    

Choose Minimum Temporary Label   Find Min by starting at the leftmost bucket and scanning right till there is a non-empty bucket.

Update  

Choose Minimum Temporary Label   6 45

Update  

Choose Minimum Temporary Label

Update

Choose Minimum Temporary Label There is nothing to update

End of Algorithm 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

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