Implementation of Dijkstra

1
Implementation of Dijkstras Algorithm
2
Dijkstras Algorithm
3
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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An Example
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Initialize
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Select the node with the minimum temporary
distance label.
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Update Step
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Decrease-Key(S,2,2) Decrease-Key(S,3,4)
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Update Step
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Extract-Min(S)
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Choose Minimum Temporary Label
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Update Step
The predecessor of node 3 is now node 2
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Decrease-Key(S,3,3) Decrease-Key(S,4,6) Decrease-K
ey(S,5,4)
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Choose Minimum Temporary Label
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Extract-Min(S)
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Choose Minimum Temporary Label
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Extract-Min(S)
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Update
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d(5) is not changed.
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Choose Minimum Temporary Label
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Extract-Min(S)
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Choose Minimum Temporary Label
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Extract-Min(S)
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14
Update
d(4) is not changed
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Decrease-Key(S,6,6)
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Choose Minimum Temporary Label
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16
Update
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d(6) is not updated
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Choose Minimum Temporary Label
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There is nothing to update
18
End of Algorithm
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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
19
Implementations

• Decrease-Key(S,x,key) Call at most m times.
• Extract-Min(S) Call n times.
• With min-priority queue, Dijkstra algorithm can
  be
• implemented in time

20
Compared with Dynamic Programming

• Dijkstras algorithm uses O(n) space.
• Dynamic programming uses O(m) space.