presentation about Fibonacci heaps and its operations

1. Fibonacci Heaps
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Created by:
Naseeba P P

2. What is a Fibonacci Heap?
• Heap ordered Trees.
• Rooted, but unordered.
• Children of a node are linked together in a Circular,
doubly linked List.
• Collection of unordered Binomial Trees.
• Support Mergeable heap operations such as Insert,
Minimum, Extract Min, and Union in constant time
O(1).
• Desirable when the number of Extract Min and
Delete operations are small relative to the number of
other operations.
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3. Cont..
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Node Pointers
- left [x]
- right [x]
- degree [x] - number of children in the child list of x
- - mark [x]

4. Fibonacci Heap Operations
• Insertion
• Linking operation
• Extract Minimum Node
• Decrease key
• Delete
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5. Insert
• Create a new singleton tree.
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6. Cont..
• Add to root list; update min pointer (if necessary).
• Increment the total number of nodes in the Heap,
n[H].
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7. Insert: Algorithm
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8. Insert Analysis
• Actual cost. O(1)
• Change in potential. +1
• Amortized cost. O(1)
• Potential of heap H
(H) = trees(H) + 2  marks(H)
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9. Union
• Combine two Fibonacci heaps
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10. Cont..
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11. Union: Algorithm
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12. Union Analysis
• Potential function
•Actual cost. O(1)
• Change in potential. 0
• Amortized cost. O(1)
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(H) = trees(H) + 2  marks(H)

13. Delete or Extract Min
• Delete minimum
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14. Cont..
• Meld its children into root list.
• Update minimum.
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15. Cont..
• Consolidate trees so that no two roots have same
rank.
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16. Cont..
current
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17. Cont..
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18. Cont..
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19. Cont..
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20. Cont..
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Link 23 into 17
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21. Cont..
Link 17 into 7
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22. Cont..
Link 24 into 7
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23. Cont..
current
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24. Cont..
current
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25. Cont..
current
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26. Cont..
current
Link 41 into 18
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27. Cont..
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28. Cont..
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29. Cont..
Stop 29

30. Delete Min: Algorithm
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31. Cont..
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32. Cont..
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33. Delete Min Analysis
Delete min.
Actual cost. O(rank(H)) + O(trees(H))
O(rank(H)) to meld min's children into root list.
O(rank(H)) + O(trees(H)) to update min.
O(rank(H)) + O(trees(H)) to consolidate trees.
Change in potential. O(rank(H)) - trees(H)
trees(H' )  rank(H) + 1 since no two trees have same rank.
(H)  rank(H) + 1 - trees(H).
Amortized cost. O(rank(H))
(H) = trees(H) + 2  marks(H)
Potential function
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34. Decrease Key
• Intuition for deceasing the key of node x.
• If heap-order is not violated, just decrease the key of x.
• Otherwise, cut tree rooted at x and meld into root list.
• To keep trees flat: as soon as a node has its second child
cut, cut it off and meld into root list (and unmark it).
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35. Cont..
Case 1. [heap order not violated]
– Decrease key of x.
– Change heap min pointer (if necessary).
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36. Cont..
Case 2a. [heap order violated]
– Decrease key of x.
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37. Cont..
Cut tree rooted at x, meld into root list, and unmark.
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38. Cont..
If parent p of x is unmarked (hasn't yet lost a child),
mark it.
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39. Cont..
Case 2b. [heap order violated]
– Decrease key of x.
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40. Cont..
Cut tree rooted at x, meld into root list, and unmark.
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41. Cont..
If parent p of x is marked
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42. Cont..
Cut p, meld into root list, and unmark
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43. Cont..
Do so recursively for all ancestors that lose a second
child.
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44. Cont..
Do so recursively for all ancestors that lose a second
child.
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45. Decrease key: Algorithm
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46. Cont..
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47. Decrease Key Analysis
Decrease-key.
Actual cost. O(c)
– O(1) time for changing the key.
– O(1) time for each of c cuts, plus melding into root list.
Change in potential. O(1) - c
– trees(H') = trees(H) + c.
– marks(H')  marks(H) - c + 2.
–   c + 2  (-c + 2) = 4 - c.
Amortized cost. O(1)
(H) = trees(H) + 2  marks(H)
Potential function
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48. Delete
Delete node x.
– decrease-key of x to -.
– delete-min element in heap.
Amortized cost. O(rank(H))
– O(1) amortized for decrease-key.
– O(rank(H)) amortized for delete-min.
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49. THANK YOU
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