1. CSN2501 DSA Presentation
Fibonacci Heap
Anshuman Biswal
PT 2012 Batch, Reg. No.: CJB0412001
M. Sc. (Engg.) in Computer Science and Networking
Module Leader: N. D. Gangadhar
Module Name: Data Structures and Algorithms
Module Code : CSN2501
M. S. Ramaiah School of Advanced Studies 1

2. Marking
Head Maximum Score
Technical Content 10
Grasp and Understanding 10
Delivery – Technical and 10
General Aspects
Handling Questions 10
Total 40
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3. Presentation Outline
• Introduction
• Fibonacci Heap Structure
• Fibonacci Heap Implementation
• Notations and Potential Function
• Fibonacci Heap Operations
– Make a Fibonacci heap
– Insert a node
– Find minimum
– Bounding the maximum degree and Linking operation
– Extract min
– Union
– Decrease key
– Delete key
• Analysis
• Fibonacci Heap Bounding the Rank
• Conclusion
• References
M. S. Ramaiah School of Advanced Studies 3

4. Introduction
• Fibonacci heap is a heap data structure consisting of collection of trees.
• It has a better amortized running time of binomial heap.
• Developed by Michael L Fredman and Robert E Tarjan in 1984 and
first published in the scientific journal in 1987.
• The name Fibonacci heap comes from the Fibonacci numbers which
are used in running time analysis.
• Fibo heaps are mostly used when the number of EXTRACT-
MIN and DELETE operations is small relative to the number of other
operations performed.
• Fibonacci heaps are loosely based on Binomial Heaps but have less
rigid structure.
• Fibonacci heaps are not good to use for operation Search.
• Original motivation: improve Dijkstra's shortest path algorithm
from O(E log V ) to O(E + V log V ). V insert, V delete-min, E decrease-key
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5. Introduction
• Binomial heap: eagerly consolidate trees after each insert.
• Fibonacci heap: lazily defer consolidation until next delete-min.
• It also used for faster method for computing minimum spanning trees
with improvements from O(m log log base (m/n+2) of n) to
O(mβ)(m, n), where β(m, n) = min{i | log to the power i of n <= m/n}.
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6. Fibonacci Heap- Structure
• Fibonacci heap. each parent larger than its children
– Set of heap-ordered trees.
– Maintain pointer to minimum element.
– Set of marked nodes.
roots heap-ordered tree
17 24 23 7 3
30 26 46
18 52 41
35
39 44
Heap H
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7. Fibonacci Heap- Structure
• Fibonacci heap.
– Set of heap-ordered trees.
– Maintain pointer to minimum element.
– Set of marked nodes. find-min takes O(1) time
min
17 24 23 7 3
30 26 46
18 52 41
35
39 44
Heap H
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8. Fibonacci Heap- Structure
• Fibonacci heap.
– Set of heap-ordered trees.
– Maintain pointer to minimum element.
– Set of marked nodes.
use to keep heaps flat
min
17 24 23 7 3
30 26 46
18 52 41
marked
35
39 44
Heap H
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9. Fibonacci Heap- Implementation
– Represent trees using left-child, right sibling pointers and circular, doubly
linked list.
• Given two such lists ,we can quickly splice off sub trees or concatenate
in O(1) time
• Can remove or insert a node in O(1) time
– Roots of trees connected with circular doubly linked list.
• fast union
– Pointer to root of tree with min element.
min
• fast find-min
17 24 23 7 3
30 26 46
18 52 41
Heap H 35
39 44
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10. Fibonacci Heap- Implementation
• Each node x has pointer p[x] to its parent & child [x] to one of its children
• Children are linked together in a doubly-linked circular list which is called the
child list of x.
• Each child y in a child list has pointers left[y] and right [y] which points to left
and right siblings.
• Left[y]==right[y]==y, then y is the only child
• Each node also has degree [x] indicating the number of children of x
min
17 24 23 7 3
30 26 46
18 52 41
Heap H 35
39 44
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11. Fibonacci Heap- Implementation
• Each node also has mark [x], a Boolean field indicating whether x has lost a
child since the last time x was made the child of another node
• Some nodes will be marked
(indicated by the marked bit set to 1).
(i) A node x will be marked if x has lost a child since the last time that x was
made a child of another node.
(ii) Newly created nodes are unmarked
(iii) When node x becomes child of another node it becomes unmarked
3 min = 3
17 24 23 7
30 26 46
18 52 41
Heap H 35
39 44
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12. Fibonacci Heap- Implementation
•The entire heap is accessed by a pointer min [H] which points to the minimum-
key root
•Min(H) = NIL => H is empty
min
17 24 23 7 3
30 26 46
18 52 41
Heap H 35
39 44
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13. Notation and Potential Function
Notation
– n = number of nodes in heap.
– rank(x) = number of children of node x.
– rank(H) = max rank of any node in heap H.
– trees(H) = number of trees in heap H.
– marks(H) = number of marked nodes in heap H.
min
trees(H) = 5 marks(H) = 3 n = 14 rank = 3
17 24 23 7 3
30 26 46
18 52 41
marked
35
39 44
Heap H
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14. Notation and Potential Function
Potential Function
(H) = trees(H) + 2  marks(H)
potential of heap H
trees(H) = 5 marks(H)=3 Ф (H) = 5 + 2.3 = 11
min
17 24 23 7 3
30 26 46
18 52 41
marked
35
39 44
Heap H
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15. Fibonacci Heap Operations
• MAKE-FIB-HEAP(): creates and returns a new empty heap.
• FIB-HEAP-INSERT(H; x): inserts node x, whose key field has
already been filled in, into heap H.
• FIB-HEAP-MINIMUM(H): returns a pointer to the node with
minimum key in heap H.
• FIB-HEAP-EXTRACT-MIN(H): deletes the node from heap H
whose key is minimum, returning a pointer to the node.
• UNION(H1;H2): creates and returns a new heap that contains all
the nodes of heaps H1 and H2. Heaps H1 and H2 are “destroyed” by
this operation.
• FIB-HEAP-DECREASE-KEY(H; x; k) :assigns to node x within
heap H the new key value k. It is assumed that key ≤ x:key.
• FIB-HEAP-DELETE(H; x) :deletes node x from heap H.
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16. Fibonacci Heap Operations
Creating a Heap
MAKE-FIB-HEAP():
• allocates and returns the Fibonacci heap object H with H.n = 0 and
H.min = NIL:
• There are no trees in H. (H) = 0
• Because t(H) = 0 and m(H) = 0, the potential of the empty Fibonacci
heap  (H) = 0
• For MAKE-FIB-HEAP: amortized cost = actual cost = O(1).
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17. Fibonacci Heap Operations
Inserting a node
FIB-HEAP-INSERT(H; x):
– Create a new singleton tree.
– Add to root list.
– Update min pointer.
21 Insert 21
min
17 24 23 7 3
30 26 46
18 52 41
35
39 44
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18. Fibonacci Heap Operations
FIB-HEAP-INSERT Analysis:
Let H = Input Fibonacci heap and H = Resulting Fibonacci heap.
Then t(H) = t(H) + 1 and m(H) = m(H)
 Increase in potential = ((t(H)+1 )+ 2m(H)) - (t(H) + 2m(H)) = 1
Since actual cost = O(1) ,so the amortized cost is O(1) + 1 = O(1)
min
17 24 23 7 21 3
30 26 46
18 52 41
35
39 44
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19. Fibonacci Heap Operations
Finding the minimum node
FIB-HEAP-MINIMUM(H):
The minimum node of Fibonacci heap is given by the pointer H.min, so
we can find the minimum node in O(1) actual time.
Since the potential of H does not change, the amortized cost of the
operation is O(1)
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20. Bounding the maximum degree
If all trees in the Fibonacci heap are unordered
binomial trees, then D(n) = log 2 n .But the
(cascading) cuts may cause the occurrence of trees
that are not unordered binomial. Therefore, a
slightly weaker result still holds: D(n) ≤ log  n
where  is the golden ratio and is defined as
(1+ 5) / 2 = 1.61803
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21. Fibonacci Heap Operations
• Linking operation. Make larger root be a child of smaller root.
larger root smaller root still heap-ordered
15 3 3
56 24 18 52 41 15 18 52 41
77 39 44 56 24 39 44
tree T1 tree T2
77
tree T'
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22. Fibonacci Heap Operations
• FIB-HEAP-EXTRACT-MIN makes a root out of each of the
minimum node-s children and removes the minimum node from the
root list.
• Next, it consolidates the root list by linking roots of equal degree until
at most one root remains of each degree.
• Consolidate(H) consolidates the root list of H by executing repeatedly
the following steps until every root in the root list has a distinct degree
value:
– Find two roots x and y from the root list with the same degree, and
with x->key y->key:
– Link y to x: remove y from the root list , and make y a child of x.
This operation is performed by FIB-HEAP-LINK.
• Consolidate(H) uses an auxiliary array A[0::D(H:n)]; if A[i] = y then
y is currently a root with y->degree = i:
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23. Fibonacci Heap Operations
FIB-HEAP-EXTRACT-MIN(H):
This is the operation where all work delayed by other operations is done.
delayed work = consolidation (or merging) of the trees in the root list.
– Extract min and concatenate its children into root list.
– Consolidate trees so that no two roots have same degree.
min
17 24 23 7 3
30 26 46
18 52 41
35
39 44
Heap H
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24. Fibonacci Heap Operations
FIB-HEAP-EXTRACT-MIN(H):
– Extract min and concatenate its children into root list.
– Consolidate trees so that no two roots have same degree.
current
min
7 24 23 17 18 52 41
30 26 46 39 44
35
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25. Fibonacci Heap Operations
FIB-HEAP-EXTRACT-MIN(H):
– Delete min and concatenate its children into root list.
– Consolidate trees so that no two roots have same degree.
0 1 2 3
current
min
7 24 23 17 18 52 41
30 26 46 39 44
35
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26. Fibonacci Heap Operations
FIB-HEAP-EXTRACT-MIN(H):
– Delete min and concatenate its children into root list.
– Consolidate trees so that no two roots have same degree.
0 1 2 3
current
min
7 24 23 17 18 52 41
30 26 46 39 44
35
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27. Fibonacci Heap Operations
FIB-HEAP-EXTRACT-MIN(H):
– Delete min and concatenate its children into root list.
– Consolidate trees so that no two roots have same degree.
0 1 2 3
min
7 24 23 17 18 52 41
30 26 46 39 44
current
35
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28. Fibonacci Heap Operations
FIB-HEAP-EXTRACT-MIN(H):
– Delete min and concatenate its children into root list.
– Consolidate trees so that no two roots have same degree.
0 1 2 3
min
7 24 23 17 18 52 41
30 26 46 39 44
current
35
Merge 17 and 23 trees.
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29. Fibonacci Heap Operations
FIB-HEAP-EXTRACT-MIN(H):
– Delete min and concatenate its children into root list.
– Consolidate trees so that no two roots have same degree.
0 1 2 3
current
min
7 24 17 18 52 41
30 26 46 23 39 44
35
Merge 7 and 17 trees.
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30. Fibonacci Heap Operations
FIB-HEAP-EXTRACT-MIN(H):
– Delete min and concatenate its children into root list.
– Consolidate trees so that no two roots have same degree.
0 1 2 3
min current
24 7 18 52 41
26 46 17 30 39 44
35 23
Merge 7 and 24 trees.
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31. Fibonacci Heap Operations
FIB-HEAP-EXTRACT-MIN(H):
– Delete min and concatenate its children into root list.
– Consolidate trees so that no two roots have same degree.
0 1 2 3
min current
7 18 52 41
24 17 30 39 44
26 46 23
35
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32. Fibonacci Heap Operations
FIB-HEAP-EXTRACT-MIN(H):
– Delete min and concatenate its children into root list.
– Consolidate trees so that no two roots have same degree.
0 1 2 3
min current
7 18 52 41
24 17 30 39 44
26 46 23
35
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33. Fibonacci Heap Operations
FIB-HEAP-EXTRACT-MIN(H):
– Delete min and concatenate its children into root list.
– Consolidate trees so that no two roots have same degree.
0 1 2 3
min current
7 18 52 41
24 17 30 39 44
26 46 23
35
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34. Fibonacci Heap Operations
FIB-HEAP-EXTRACT-MIN(H):
– Delete min and concatenate its children into root list.
– Consolidate trees so that no two roots have same degree.
0 1 2 3
min current
7 18 52 41
24 17 30 39 44
26 46 23
Merge 41 and 18 trees.
35
M. S. Ramaiah School of Advanced Studies 34

35. Fibonacci Heap Operations
FIB-HEAP-EXTRACT-MIN(H):
– Delete min and concatenate its children into root list.
– Consolidate trees so that no two roots have same degree.
0 1 2 3
min current
7 52 18
24 17 30 41 39
26 46 23 44
35
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36. Fibonacci Heap Operations
FIB-HEAP-EXTRACT-MIN(H):
– Delete min and concatenate its children into root list.
– Consolidate trees so that no two roots have same degree.
0 1 2 3
min current
7 52 18
24 17 30 41 39
26 46 23 44
35
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37. Fibonacci Heap Operations
FIB-HEAP-EXTRACT-MIN(H):
– Delete min and concatenate its children into root list.
– Consolidate trees so that no two roots have same degree.
min
7 52 18
24 17 30 41 39
26 46 23 44
Stop.
35
M. S. Ramaiah School of Advanced Studies 37

38. Fibonacci Heap Operations
FIB-HEAP-EXTRACT-MIN(H): Example 2 [1]
min
23 7 21 3 17 24
30 26 46
18 52 38
35
39 41
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39. Fibonacci Heap Operations
FIB-HEAP-EXTRACT-MIN(H): Example 2 [1]
23 7 21 18 52 38 17 24
39 41 30 26 46
35
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40. Fibonacci Heap Operations
FIB-HEAP-EXTRACT-MIN(H): Example 2 [1]
0 1 2 3
x
23 7 21 18 52 38 17 24
39 41 30 26 46
35
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41. Fibonacci Heap Operations
FIB-HEAP-EXTRACT-MIN(H): Example 2 [1]
0 1 2 3
x
23 7 21 18 52 38 17 24
39 41 30 26 46
35
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42. Fibonacci Heap Operations
FIB-HEAP-EXTRACT-MIN(H): Example 2 [1]
0 1 2 3
x
23 7 21 18 52 38 17 24
39 41 30 26 46
35
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43. Fibonacci Heap Operations
FIB-HEAP-EXTRACT-MIN(H): Example 2 [1]
0 1 2 3
x
23 7 21 18 52 38 17 24
39 41 30 26 46
35
Merge 7,23
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44. Fibonacci Heap Operations
FIB-HEAP-EXTRACT-MIN(H): Example 2 [1]
0 1 2 3
x
7 21 18 52 38 17 24
39 41 30 26 46
23
35
Merge 7,17
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45. Fibonacci Heap Operations
FIB-HEAP-EXTRACT-MIN(H): Example 2 [1]
0 1 2 3
x
7 21 18 52 38 24
17
39 41 26 46
23
30 35
Merge 7,24
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46. Fibonacci Heap Operations
FIB-HEAP-EXTRACT-MIN(H): Example 2 [1]
0 1 2 3
x
7 21 18 52 38
17
39 41
23 24
30
26 46
35
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47. Fibonacci Heap Operations
FIB-HEAP-EXTRACT-MIN(H): Example 2 [1]
0 1 2 3
x
7 21 18 52 38
17
39 41
23 24
30
26 46
35
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48. Fibonacci Heap Operations
FIB-HEAP-EXTRACT-MIN(H): Example 2 [1]
0 1 2 3
x
7 21 18 52 38
17
39 41
23 24
30
26 46
35
Merge 21,52
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49. Fibonacci Heap Operations
FIB-HEAP-EXTRACT-MIN(H): Example 2 [1]
0 1 2 3
x
7 21 18 38
17
52 39 41
23 24
30
26 46
35
Merge 18,21
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50. Fibonacci Heap Operations
FIB-HEAP-EXTRACT-MIN(H): Example 2 [1]
0 1 2 3
x
7 18 38
17 21
39 41
23 24
30 52
26 46
35
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51. Fibonacci Heap Operations
FIB-HEAP-EXTRACT-MIN(H): Example 2 [1]
0 1 2 3
x
7 18 38
17 21
39 41
23 24
30 52
26 46
35
Stop
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52. Fibonacci Heap Operations
FIB-HEAP-EXTRACT-MIN Analysis
• Notation.
– D(n) = max degree of any node in Fibonacci heap with n nodes.
– t(H) = # trees in heap H.
– (H) = t(H) + 2m(H).
• Actual cost. O(D(n) + t(H))
– O(D(n)) work adding min's children into root list and updating
min.
• at most D(n) children of min node
– O(D(n) + t(H)) work consolidating trees.
• work is proportional to size of root list since number of roots
decreases by one after each merging
•  D(n) + t(H) - 1 root nodes at beginning of consolidation
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53. Fibonacci Heap Operations
FIB-HEAP-EXTRACT-MIN Analysis
• The potential before extracting the minimum node is t(H) + 2m(H).
• At most D(n) + 1 roots remain and no nodes become marked during
the operation => the potential after extracting the minimum node is
≤(D(n) + 1) + 2m(H).
=> the amortized cost is at most
P(D(n) + t(H)) + ((D(n) + 1) + 2m(H)) - (t(H) + 2m(H))
= O(D(n)) + O(t(H)) - t(H)
= O(D(n))
because the units of potential can be scaled to dominate the constant
hidden in O(t(H)):
M. S. Ramaiah School of Advanced Studies 53

54. Fibonacci Heap Operations
FIB-HEAP-UNION(H1,H2)
1. Concatenate the root list of H1 and H2 into new root list H
2. Set the minimum node of H
3. Set n[H] to total number of nodes
min min
23 24 17 7 3 21
30 26 46
18 52 41
35
39 44
Heap H2
Heap H1
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55. Fibonacci Heap Operations
Analysis: The change in potential is
Ф(H) - (Ф(H1) + Ф(H2))
= (t(H) + 2m(H)) - ((t(H1) + 2m(H1)) + ((t(H2) + 2m(H2))
= 0; ( since, t(H) = t(H1) + t(H2) and m(H) = m(H1) + m(H2)
=> amortized cost of FIB-HEAP-UNION = actual cost = O(1).
min
23 24 17 7 3 21
30 26 46
18 52 41
35
39 44
Heap H
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56. Fibonacci Heap Operations
• FIB-HEAP-DECREASE-KEY(H; x; k)
– Case 1: min-heap property not violated.
• decrease key of x to k
• change heap min pointer if necessary
min
7 18 38
24 17 23 21 39 41
26 45
46 30 52
35 88 72 Decrease 46 to 45.
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57. Fibonacci Heap Operations
• FIB-HEAP-DECREASE-KEY(H; x; k)
– Case 2: parent of x is unmarked.
• decrease key of x to k
• cut off link between x and its parent
• mark parent
• add tree rooted at x to root list, updating heap min pointer
min
7 18 38
24 17 23 21 39 41
26 15
45 30 52
35 88 72 Decrease 45 to 15.
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58. Fibonacci Heap Operations
• FIB-HEAP-DECREASE-KEY(H; x; k)
– Case 2: parent of x is unmarked.
• decrease key of x to k
• cut off link between x and its parent
• mark parent
• add tree rooted at x to root list, updating heap min pointer
min
7 18 38
24 17 23 21 39 41
26 45
15 30 52
35 88 72 Decrease 45 to 15.
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59. Fibonacci Heap Operations
• FIB-HEAP-DECREASE-KEY(H; x; k)
– Case 2: parent of x is unmarked.
• decrease key of x to k
• cut off link between x and its parent
• mark parent
• add tree rooted at x to root list, updating heap min pointer
min
15 7 18 38
72 24 17 23 21 39 41
26 30 52
35 88
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60. Fibonacci Heap Operations
• FIB-HEAP-DECREASE-KEY(H; x; k)
– Case 3: parent of x is marked.
• decrease key of x to k
• cut off link between x and its parent p[x], and add x to root list
• cut off link between p[x] and p[p[x]], add p[x] to root list
– If p[p[x]] unmarked, then mark it.
– If p[p[x]] marked, cut off p[p[x]], unmark, and repeat
min
15 7 18 38
72 24 17 23 21 39 41
26 30 52
35
5 88 Decrease 35 to 5.
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61. Fibonacci Heap Operations
• FIB-HEAP-DECREASE-KEY(H; x; k)
– Case 3: parent of x is marked.
• decrease key of x to k
• cut off link between x and its parent p[x], and add x to root list
• cut off link between p[x] and p[p[x]], add p[x] to root list
– If p[p[x]] unmarked, then mark it.
– If p[p[x]] marked, cut off p[p[x]], unmark, and repeat
min
15 5 7 18 38
72 24 17 23 21 39 41
26 30 52
88 Decrease 35 to 5.
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62. Fibonacci Heap Operations
• FIB-HEAP-DECREASE-KEY(H; x; k)
– Case 3: parent of x is marked.
• decrease key of x to k
• cut off link between x and its parent p[x], and add x to root list
• cut off link between p[x] and p[p[x]], add p[x] to root list
– If p[p[x]] unmarked, then mark it.
– If p[p[x]] marked, cut off p[p[x]], unmark, and repeat
min
15 5 26 7 18 38
72 88 24 17 23 21 39 41
30 52
Decrease 35 to 5.
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63. Fibonacci Heap Operations
• FIB-HEAP-DECREASE-KEY(H; x; k)
– Case 3: parent of x is marked.
• decrease key of x to k
• cut off link between x and its parent p[x], and add x to root list
• cut off link between p[x] and p[p[x]], add p[x] to root list
– If p[p[x]] unmarked, then mark it.
– If p[p[x]] marked, cut off p[p[x]], unmark, and repeat
min
15 5 26 24 7 18 38
72 88 17 23 21 39 41
30 52
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64. Fibonacci Heap Operations
FIB-HEAP-DECREASE-KEY(H; x; k) Analysis
• Notation.
– t(H) = # trees in heap H.
– m(H) = # marked nodes in heap H.
– (H) = t(H) + 2m(H).
• Actual cost. O(c)
– O(1) time for decrease key.
– O(1) time for each of c cascading cuts, plus reinserting in root list.
• Amortized cost. O(1)
– t(H') = t(H) + c
– m(H')  m(H) - c + 2
• each cascading cut unmark a node
• last cascading cut could potentially mark a node
–   c + 2(-c + 2) = 4 - c.
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65. Fibonacci Heap Operations
FIB-HEAP-DELETE(H; x)
– Decrease key of x to -.
– Extract Min or Delete min element in heap.
6 2 1
5 3 4 7
8
9 Delete 9
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66. Fibonacci Heap Operations
FIB-HEAP-DELETE(H; x)
– Decrease key of x to -.
– Extract Min or Delete min element in heap.
min
6 2 1
5 3 4 7
8
- Delete 9
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67. Fibonacci Heap Operations
FIB-HEAP-DELETE(H; x)
– Decrease key of x to -.
– Extract Min or Delete min element in heap.
min
6 2 1 -
5 3 4 7
8
Delete 9
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68. Fibonacci Heap Operations
FIB-HEAP-DELETE(H; x)
– Decrease key of x to -.
– Extract Min or Delete min element in heap.
min
6 2 1 - 8 7
5 3 4
Delete 9
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69. Fibonacci Heap Operations
FIB-HEAP-DELETE(H; x)
– Decrease key of x to -.
– Extract Min or Delete min element in heap.
min
6 2 1 - 8 7
5 3 4
Delete 9
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70. Fibonacci Heap Operations
FIB-HEAP-DELETE(H; x)
– Decrease key of x to -.
– Extract Min or Delete min element in heap.
min
6 1
2 3 4
5 7
8 Delete 9
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71. Fibonacci Heap Operations
FIB-HEAP-DELETE(H; x) Analysis
• The amortized execution time of FIB-HEAP-DELETE(H; x) is the
sum of the amortized time of FIB-HEAP-DECREASE i.e. O(1) and
the amortized time of FIB-HEAP-EXTRACT-MIN(H) i.e. O(D(n))
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72. Conclusion
• Fibonacci heap is a heap data structure consisting of collection of trees.
• It has a better amortized running time of binomial heap.
• It improve Dijkstra's shortest path algorithm
from O(E log V ) to O(E + V log V ).
• It is a heap ordered trees with the minimum node pointed to by a
pointer and some nodes marked.
• Fibonacci heap takes a amortized time of O(1) for make heap, insert,
find min ,union , decrease key and is empty operations and a O(log n)
amortized time for delete-min and delete operation.
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73. References
[1] Cormen T.H, Leiserson C.E., Rivest R.L., and Stein
C,(2009),IntroductiontoAlgorithms,3rdedn,NewDelhi:
Prentice Hall of India
[2] Fredman M and Tarjan R.E.,(July 1987), Fibonacci Heaps
and Their Uses in Improved Network Optimization
Algorithms. Journal of the Association for Computing
Machinery, Vol. 34, No. 3, July 1987, Pages 596-615.
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