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Fibonacci Heaps

Created by:Naseeba P P

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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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Node Pointers - left [x] - right [x] - degree [x] - number of children in the child list of x - - mark [x]

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Fibonacci Heap Operations

• Insertion• Linking operation• Extract Minimum Node• Decrease key• Delete

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Insert

• Create a new singleton tree.

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• Add to root list; update min pointer (if necessary).• Increment the total number of nodes in the Heap,

n[H].

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Insert: Algorithm

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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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Union

• Combine two Fibonacci heaps

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Cont..

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Union: Algorithm

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Union Analysis

• Potential function

•Actual cost. O(1) • Change in potential. 0• Amortized cost. O(1)

(H)  = trees(H) + 2 marks(H)

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Delete or Extract Min

• Delete minimum

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• Meld its children into root list.• Update minimum.

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• Consolidate trees so that no two roots have same rank.

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current

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current

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current

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current

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current

Link 23 into 17

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Link 17 into 7

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Link 24 into 7

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current

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current

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current

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current

Link 41 into 18

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Stop

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Delete Min: Algorithm

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Delete Min AnalysisDelete 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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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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Cont..

Case 1. [heap order not violated]– Decrease key of x.– Change heap min pointer (if necessary).

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Cont..Case 2a. [heap order violated]– Decrease key of x.

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Cut tree rooted at x, meld into root list, and unmark.

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If parent p of x is unmarked (hasn't yet lost a child), mark it.

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Case 2b. [heap order violated]– Decrease key of x.

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Cut tree rooted at x, meld into root list, and unmark.

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If parent p of x is marked

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Cut p, meld into root list, and unmark

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Do so recursively for all ancestors that lose a second child.

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Do so recursively for all ancestors that lose a second child.

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Decrease key: Algorithm

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Decrease Key AnalysisDecrease-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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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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