Fibonacci Heap

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Fibonacci Heap

• BHFH

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Why BHFH -- ????
MaxPQ
MinPQ
Symmetric Max Data Structures
DEPQ
Min Heap
Deap
MinMax
Mergeable MinPQ
Min-Leftist
Min-Skew
Min-B-Heap
Min-F-Heap
Inheritance
ADT
????
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Why BHFH -- ??

• Server for one priority shuts down?
• Efficient algorithms for Shortest-path problem
  and finding minimum spanning trees?(Improved
  network optimization)

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Whats BHFH -- Definition

• A binomial heap is a collection of binomial
  trees.
• The binomial tree Bk is an ordered tree defined
  recursively. the binomial tree B0 consists of a
  single node. The binomial tree Bk consists of two
  binomial trees Bk-1 that are linked together the
  root of one is the leftmost child of the root
• of the other.
• Like a binomial heap, a Fibonacci heap is a
  collection of min-heap-ordered trees. The trees
  in a Fibonacci heap are not constrained to be
  binomial trees.

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BH ??
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FH??
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?????

• MakeHeap()
• Insert(H,x)
• Minimum(H)
• Extract-min(H)
• Union(H1,H2)
• DecreaceKey(H,x,k)
• Delete(H,x)
• ????

Binary Min-Heap
Binomial Heap
Fibonacci Heap
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How Fibonacci Insert

• FIB-HEAP-INSERT(H, x)
• 1 degreex ? 0
• 2 px ? NIL
• 3 childx ? NIL
• 4 leftx ? x
• 5 rightx ? x
• 6 markx ? FALSE
• 7 concatenate the root list containing x with
  root list H
• 8 if minH NIL or keyx lt keyminH
• 9 then minH ? x
• 10 nH ? nH 1

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Fibonacci Insert 21
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How Fibonacci Union

• FIB-HEAP-UNION(H1, H2)
• 1 H ? MAKE-FIB-HEAP()
• 2 minH ? minH1
• 3 concatenate the root list of H2 with the root
  list of H
• 4 if (minH1 NIL) or (minH2 ? NIL and
  minH2 lt minH1)
• 5 then minH ? minH2
• 6 nH ? nH1 nH2
• 7 free the objects H1 and H2
• 8 return H

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How Fibonacci Extract-min

• FIB-HEAP-EXTRACT-MIN(H)
• 1 z ? minH
• 2 if z ? NIL
• 3 then for each child x of z
• 4 do add x to the root list of H
• 5 px ? NIL
• 6 remove z from the root list of H
• 7 if z rightz
• 8 then minH ? NIL
• 9 else minH ? rightz
• 10 CONSOLIDATE(H)
• 11 nH ? nH - 1
• 12 return z

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Extract-min 1
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Extract-min 2
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Extract-min 3
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Extract-min Final
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How Fibonacci Consolidate

• CONSOLIDATE(H)
• 1 for i ? 0 to D(nH)
• 2 do Ai ? NIL
• 3 for each node w in the root list of H
• 4 do x ? w
• 5 d ? degreex
• 6 while Ad ? NIL
• 7 do y ? Ad ? Another node with the
  same degree as x.
• 8 if keyx gt keyy
• 9 then exchange x ? y
• 10 FIB-HEAP-LINK(H, y, x)
• 11 Ad ? NIL
• 12 d ? d 1
• 13 Ad ? x
• 14 minH ? NIL
• 15 for i ? 0 to D(nH)
• 16 do if Ai ? NIL
• 17 then add Ai to the root list of H
• 18 if minH NIL or
  keyAi lt keyminH

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How Fibonacci Decrease-Key

• FIB-HEAP-DECREASE-KEY(H, x, k)
• 1 if k gt keyx
• 2 then error "new key is greater than current
  key"
• 3 keyx ? k
• 4 y ? px
• 5 if y ? NIL and keyx lt keyy
• 6 then CUT(H, x, y)
• 7 CASCADING-CUT(H, y)
• 8 if keyx lt keyminH
• 9 then minH ? x
• CUT(H, x, y)
• 1 remove x from the child list of y, decrementing
  degreey
• 2 add x to the root list of H
• 3 px ? NIL
• 4 markx ? FALSE
• CASCADING-CUT(H, y)
• 1 z ? py
• 2 if z ? NIL
• 3 then if marky FALSE

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Fibonacci Decrease-Key
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How Fibonacci Delete

• FIB-HEAP-DELETE(H, x)
• 1 FIB-HEAP-DECREASE-KEY(H, x, -8)
• 2 FIB-HEAP-EXTRACT-MIN(H)

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How Binomial Delete

• BINOMIAL-HEAP-DELETE(H, x)
• 1 BINOMIAL-HEAP-DECREASE-KEY(H, x, -8)
• 2 BINOMIAL-HEAP-EXTRACT-MIN(H)

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Note -- Time Complexity
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Note -- Difference

• 1 Delete
• DecreaseKey
• 2 BinomialOrdered
• FibonacciUnordered

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Note -- Application

• Shortest-path
• Combine PQ

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Fibonacci Heap

• ??

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• 1.???????Fibonacci Heap,?
• insert??????key???
• 70?80?90?30?40?50?20?60
• (hint insert?????????)
• 2.extractMin
• 3.?80?key,Decrease Key?10
• 4.??Fibonacci Heap??,????Union