Fibonacci Heaps

1
Fibonacci Heaps

• Especially desirable when the number of calls to
  Extract-Min Delete is small (note that all
  other operations run in O(1)
• This arises in many applications. Some graph
  problems, like minimum spanning tree and
  single-source-shortest-path problems call
  decrease-key much more often than other operations

2
Fibonacci Heaps

• Loosely based on binomial heaps
• If Decrease-key and Delete are never called, each
  tree is like a binomial tree

3
Fibonacci Heaps

• A Fibonacci heap is a collection of heap-ordered
  trees.
• Each tree is rooted but unordered.
• Each node x has pointed px to its parent
  child x to one of its children
• Children are linked together in a doubly-linked
  circular list.
• Doubly-linked circular list Removing a node is
  O(1) and splicing two lists is O(1)

4
Fibonacci Heaps

• Each node also has degree x indicating the
  number of children of x
• 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
• The entire heap is accessed by a pointer min H
  which points to the minimum-key root

5
Amortized Analysis

• Potential function ? (H) t (H) 2 m (H)
• The amortized analysis will depend on there being
  a known bound D(n) on the maximum degree of any
  node in an n-node heap
• When Decrease-Key and Delete are not used,
  D(n)log (n)

6
Fibonacci Heap

• An unordered binomial tree is the same as a
  binomial tree except that Uk 2 copies of Uk-1
  where the root of one tree is made any child of
  the root of the other
• If an n-node Fibonacci heap is a collection of
  unordered binomial trees, then D(n)log(n)

7
Fibonacci Heaps

• Called a lazy data structure
• Key idea Delay work for as long as possible

8
Fibonacci Heap Operations

• Insert
• Find-Min
• Union
• Extract-Min
• Consolidate

9
Analysis of Extract-Min

• Observe that if all trees are unordered binomial
  trees before the operation, then they are
  unordered binomial trees afterwards
• Only changes to structure of trees
• Children of minimum-root become roots
• Two Uk trees are linked to form Uk1

10
Analysis of Cost Actual Cost

• O(D(n)) children of minimum node.
• Plus cost of main consolidate loop
• Size of root list upon calling consolidate is
  D(n) t(H) - 1
• Every time through loop, one root is linked to
  another
• Therefore, total amount of work is D(n)t(H)

11
Analysis of Cost

• Actual Cost D(n) t(H)
• Potential Before t(H) 2m(H)
• Potential After At most (D(n)1) 2 m(H) since
  at most D(n) 1 roots remain.

12
Decrease Key

• If heap order is not violated (by the change in
  key value), then nothing happens
• If heap order is violated, we cut x from its
  parent making it a root
• The mark field has the following effect as soon
  as a node loses two children, it is cut from its
  parent
• Therefore, when we cut a node x from its parent
  y, we also set mark y

13
Decrease Key

• If mark x is already set, we cut y as well.
• Therefore, we might have a series of cascading
  cuts up the tree.

14
Actual Cost of Decrease Key

• Actual Cost 1 cost of cascading cuts
• Let c be number of times cascading cut is
  recursively called.
• Therefore, actual cost is O(c)

15
Change in Potential

• Let H be original heap. Each recursive call,
  except for last one, cuts a marked node and
  clears a marked bit.
• Afterwards, there are t(H) c trees at most
  m(H) - c 2 marked nodes ( c - 1 unmarked by
  cascading cuts last call may have marked one)

16
Cost of Decrease Key

• Intuitively m(H) is weighted twice t(H) because
  when a marked node is cut, the potential is
  reduced by 2. One unit pays for the cut the
  other compensates for the increase in potential
  due to t(H)

17
Bounding Maximum Degree

• To show that Extract-Min Delete are O(log n),
  we must show that the upper bound D(n) is O(log
  n)
• When all trees are unordered binomial trees, then
  D(n)?log n?
• But, cuts that take place in Decrease-Key may
  violate the unordered binomial tree properties.