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 p[x] 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 U k = 2 copies of U k-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 U k trees are linked to form U k+1

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.