1. 1 Fibonacci heaps: idea List of multiway trees which are all heap-ordered. Definition: A tree is called heap-ordered if the key of each node is greater than or equal to the key of its parent (if there is a parent). - Fibonacci heaps is a collection of trees with the heap property; these trees are notnecessarily binomial). -Tree roots are connected in a circular doubly linked list. -The FH is accessible by the min pointer (H) -The order of the trees is arbitrary 219 13458 3621 24 158352 79 117

2. 2 Theorem. Starting from empty Fibonacci heap, any sequence of a 1 insert, a 2 delete-min, and a 3 decrease-key operations takes O(a 1 + a 2 log n + a 3 ) time. make-heap Operation insert find-min delete-min union decrease-key delete 1 Binary Heap log n 1 n 1 Binomial Heap log n 1 Fibonacci Heap † 1 1 log n 1 1 1 Relaxed Heap 1 1 log n 1 1 1 Linked List 1 n n 1 n n is-empty11111 Priority Queues Performance Cost Summary † amortizedn = number of elements in priority queue

3. 3 Hopeless challenge. O(1) insert, delete-min and decrease-key. Why? Priority Queues Performance Cost Summary make-heap Operation insert find-min delete-min union decrease-key delete 1 Binary Heap log n 1 n 1 Binomial Heap log n 1 Fibonacci Heap † 1 1 log n 1 1 1 Relaxed Heap 1 1 log n 1 1 1 Linked List 1 n n 1 n n is-empty11111 † amortizedn = number of elements in priority queue

4. 4 Fibonacci Heaps History. [Fredman and Tarjan, 1986] n Ingenious data structure and analysis. Original motivation: improve Dijkstra's shortest path algorithm from O(E log V ) to O(E + V log V ). Basic idea. n Similar to binomial heaps, but less rigid structure. Binomial heap: eagerly consolidate trees after each insert. Fibonacci heap: lazily defer consolidation until next delete-min. V insert, V delete-min, E decrease-key

5. 5 723 30 17 35 2646 24 Heap H 39 41 1852 3 44 Fibonacci Heaps: Structure Fibonacci heap. n Set of heap-ordered trees. n Maintain pointer to minimum element. n Set of marked nodes. roots heap-ordered tree each parent larger than its children

6. 6 723 30 17 35 2646 24 Heap H 39 41 1852 3 44 Fibonacci Heaps: Structure Fibonacci heap. n Set of heap-ordered trees. n Maintain pointer to minimum element. n Set of marked nodes. min find-min takes O(1) time

7. 7 723 30 17 35 2646 24 Heap H 39 41 1852 3 44 Fibonacci Heaps: Structure Fibonacci heap. n Set of heap-ordered trees. n Maintain pointer to minimum element. n Set of marked nodes. min use to keep heaps flat (stay tuned) marked

8. 8 Fibonacci Heaps: Notation 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. 723 30 17 35 2646 24 39 41 1852 3 44 rank = 3 min Heap H trees(H) = 5 marks(H) = 3 marked n = 14

9. 9 Fibonacci Heaps: Potential Function 723 30 17 35 2646 24  (H) = 5 + 2  3 = 11 39 41 1852 3 44 min Heap H  (H) = trees(H) + 2  marks(H) potential of heap H trees(H) = 5 marks(H) = 3 marked

10. 10 Insert

11. 11 Fibonacci Heaps: Insert Insert. n Create a new singleton tree. n Add to root list; update min pointer (if necessary). 723 30 17 35 2646 24 39 41 1852 3 44 21 insert 21 min Heap H

12. 12 Fibonacci Heaps: Insert Insert. n Create a new singleton tree. n Add to root list; update min pointer (if necessary). 39 41 723 1852 3 30 17 35 2646 24 44 21 min Heap H insert 21

13. 13 Fibonacci Heaps: Insert Analysis Actual cost. O(1) Change in potential. +1 Amortized cost. O(1) 39 41 7 1852 3 30 17 35 2646 24 44 21 23 min Heap H  (H) = trees(H) + 2  marks(H) potential of heap H

14. 14 Delete Min

15. 15 Linking Operation Linking operation. Make larger root be a child of smaller root. 39 411852 3 44 77 5624 15 tree T 1 tree T 2 39 411852 3 44 77 5624 15 tree T' smaller root larger rootstill heap-ordered

16. 16 Fibonacci Heaps: Delete Min Delete min. n Delete min; meld its children into root list; update min. n Consolidate trees so that no two roots have same rank. 39 411852 3 44 1723 30 7 35 2646 24 min

17. 17 Fibonacci Heaps: Delete Min Delete min. n Delete min; meld its children into root list; update min. n Consolidate trees so that no two roots have same rank. 39 4117231852 30 7 35 2646 24 44 min

18. 18 Fibonacci Heaps: Delete Min Delete min. n Delete min; meld its children into root list; update min. n Consolidate trees so that no two roots have same rank. 39 4117231852 30 7 35 2646 24 44 min current

19. 19 Fibonacci Heaps: Delete Min Delete min. n Delete min; meld its children into root list; update min. n Consolidate trees so that no two roots have same rank. 39 4117231852 30 7 35 2646 24 44 0123 current min rank

20. 20 Fibonacci Heaps: Delete Min Delete min. n Delete min; meld its children into root list; update min. n Consolidate trees so that no two roots have same rank. 39 4117231852 30 7 35 2646 24 44 0123 min current rank

21. 21 Fibonacci Heaps: Delete Min Delete min. n Delete min; meld its children into root list; update min. n Consolidate trees so that no two roots have same rank. 39 4117231852 30 7 35 2646 24 44 0123 min current rank

22. 22 Fibonacci Heaps: Delete Min Delete min. n Delete min; meld its children into root list; update min. n Consolidate trees so that no two roots have same rank. 39 4117231852 30 7 35 2646 24 44 0123 min current rank link 23 into 17

23. 23 Fibonacci Heaps: Delete Min Delete min. n Delete min; meld its children into root list; update min. n Consolidate trees so that no two roots have same rank. 39 4117 23 1852 30 7 35 2646 24 44 0123 min current rank link 17 into 7

24. 24 Fibonacci Heaps: Delete Min Delete min. n Delete min; meld its children into root list; update min. n Consolidate trees so that no two roots have same rank. 39 417 30 1852 17 35 2646 24 44 0123 23 current min rank link 24 into 7

25. 25 Fibonacci Heaps: Delete Min Delete min. n Delete min; meld its children into root list; update min. n Consolidate trees so that no two roots have same rank. 39 417 30 1852 23 17 35 2646 2444 0123 min current rank

26. 26 Fibonacci Heaps: Delete Min Delete min. n Delete min; meld its children into root list; update min. n Consolidate trees so that no two roots have same rank. 39 417 30 1852 23 17 35 2646 2444 0123 min current rank

27. 27 Fibonacci Heaps: Delete Min Delete min. n Delete min; meld its children into root list; update min. n Consolidate trees so that no two roots have same rank. 39 417 30 1852 23 17 35 2646 2444 0123 min current rank

28. 28 Fibonacci Heaps: Delete Min Delete min. n Delete min; meld its children into root list; update min. n Consolidate trees so that no two roots have same rank. 39 417 30 1852 23 17 35 2646 2444 0123 min current rank link 41 into 18

29. 29 Fibonacci Heaps: Delete Min Delete min. n Delete min; meld its children into root list; update min. n Consolidate trees so that no two roots have same rank. 39 41 7 30 1852 23 17 35 2646 24 44 0123 min current rank

30. 30 Fibonacci Heaps: Delete Min Delete min. n Delete min; meld its children into root list; update min. n Consolidate trees so that no two roots have same rank. 7 30 52 23 17 35 2646 24 0123 min rank 39 41 18 44 current

31. 31 Fibonacci Heaps: Delete Min Delete min. n Delete min; meld its children into root list; update min. n Consolidate trees so that no two roots have same rank. 7 30 52 23 17 35 2646 24 min 39 41 18 44 stop

32. 32 Fibonacci Heaps: Delete Min Analysis Delete 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. Amortized cost. O(rank(H))  (H) = trees(H) + 2  marks(H) potential function

33. 33 Decrease Key

34. 34 Intuition for decreasing 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. n 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). 24 46 17 30 23 7 88 26 21 52 39 18 41 38 72 Fibonacci Heaps: Decrease Key 35 min marked node: one child already cut

35. 35 Case 1. [heap order not violated] Decrease key of x. n Change heap min pointer (if necessary). 24 46 17 30 23 7 88 26 21 52 39 18 41 38 72 Fibonacci Heaps: Decrease Key 29 35 min x decrease-key of x from 46 to 29

36. 36 Case 1. [heap order not violated] Decrease key of x. n Change heap min pointer (if necessary). 24 29 17 30 23 7 88 26 21 52 39 18 41 38 72 Fibonacci Heaps: Decrease Key 35 min x decrease-key of x from 46 to 29

37. 37 Case 2a. [heap order violated] Decrease key of x. Cut tree rooted at x, meld into root list, and unmark. If parent p of x is unmarked (hasn't yet lost a child), mark it; Otherwise, cut p, meld into root list, and unmark (and do so recursively for all ancestors that lose a second child). 24 29 17 30 23 7 88 26 21 52 39 18 41 38 72 Fibonacci Heaps: Decrease Key 15 35 min decrease-key of x from 29 to 15 p x

38. 38 Case 2a. [heap order violated] Decrease key of x. Cut tree rooted at x, meld into root list, and unmark. If parent p of x is unmarked (hasn't yet lost a child), mark it; Otherwise, cut p, meld into root list, and unmark (and do so recursively for all ancestors that lose a second child). 24 15 17 30 23 7 88 26 21 52 39 18 41 38 72 Fibonacci Heaps: Decrease Key 35 min decrease-key of x from 29 to 15 p x

39. 39 Case 2a. [heap order violated] Decrease key of x. Cut tree rooted at x, meld into root list, and unmark. If parent p of x is unmarked (hasn't yet lost a child), mark it; Otherwise, cut p, meld into root list, and unmark (and do so recursively for all ancestors that lose a second child). 2417 30 23 7 88 26 21 52 39 18 41 38 Fibonacci Heaps: Decrease Key 35 min decrease-key of x from 29 to 15 p 15 72 x

40. 40 Case 2a. [heap order violated] Decrease key of x. Cut tree rooted at x, meld into root list, and unmark. If parent p of x is unmarked (hasn't yet lost a child), mark it; Otherwise, cut p, meld into root list, and unmark (and do so recursively for all ancestors that lose a second child). 2417 30 23 7 88 26 21 52 39 18 41 38 Fibonacci Heaps: Decrease Key 35 min decrease-key of x from 29 to 15 p 15 72 x mark parent 24

41. 41 35 Case 2b. [heap order violated] Decrease key of x. Cut tree rooted at x, meld into root list, and unmark. If parent p of x is unmarked (hasn't yet lost a child), mark it; Otherwise, cut p, meld into root list, and unmark (and do so recursively for all ancestors that lose a second child). 24 15 17 30 23 7 88 26 21 52 39 18 41 38 72 24 Fibonacci Heaps: Decrease Key 5 min x p decrease-key of x from 35 to 5

42. 42 5 Case 2b. [heap order violated] Decrease key of x. Cut tree rooted at x, meld into root list, and unmark. If parent p of x is unmarked (hasn't yet lost a child), mark it; Otherwise, cut p, meld into root list, and unmark (and do so recursively for all ancestors that lose a second child). 24 15 17 30 23 7 88 26 21 52 39 18 41 38 72 24 Fibonacci Heaps: Decrease Key min x p decrease-key of x from 35 to 5

43. 43 Fibonacci Heaps: Decrease Key 2417 30 23 7 26 21 52 39 18 41 38 24 5 88 15 72 decrease-key of x from 35 to 5 x p min Case 2b. [heap order violated] Decrease key of x. Cut tree rooted at x, meld into root list, and unmark. If parent p of x is unmarked (hasn't yet lost a child), mark it; Otherwise, cut p, meld into root list, and unmark (and do so recursively for all ancestors that lose a second child).

44. 44 Case 2b. [heap order violated] Decrease key of x. Cut tree rooted at x, meld into root list, and unmark. If parent p of x is unmarked (hasn't yet lost a child), mark it; Otherwise, cut p, meld into root list, and unmark (and do so recursively for all ancestors that lose a second child). 2417 30 23 7 26 21 52 39 18 41 38 24 5 Fibonacci Heaps: Decrease Key 88 15 72 decrease-key of x from 35 to 5 x p second child cut min

45. 45 Case 2b. [heap order violated] Decrease key of x. Cut tree rooted at x, meld into root list, and unmark. If parent p of x is unmarked (hasn't yet lost a child), mark it; Otherwise, cut p, meld into root list, and unmark (and do so recursively for all ancestors that lose a second child). 24 26 17 30 23 7 21 52 39 18 41 38 88 24 5 Fibonacci Heaps: Decrease Key 15 72 decrease-key of x from 35 to 5 xp min

46. 46 Case 2b. [heap order violated] Decrease key of x. Cut tree rooted at x, meld into root list, and unmark. If parent p of x is unmarked (hasn't yet lost a child), mark it; Otherwise, cut p, meld into root list, and unmark (and do so recursively for all ancestors that lose a second child). 24 26 17 30 23 7 21 52 39 18 41 38 88 24 5 Fibonacci Heaps: Decrease Key 15 72 decrease-key of x from 35 to 5 xp p' second child cut min

47. 47 Case 2b. [heap order violated] Decrease key of x. Cut tree rooted at x, meld into root list, and unmark. If parent p of x is unmarked (hasn't yet lost a child), mark it; Otherwise, cut p, meld into root list, and unmark (and do so recursively for all ancestors that lose a second child). 26 17 30 23 7 21 52 39 18 41 38 88 5 Fibonacci Heaps: Decrease Key 1524 72 decrease-key of x from 35 to 5 xpp' min don't mark parent if it's a root p''

48. 48 Decrease-key. Actual cost. O(c) n 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. Amortized cost. O(1) Fibonacci Heaps: Decrease Key Analysis  (H) = trees(H) + 2  marks(H) potential function

49. 49 Analysis

50. 50 Union

51. 51 Fibonacci Heaps: Union Union. Combine two Fibonacci heaps. Representation. Root lists are circular, doubly linked lists. 39 41 717 1852 3 30 23 35 2646 24 44 21 min Heap H'Heap H''

52. 52 Fibonacci Heaps: Union Union. Combine two Fibonacci heaps. Representation. Root lists are circular, doubly linked lists. 39 41 717 1852 3 30 23 35 2646 24 44 21 min Heap H

53. 53 Fibonacci Heaps: Union Actual cost. O(1) Change in potential. 0 Amortized cost. O(1)  (H) = trees(H) + 2  marks(H) potential function 39 41 717 1852 3 30 23 35 2646 24 44 21 min Heap H

54. 54 Delete

55. 55 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. Fibonacci Heaps: Delete  (H) = trees(H) + 2  marks(H) potential function

56. 56 make-heap Operation insert find-min delete-min union decrease-key delete 1 Binary Heap log n 1 n 1 Binomial Heap log n 1 Fibonacci Heap † 1 1 log n 1 1 1 Relaxed Heap 1 1 log n 1 1 1 Linked List 1 n n 1 n n is-empty11111 Priority Queues Performance Cost Summary † amortizedn = number of elements in priority queue