1. Fibonacci Heaps

2. Fibonacci Binary insert O(1) O(log(n)) find O(1) N/A union O(1) N/A minimum O(1) O(1) decrease key O(1) O(log(n)) delete O(log(n) O(log(n)) delete minimum O(log(n)) N/A

3. Fibonacci Heaps Binomial Tree: A binomial tree of order 0 is a single node A binomial tree of order k has a root of degree k and its children are roots of binomial trees of orders k-1, k-2,..., 2, 1, 0 (in order). A binomial tree of order k has 2 k nodes

4. Fibonacci Heaps Data Structures: Circular doubly linked list of siblings ( ) All nodes have pointers to their parents One pointer to a child

5. Fibonacci Heaps Forest of binomial trees - key of node is less than keys of children

6. Fibonacci Heaps Forest of binomial trees - key of node is less than keys of children Node with minimum key is a root of one of the trees

7. Fibonacci Heaps Forest of binomial trees - key of node is less than keys of children Node with minimum key is a root of one of the trees A node may have degree greater than 2 but no larger than O(log(n))

8. Fibonacci Heaps Forest of binomial trees - key of node is less than keys of children Node with minimum key is a root of one of the trees A node may have degree greater than 2 but no larger than O(log(n)) Size of a subtree rooted in node of degree k is > F k+2 where F k is the kth Fibonacci number

9. Fibonacci Heaps Forest of binomial trees - key of node is less than keys of children Node with minimum key is a root of one of the trees A node may have degree greater than 2 but no larger than O(log(n)) Size of a subtree rooted in node of degree k is > F k+2 where F k is the kth Fibonacci number 6 6 2 2 5 5 3 3 4 4 7 7 8 8 9 9 1 1 Minimum Node

10. Fibonacci Heaps Operation: Find Minimum Simple lookup using Min Node pointer - O(1) 6 6 2 2 5 5 3 3 4 4 7 7 8 8 9 9 1 1 Minimum Node

11. Fibonacci Heaps Operation: Union of two heaps Attach higher numbered node to smaller Remove former root from linked list Add former root to children linked list - O(1) 6 6 2 2 5 5 3 3 4 4 7 7 8 8 9 9 1 1 Minimum Node

12. Fibonacci Heaps Operation: Union of two heaps Attach higher numbered node to smaller Remove former root from linked list Add former root to children linked list - O(1) 6 6 2 2 5 5 3 3 4 4 7 7 8 8 9 9 1 1 Minimum Node

13. Fibonacci Heaps 6 6 2 2 5 5 3 3 4 4 7 7 8 8 9 9 1 1 Minimum Node Operation: Insert Add new node as a heap Attach to the root linked list - O(1)

14. Fibonacci Heaps Operation: Insert Add new node as a heap Attach to the root linked list - O(1) 6 6 2 2 5 5 3 3 4 4 7 7 8 8 9 9 1 1 Minimum Node 1010 1010

15. Fibonacci Heaps Operation: Delete Minimum 6 6 2 2 5 5 3 3 4 4 7 7 8 8 9 9 1 1 Minimum Node

16. Fibonacci Heaps Operation: Delete Minimum Remove minimum node and make children roots 6 6 2 2 5 5 3 3 4 4 7 7 8 8 9 9 Minimum Node

17. Fibonacci Heaps Operation: Delete Minimum Remove minimum node and make children roots Union roots of same degree until all roots have different degree 6 6 2 2 5 5 3 3 4 4 7 7 8 8 9 9 Minimum Node

18. Fibonacci Heaps Operation: Delete Minimum Remove minimum node and make children roots Union roots of same degree until all roots have different degree 6 6 2 2 5 5 3 3 4 4 7 7 8 8 9 9 Minimum Node

19. Fibonacci Heaps Operation: Delete Minimum Remove minimum node and make children roots Union roots of same degree until all roots have different degree 6 6 2 2 5 5 3 3 4 4 7 7 8 8 9 9 Minimum Node

20. Fibonacci Heaps Operation: Delete Minimum Remove minimum node and make children roots Union roots of same degree until all roots have different degree Reset Minimum Node pointer 6 6 2 2 5 5 3 3 4 4 7 7 8 8 9 9 Minimum Node

21. Fibonacci Heaps Operation: Decrease Key 6 6 2 2 5 5 3 3 4 4 7 7 8 8 9 9 1 1 Minimum Node

22. Fibonacci Heaps Operation: Decrease Key Marked nodes are those having had exactly one child promoted to a root previously 6 6 2 2 5 5 3 3 4 4 7 7 8 8 9 9 1 1 Minimum Node

23. Fibonacci Heaps Operation: Decrease Key Decrease the key - if violation, cut from the tree, - promote it to a root - mark its parent if it is unmarked - if the parent had been marked cut it from its tree, and promote it to a root, and unmark it. 6 6 2 2 5 5 3 3 4 4 7 7 8 8 0 0 1 1 Minimum Node

24. Fibonacci Heaps Operation: Decrease Key Decrease the key - if violation, cut from the tree, - promote it to a root - mark its parent if it is unmarked - if the parent had been marked cut it from its tree, and promote it to a root, and unmark it. 6 6 2 2 5 5 3 3 4 4 7 7 8 8 0 0 1 1 Minimum Node

25. Fibonacci Heaps Operation: Decrease Key Decrease the key - if violation, cut from the tree, - promote it to a root - mark its parent if it is unmarked - if the parent had been marked cut it from its tree, and promote it to a root, and unmark it. - O(1) !!! 6 6 2 2 5 5 3 3 4 4 7 7 8 8 0 0 1 1 Minimum Node

26. Fibonacci Heaps Operation: Decrease Key Decrease the key - Change the Minimum Node pointer 6 6 2 2 5 5 3 3 4 4 7 7 8 8 0 0 1 1 Minimum Node

27. Fibonacci Heaps Operation: Delete Key of node to be deleted changed to minus infinity (decrease key operation) Followed by simple delete minimum 6 6 2 2 5 5 3 3 4 4 7 7 8 8 9 9 1 1 Minimum Node

28. Fibonacci Heaps Operation: Delete Key of node to be deleted changed to minus infinity (decrease key operation) Followed by simple delete minimum 6 6 2 2 5 5 3 3 4 4 7 7 8 8 -∞-∞ -∞-∞ 1 1 Minimum Node

29. Fibonacci Heaps Operation: Delete Key of node to be deleted changed to minus infinity (decrease key operation) Followed by simple delete minimum -∞-∞ -∞-∞ 2 2 5 5 3 3 4 4 7 7 8 8 6 6 1 1 Minimum Node

30. Fibonacci Heaps Operation: Delete Key of node to be deleted changed to minus infinity (decrease key operation) Followed by simple delete minimum 2 2 5 5 3 3 4 4 7 7 8 8 6 6 1 1 Minimum Node

31. Fibonacci Heaps Operation: Delete Key of node to be deleted changed to minus infinity (decrease key operation) Followed by simple delete minimum - O(log(n)) 2 2 5 5 3 3 4 4 7 7 8 8 6 6 1 1 Minimum Node

32. Fibonacci Heaps Result: Complexity of algorithms using priority queues is reduced! Example: Shortest Path – O((m+n)log(n)) with binary heap O(m +nlog(n)) with Fibonacci heap