1. Leftist Heaps Text Leftist Heap Building a Leftist Heap
Read Weiss, §6.6
Leftist Heap
Definition of null path length
Definition of leftist heap
Building a Leftist Heap
Sequence of inserts
Re-heapification if leftist heap property is violated

2. Motivation
A binary heap provides O(log n) inserts and O(log n) deletes but suffers from O(n log n) merges
A leftist heap offers O(log n) inserts and O(log n) deletes and O(log n) merges
Note, however, leftist heap inserts and deletes are more expensive than Binary Heap inserts and deletes

3. Definition
A Leftist (min)Heap is a binary tree that satisfies the follow-ing conditions. If X is a node and L and R are its left and right children, then:
X.value ≤ L.value
X.value ≤ R.value
null path length of L ≥ null path length of R
where the null path length of a node is the shortest distance between from that node to a descendant with 0 or 1 child.
If a node is null, its null path length is −1.

4. Example: Null Path Length4 8 19 12 27 20 15 25 43
node
npl

5. Example: Null Path Length4 8 19 1
What is the npl of the right child of 8? 12 1 27 20
What are the npl’s of the children of 20 and the right child of 27? 15 25 43
node
npl

6. Example: Null Path Length4
node 4 violates leftist heap property  fix by swapping children 8 19 1 12 1 27 20 15 25 43
node
npl

7. Leftist Heap 4 19 20 27 43 1 25 12 15 8 1
node
npl

8. Merging Leftist Heaps Consider two leftist heaps …4 6 19 8 8 7 27 20 12 14 15 25 43
Task: merge them into a single leftist heap

9. Merging Leftist Heaps First, instantiate a Stack 4 6 19 8 8 7 27 20 1214 15 25 43
First, instantiate a Stack

10. Merging Leftist Heaps Compare root nodes x y merge(x,y) 4 6 19 8 8 727 20 12 14 15 25 43
Compare root nodes
merge(x,y)

11. Merging Leftist Heaps Remember smaller value x y 4 4 6 19 8 8 7 27 2012 14 15 25 43
Remember smaller value

12. Merging Leftist Heaps y 4 4 x 6 19 8 8 7 27 20 12 14 15 25 43
Repeat the process with the right child of the smaller value

13. Merging Leftist Heaps Remember smaller value y x 6 4 4 6 19 8 8 7 2720 12 14 15 25 43
Remember smaller value

14. Merging Leftist Heaps 6 4 4 x 6 y 19 8 8 7 27 20 12 14 15 25 43
Repeat the process with the right child of the smaller value

15. Merging Leftist Heaps Remember smaller value x y 7 6 4 4 6 19 8 8 7 2720 12 14 15 25 43
Remember smaller value

16. Merging Leftist Heaps 7 6 4 4 x 6 19 8 8 7 y 27 20 12 14
null 15 25 43
Repeat the process with the right child of the smaller value

17. Merging Leftist Heaps 7 6 4 4 x 6 19 8 8 7 27 20 12 14 15 25 43 8
Because one of the arguments is null, return the other argument

18. Merging Leftist Heaps Make 8 the right child of 7 x Refers to node 8 76 4 4 6 x 19
Refers to node 8 8 7 27 20 14 8 12 43 8 15 25
Make 8 the right child of 7

19. Merging Leftist Heaps Make 7 leftist (by swapping children)6 4 4 6 19
Refers to node 8 8 7 8 27 20 14 12 43 8 15 25
Make 7 leftist (by swapping children)

20. Merging Leftist Heaps Return node 7 Refers to node 8 6 4 7 4 6 19 8 727 20 14 12 43 15 25 7
Return node 7

21. Make 7 the right child of 6 (which it already is)
Merging Leftist Heaps 6 4 4 6 19
Refers to node 8 8 7 8 27 20 14 12 43 15 25 7
Make 7 the right child of 6 (which it already is)

22. Merging Leftist Heaps Make 6 leftist (it already is) Refers to node 84 4 6 19
Refers to node 8 8 7 8 27 20 14 12 43 15 25 7
Make 6 leftist (it already is)

23. Merging Leftist Heaps Return node 6 Refers to node 8 4 6 4 6 19 8 7 827 20 14 12 43 15 25 6
Return node 6

24. Merging Leftist Heaps Make 6 the right child of 4 4 6 4 19 6 27 20 8 714 8 43 12 15 25 6
Make 6 the right child of 4

25. Merging Leftist Heaps Make 4 leftist (it already is) 4 6 4 19 6 27 208 7 14 8 43 12 15 25 6
Make 4 leftist (it already is)

26. Final Leftist Heap Verify that the tree is heap4 19 6 27 20 8 7 14 8 43 12 15 25
Verify that the tree is heap
Verify that the heap is leftist 4
Return node 4

27. Analysis Height of a leftist heap ≈ O(log n)
Maximum number of values stored in Stack ≈ 2 * O(log n) ≈ O(log n)
Total cost of merge ≈ O(log n)

28. Inserts and Deletes
To insert a node into a leftist heap, merge the leftist heap with the node
After deleting root X from a leftist heap, merge its left and right subheaps
In summary, there is only one operation, a merge.

29. Skew Heaps Text Skew Heap Building a Skew Heap Read Weiss, §6.7
No need for null path length
Definition of skew heap
Building a Skew Heap
Sequence of inserts
Fix heap if leftist heap property is violated

30. Motivation Simplify leftist heap by not maintaining null path lengths
swapping children at every step

31. Definition
A Skew (min)Heap is a binary tree that satisfies the follow-ing conditions. If X is a node and L and R are its left and right children, then:
X.value ≤ L.value
X.value ≤ R.value
A Skew (max)Heap is a binary tree that satisfies the follow-ing conditions. If X is a node and L and R are its left and right children, then:
X.value ≥ L.value
X.value ≥ R.value

32. Merging Skew Heaps Consider two skew heaps …4 6 19 8 8 7 27 20 12 14 15 25 43
Task: merge them into a single skew heap

33. Merging Skew Heaps First, instantiate a Stack 4 6 19 8 8 7 27 20 12 1415 25 43
First, instantiate a Stack

34. Merging Skew Heaps Compare root nodes x y merge(x,y) 4 6 19 8 8 7 2720 12 14 15 25 43
Compare root nodes
merge(x,y)

35. Merging Skew Heaps Remember smaller value x y 4 4 6 19 8 8 7 27 20 1214 15 25 43
Remember smaller value

36. Merging Skew Heaps y 4 4 x 6 19 8 8 7 27 20 12 14 15 25 43
Repeat the process with the right child of the smaller value

37. Merging Skew Heaps Remember smaller value y x 6 4 4 6 19 8 8 7 27 2012 14 15 25 43
Remember smaller value

38. Merging Skew Heaps 6 4 4 x 6 y 19 8 8 7 27 20 12 14 15 25 43
Repeat the process with the right child of the smaller value

39. Merging Skew Heaps Remember smaller value x y 7 6 4 4 6 19 8 8 7 27 2012 14 15 25 43
Remember smaller value

40. Merging Skew Heaps 7 6 4 4 x 6 19 8 8 7 y 27 20 12 14
null 15 25 43
Repeat the process with the right child of the smaller value

41. Merging Skew Heaps 7 6 4 4 x 6 19 8 8 7 27 20 12 14 15 25 43 8
Because one of the arguments is null, return the other argument

42. Merging Skew Heaps Make 8 the right child of 7 x Refers to node 8 7 64 4 6 x 19
Refers to node 8 8 7 27 20 14 8 12 43 8 15 25
Make 8 the right child of 7

43. Merging Skew Heaps Swap children of node 7 Refers to node 8 7 6 4 8 419
Refers to node 8 8 7 8 27 20 14 12 43 8 15 25
Swap children of node 7

44. Merging Skew Heaps Return node 7 Refers to node 8 6 4 7 4 6 19 8 7 827 20 14 12 43 15 25 7
Return node 7

45. Make 7 the right child of 6 (which it already is)
Merging Skew Heaps 6 4 4 6 19
Refers to node 8 8 7 8 27 20 14 12 43 15 25 7
Make 7 the right child of 6 (which it already is)

46. Merging Skew Heaps Swap children of node 6 Refers to node 8 6 4 7 4 619
Refers to node 8 7 8 8 27 20 14 12 43 15 25 7
Swap children of node 6

47. Merging Skew Heaps Return node 6 Refers to node 8 4 6 4 6 19 7 8 8 2720 14 12 43 15 25 6
Return node 6

48. Merging Skew Heaps Make 6 the right child of 4 4 6 4 19 6 7 8 27 20 814 43 12 15 25 6
Make 6 the right child of 4

49. Merging Skew Heaps Swap children of node 4 4 6 4 6 19 7 8 27 20 8 1443 12 15 25 6
Swap children of node 4

50. Final Skew Heap Verify that the tree is heap4 6 19 7 8 27 20 8 14 43 12 15 25
Verify that the tree is heap
Verify that the heap is skew 4
Return node 4

51. Analysis Height of a leftist heap ≈ O(log n)
Maximum number of values stored in Stack ≈ 2 * O(log n) ≈ O(log n)
Total cost of merge ≈ O(log n)

52. Inserts and Deletes
To insert a node into a leftist heap, merge the leftist heap with the node
After deleting root X from a leftist heap, merge its left and right subheaps
In summary, there is only one operation, a merge.