Leftist Heaps and Skew Heaps. (Leftist Heaps and Skew Heaps) Data Structures and Programming Spring / 41
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1 Leftist Heaps and Skew Heaps (Leftist Heaps and Skew Heaps) Data Structures and Programming Spring / 41
2 Leftist Heaps 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(logn) merges Note, however, leftist heap inserts and deletes are more expensive than Binary Heap inserts and deletes (Leftist Heaps and Skew Heaps) Data Structures and Programming Spring / 41
3 Leftist Heaps: Definition A Leftist (min)heap is a binary tree that satisfies the following conditions. If X is a node and L and R are its left and right children, then: 1 X.value L.value 2 X.value R.value 3 null path length of L null path length of R where the null path length of a node is the shortest between from that node to a descendant with 0 or 1 child. If a node is null, its null path length is -1. (Leftist Heaps and Skew Heaps) Data Structures and Programming Spring / 41
4 Example: Null Path Length (Leftist Heaps and Skew Heaps) Data Structures and Programming Spring / 41
5 Example: Null Path Length (Leftist Heaps and Skew Heaps) Data Structures and Programming Spring / 41
6 Example: Null Path Length (Leftist Heaps and Skew Heaps) Data Structures and Programming Spring / 41
7 Leftist Heaps (Leftist Heaps and Skew Heaps) Data Structures and Programming Spring / 41
8 Merging Leftist Heaps (Leftist Heaps and Skew Heaps) Data Structures and Programming Spring / 41
9 Merging Leftist Heaps (Leftist Heaps and Skew Heaps) Data Structures and Programming Spring / 41
10 Merging Leftist Heaps (Leftist Heaps and Skew Heaps) Data Structures and Programming Spring / 41
11 Merging Leftist Heaps (Leftist Heaps and Skew Heaps) Data Structures and Programming Spring / 41
12 Merging Leftist Heaps (Leftist Heaps and Skew Heaps) Data Structures and Programming Spring / 41
13 Merging Leftist Heaps (Leftist Heaps and Skew Heaps) Data Structures and Programming Spring / 41
14 Merging Leftist Heaps (Leftist Heaps and Skew Heaps) Data Structures and Programming Spring / 41
15 Merging Leftist Heaps (Leftist Heaps and Skew Heaps) Data Structures and Programming Spring / 41
16 Merging Leftist Heaps (Leftist Heaps and Skew Heaps) Data Structures and Programming Spring / 41
17 Merging Leftist Heaps (Leftist Heaps and Skew Heaps) Data Structures and Programming Spring / 41
18 Merging Leftist Heaps (Leftist Heaps and Skew Heaps) Data Structures and Programming Spring / 41
19 Merging Leftist Heaps (Leftist Heaps and Skew Heaps) Data Structures and Programming Spring / 41
20 Merging Leftist Heaps (Leftist Heaps and Skew Heaps) Data Structures and Programming Spring / 41
21 Merging Leftist Heaps (Leftist Heaps and Skew Heaps) Data Structures and Programming Spring / 41
22 Merging Leftist Heaps (Leftist Heaps and Skew Heaps) Data Structures and Programming Spring / 41
23 Merging Leftist Heaps (Leftist Heaps and Skew Heaps) Data Structures and Programming Spring / 41
24 Merging Leftist Heaps (Leftist Heaps and Skew Heaps) Data Structures and Programming Spring / 41
25 Merging Leftist Heaps (Leftist Heaps and Skew Heaps) Data Structures and Programming Spring / 41
26 Final Leftist Heap (Leftist Heaps and Skew Heaps) Data Structures and Programming Spring / 41
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) (Leftist Heaps and Skew Heaps) Data Structures and Programming Spring / 41
28 Insert and Delete 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. (Leftist Heaps and Skew Heaps) Data Structures and Programming Spring / 41
29 Skew Heaps Simplify leftist heap by not maintaining null path lengths swapping children at every step A Skew (min)heap is a binary tree that satisfies the following conditions. If X is a node and L and R are its left and right children, then: 1 X.value L.value 2 X.value R.value A Skew (max)heap is a binary tree that satisfies the following conditions. If X is a node and L and R are its left and right children, then: 1 X.value L.value 2 X.value R.value (Leftist Heaps and Skew Heaps) Data Structures and Programming Spring / 41
30 Merging Skew Heaps (Leftist Heaps and Skew Heaps) Data Structures and Programming Spring / 41
31 Merging Skew Heaps (Leftist Heaps and Skew Heaps) Data Structures and Programming Spring / 41
32 Merging Skew Heaps (Leftist Heaps and Skew Heaps) Data Structures and Programming Spring / 41
33 Merging Skew Heaps (Leftist Heaps and Skew Heaps) Data Structures and Programming Spring / 41
34 Merging Skew Heaps (Leftist Heaps and Skew Heaps) Data Structures and Programming Spring / 41
35 Merging Skew Heaps (Leftist Heaps and Skew Heaps) Data Structures and Programming Spring / 41
36 Merging Skew Heaps (Leftist Heaps and Skew Heaps) Data Structures and Programming Spring / 41
37 Merging Skew Heaps (Leftist Heaps and Skew Heaps) Data Structures and Programming Spring / 41
38 Merging Skew Heaps (Leftist Heaps and Skew Heaps) Data Structures and Programming Spring / 41
39 Merging Skew Heaps (Leftist Heaps and Skew Heaps) Data Structures and Programming Spring / 41
40 Merging Skew Heaps (Leftist Heaps and Skew Heaps) Data Structures and Programming Spring / 41
41 Final Skew Heap (Leftist Heaps and Skew Heaps) Data Structures and Programming Spring / 41
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