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Leftist HeapsText• Read Weiss, §23.1 (Skew Heaps)
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
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
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:
1. X.value ≤ L.value2. X.value ≤ R.value3. 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.
Example: Null Path Length
19
2027
4325
12
15
8
4
node 4 8 19 12 15 25 27 20 43
npl 1 0 1 1 0 0 0 0 0
Example: Null Path Length
19
2027
4325
12
15
8
4
node 4 8 19 12 15 25 27 20 43
npl 1 0 1 1 0 0 0 0 0
0 1
1
0
00
0 0
What are the npl’s of the children of 20 and the right child of 27?
What is the npl of the right child of 8?
Example: Null Path Length
19
2027
4325
12
15
8
4
node 4 8 19 12 15 25 27 20 43
npl 1 0 1 1 0 0 0 0 0
•node 4 violates leftist heap property fix by swapping children
0 1
1
0
00
0 0
Leftist Heap
4
node 4 8 19 12 15 25 27 20 43
npl 1 0 1 1 0 0 0 0 0
25
12
15
8 0
1
0 0
19
2027
43
1
00
0
Merging Leftist Heaps
4
25
12
15
819
2027
43
6
78
14
Consider two leftist heaps …
Task: merge them into a single leftist heap
Merging Leftist Heaps
4
25
12
15
819
2027
43
6
78
14
First, instantiate a Stack
Merging Leftist Heaps
4
25
12
15
819
2027
43
6
78
14
Compare root nodesmerge(x,y)
x y
Merging Leftist Heaps
4
25
12
15
819
2027
43
6
78
14
Remember smaller value
4
x y
Merging Leftist Heaps
4
25
12
15
819
2027
43
6
78
14
Repeat the process with the right child of the smaller value
4
x
y
Merging Leftist Heaps
4
25
12
15
819
2027
43
6
78
14
Remember smaller value
6
4
x
y
Merging Leftist Heaps
4
25
12
15
819
2027
43
6
78
14
Repeat the process with the right child of the smaller value
6
4
x y
Merging Leftist Heaps
4
25
12
15
819
2027
43
6
78
14
Remember smaller value
7
6
4
x y
Merging Leftist Heaps
4
25
12
15
819
2027
43
6
78
14
Repeat the process with the right child of the smaller value
7
6
4
x
y
null
Merging Leftist Heaps
4
25
12
15
819
2027
43
6
78
14
Because one of the arguments is null, return the other argument
7
6
4
x
8
Merging Leftist Heaps
4
25
12
15
19
2027
43
6
8
14
Make 8 the right child of 7
7
6
4
x
8
8
7Refers to node 8
Merging Leftist Heaps
4
25
12
15
19
2027
43
6
8
14
Make 7 leftist (by swapping children)
7
6
4
8
8
7Refers to node 8
Merging Leftist Heaps
4
25
12
15
19
2027
43
6
8
14
Return node 7
6
4
7
8
7Refers to node 8
Merging Leftist Heaps
4
25
12
15
19
2027
43
6
8
14
Make 7 the right child of 6 (which it already is)
6
4
7
8
7Refers to node 8
Merging Leftist Heaps
4
25
12
15
19
2027
43
6
8
14
Make 6 leftist (it already is)
6
4
7
8
7Refers to node 8
Merging Leftist Heaps
4
25
12
15
19
2027
43
6
8
14
Return node 6
46
8
7Refers to node 8
Merging Leftist Heaps
4
25
12
15
19
2027
43
6
8
14
Make 6 the right child of 4
46
8
7
Merging Leftist Heaps
4
25
12
15
19
2027
43
6
8
14
Make 4 leftist (it already is)
46
8
7
Final Leftist Heap
4
25
12
15
19
2027
43
6
8
14
Return node 4
4
8
7
• Verify that the tree is heap
• Verify that the heap is leftist
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)
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.
Skew HeapsText• Read Weiss, §6.7
Skew Heap• No need for null path length• Definition of skew heap
Building a Skew Heap• Sequence of inserts• Swap children at every merge step
Motivation
• Simplify leftist heap by
– not maintaining null path lengths
– swapping children at every merge step
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:
1. X.value ≤ L.value2. 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:
1. X.value ≥ L.value2. X.value ≥ R.value
Merging Skew Heaps
4
25
12
15
819
2027
43
6
78
14
Consider two skew heaps …
Task: merge them into a single skew heap
Merging Skew Heaps
4
25
12
15
819
2027
43
6
78
14
First, instantiate a Stack
Merging Skew Heaps
4
25
12
15
819
2027
43
6
78
14
Compare root nodesmerge(x,y)
x y
Merging Skew Heaps
4
25
12
15
819
2027
43
6
78
14
Remember smaller value
4
x y
Merging Skew Heaps
4
25
12
15
819
2027
43
6
78
14
Repeat the process with the right child of the smaller value
4
x
y
Merging Skew Heaps
4
25
12
15
819
2027
43
6
78
14
Remember smaller value
6
4
x
y
Merging Skew Heaps
4
25
12
15
819
2027
43
6
78
14
Repeat the process with the right child of the smaller value
6
4
x y
Merging Skew Heaps
4
25
12
15
819
2027
43
6
78
14
Remember smaller value
7
6
4
x y
Merging Skew Heaps
4
25
12
15
819
2027
43
6
78
14
Repeat the process with the right child of the smaller value
7
6
4
x
y
null
Merging Skew Heaps
4
25
12
15
819
2027
43
6
78
14
Because one of the arguments is null, return the other argument
7
6
4
x
8
Merging Skew Heaps
4
25
12
15
19
2027
43
6
8
14
Make 8 the right child of 7
7
6
4
x
8
8
7Refers to node 8
Merging Skew Heaps
4
25
12
15
19
2027
43
6
8
14
Swap children of node 7
7
6
4
8
8
7Refers to node 8
Merging Skew Heaps
4
25
12
15
19
2027
43
6
8
14
Return node 7
6
4
7
8
7Refers to node 8
Merging Skew Heaps
4
25
12
15
19
2027
43
6
8
14
Make 7 the right child of 6 (which it already is)
6
4
7
8
7Refers to node 8
Merging Skew Heaps
4
25
12
15
19
2027
43
6
8
14
Swap children of node 6
6
4
7
8
7Refers to node 8
Merging Skew Heaps
4
19
2027
43
Return node 6
46
Refers to node 8
25
12
15
6
8
148
7
Merging Skew Heaps
Make 6 the right child of 4
46
4
19
2027
43
25
12
15
6
8
148
7
Merging Skew Heaps
Swap children of node 4
46
4
19
2027
43
25
12
15
6
8
148
7
Final Skew Heap
Return node 4
4
• Verify that the tree is heap
• Verify that the heap is skew
4
19
2027
43
25
12
15
6
8
148
7
Analysis
• Height of a skew 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)
Inserts and Deletes
• To insert a node into a skew heap, merge the leftist heap with the node
• After deleting root X from a skew heap, merge its left and right subheaps
• In summary, there is only one operation, a merge.
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