CS2420: Lecture 20

1
CS2420 Lecture 20

• Vladimir Kulyukin
• Computer Science Department
• Utah State University

2
Outline

• HeapSort (Chapter 7)

3
The Heap Property

• A Heap is a Complete Binary Tree. All levels are
  filled except possibly the last one.
• The Heap Property
• Minimal heap property for every node X with
  parent P, the key in P is smaller than or equal
  to the key in X.
• Maximum heap property for every node X with
  parent P, the key in P is greater than or equal
  to the key in X.

4
The Heap Property
P
P
X
X
The minimum heap property
The maximum heap property
5
Packing Heaps into Arrays
1
16
2
3
14
10
We can convert this heap into a one dimensional
array by doing a level-order traversal on it.
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5
6
7
9
3
7
8
8
9
10
1
4
2
6
Packing Heaps into Arrays
1
16
We can convert this heap into an array by doing
a level-order traversal on it.
2
3
14
10
4
5
6
7
9
3
7
8
8
9
10
Note that the index count of the array starts
with 1.
1
4
2
1
2 3 4 5 6 7 8
9 10
0
7
8
14
10
16
9
3
2
4
1
7
Packing Heaps into Arrays
1
16
2
3
14
10
4
5
6
7
9
3
7
8
8
9
10
1
4
2
1
2 3 4 5 6 7 8
9 10
0
7
8
14
10
16
9
3
2
4
1
8
Interval Nodes and Leaves
1
16
2
3
14
10
4
5
6
7
9
3
7
8
8
9
10
1
4
2
1
2 3 4 5 6 7 8
9 10
0
7
8
14
10
16
9
3
2
4
1
Internal Nodes
Leaves
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Leaves vs. Non-Leaves
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Maintaining the Heap Property

• A heap is a container so items can be inserted
  and deleted at any time.
• The problem is that if we insert/delete an item,
  the heap property may be broken after the
  insertion/deletion.
• We need to make sure that the heap property is
  restored after every insertion and deletion.

11
Maintaining the Property Max Heap Example
Heap Property is broken.
4
7
14
8
2
1
0 1 2 3 4 5
6 7
4
14
7
2
8
1
12
Maintaining the Property Example
4
We swap 4 with the maximum of 14 and 7.
7
14
8
2
1
0 1 2 3 4 5
6 7
4
14
7
2
8
1
13
Maintaining the Property Example
14
We swap 4 with the maximum of 2 and 8.
7
4
8
2
1
0 1 2 3 4 5
6 7
14
4
7
2
8
1
14
Maintaining the Property Example
14
7
The heap property is now restored at every node.
8
4
2
1
0 1 2 3 4 5
6 7
14
8
7
2
4
1
15
RestoreHeap Maintaining The Heap Property
RestoreHeap(A, i) int Max 0 int LCH
LeftChild(i) int RCH RightChild(i) If ( LCH
lt A.size() ALCH gt Ai ) Max LCH
Else Max i If ( RCH lt A.size()
ARCH gt AMAX ) Max RCH If ( Max
! i ) Swap(Ai, AMax)
RestoreHeap(A, Max)
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RestoreHeap Asymptotic Analysis
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RestoreHeap Asymptotic Analysis
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Building a Heap

• Given the dimensions of the array A, discard all
  the leaves. The leaf indices are N/2 1, N.
• Start from the Rightmost Inner (Non-Leaf) Node
  and go up the tree restoring the heap property at
  every inner node until you reach and restore the
  heap property at the root node.

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Building a Heap
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Building a Max Heap Example
1
10
2
3
5
20
4
5
6
7
11
15
21
34
There are 7 nodes in the heap so the inner node
range is 1, 3. Thus, we start with node 3.
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Building a Max Heap Example
1
10
RestoreHeap(3)
5
20
2
3
5
4
6
7
11
15
21
34
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Building a Max Heap Example
1
10
RestoreHeap(2)
5
34
2
3
5
4
6
7
11
15
21
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Building a Max Heap Example
RestoreHeap(1)
1
10
15
34
2
3
5
4
6
7
11
5
21
20
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Building a Max Heap Example
1
34
RestoreHeap(3)
3
15
10
2
5
4
6
7
11
5
21
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Building a Max Heap Example
1
34
3
15
21
2
5
4
6
7
11
5
10
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