Heap And Heap Sort

1
Heap And Heap Sort

• Ethan Coulter
• 4/22/2004
• CS 146 Dr. Lee

2
What is a heap?

• A heap data structure is a data structure
  that stores a collection of objects (with keys),
  and has the following properties
• Complete Binary tree
• Heap Order
• It is implemented as an array where each node in
  the tree corresponds to an element of the array.

3
A Complete Binary Tree

• A Complete Binary Tree is a binary tree that is
  completely filled on all levels with a possible
  exception where the lowest level is filled from
  left to right.

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Heap Order Property

• For every node v, other than the root, the key
  stored in v is greater or equal (smaller or equal
  for max heap) than the key stored in the parent
  of v.
• In this case the maximum value is stored
• in the root

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Max Heap Example
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Properties Of The Heap

• The parent of node v, stored in Ai, is
• stored in Ai/2
• The left child of node v, stored in Ai, is
  stored in A2i
• The right child of node v, stored in Ai, is
  stored in A2i1

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Maintaining The Heap Order

• In order to maintain the heap order
• the value of all children of Ai must
• be less than the value of Ai.
• Ai.child.value lt Ai.value
• To keep this property we use a function called
  Heapify()

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Heapify()

• Input an array A and an index i.
• Assumption
• the binary trees rooted at left and right
• children of Ai are heaps
• Ai may violate the heap order property
• Purpose Push the value at Ai down
• the heap until the tree rooted at Ai is
• a heap.

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The Algorithm

• Heapify(A, i)
• l ? Left_child(i)
• r ? Right_child(i)
• if (l heap-sizeA and A l gt A i then
• Largest ? l
• else
• Largest ? i
• if (r heap-sizeA and Ar gt A largest then
• Largest ? r
• if (largest i) then
• swap(A i, A largest) A i ? A largest
• Heapify(A, largest)

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4 lt 14 so we need to swap
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Now Check 4s New Children and Swap
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The Heap Property Is Now Maintained
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Heap Sort

• 1. Build_heap(A)
• 2. For i ? lengthA down to 2 do
• 3. Swap(A1, Ai)
• 4. heap-sizeA ? heap-sizeA-1
• 5. Heapify(A,1)

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Time Cost Analysis

• Line 1 takes O(n) time
• There are n-1 calls to Heapify each call
• requires O(log n) time.
• Total cost O(n log n).

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Example
19
12
16
1
4
7
Array A
19
12
16
1
4
7
16
Example
Take out biggest
19
12
16
Move the last element to the root
1
4
7
Sorted
Array A
12
16
1
4
7
19
17
Example
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swap
HEAPIFY()
12
16
1
4
Sorted
Array A
12
16
1
4
7
19
18
Example
16
12
7
1
4
Sorted
Array A
12
16
1
4
7
19
19
Example
Take out biggest
16
Move the last element to the root
12
7
1
4
Sorted
Array A
12
1
4
7
19
16
20
Example
4
12
7
1
Sorted
Array A
12
1
4
7
19
16
21
Example
4
swap
HEAPIFY()
12
7
1
Sorted
Array A
12
1
4
7
19
16
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Example
12
7
4
1
Sorted
Array A
12
1
4
7
19
16
23
Example
Take out biggest
12
Move the last element to the root
7
4
1
Sorted
Array A
1
4
7
19
12
16
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Example
1
swap
HEAPIFY()
7
4
Sorted
Array A
1
4
7
19
12
16
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Example
7
4
1
Sorted
Array A
1
4
7
19
12
16
26
Example
Take out biggest
7
Move the last element to the root
4
1
Sorted
Array A
1
4
19
12
16
7
27
Example
1
swap
HEAPIFY()
4
Sorted
Array A
4
1
19
12
16
7
28
Example
Take out biggest
4
Move the last element to the root
1
Sorted
Array A
1
4
19
12
16
7
29
Example
Take out biggest
1
Sorted
Array A
1
4
19
12
16
7
30
Example
Sorted
19
12
16
1
4
7