HEAPSORT

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HEAPSORT

• The array A1 .. An is sorted by treating the
  sub-array A1 .. Ap as a heap1. Build A1
  .. Ap into a heap.2. Exchange A1 and Ap,
  so that Ap is now the maximum element.Repeat 1
  and 2 for p n, n-1, ... , 3, 2.

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COMPLETE BINARY TREES

• A complete binary tree is a rooted binary tree in
  which each parent node has exactly two children
  and all the leaves are at the last level.

3
Complete Binary Trees

• Example

Level 0
Level 1
Level 2
Level 3
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ALMOST COMPLETE BINARY TREES

• An almost complete binary tree is a rooted binary
  tree with levels 0 to k, in which all nodes in
  levels 0 to (k-1) are present and only the
  rightmost nodes in level k may be absent.
• A complete binary tree is an almost complete
  binary tree.

5
Almost Complete Binary Trees

• Example

Level 0
Level 1
Level 2
Level 3
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HEAPS

• A heap is an almost complete binary tree in which
  each node x is assigned a value v(x) that
  satisfies the heap propertyv(x) max
  v(lchild(x)), v(rchild(x))

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Heaps

• Example

100
100
98
75
85
15
99
92
83
20
74
50
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ARRAYS AS ALMOST COMPLETE BINARY TREES

• An array A1 .. An can be treated as an almost
  complete binary tree, as followsA1 is the
  rootA2 and A3 are the children of
  A1A4 and A5 are the children of
  A2A6 and A7 are the children of A3etc.

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Arrays as Almost Complete Binary Trees

• Array elements as nodes

A1
A3
A2
A4
A5
A7
A6
A10
A12
A11
A9
A8
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Arrays as Almost Complete Binary Trees

• Assume an array A1 .. An.
• For a parent node at array index klchild(k) 2
  krchild(k) 2 k 1provided that 2k n
  and 2k1 n.
• For a child node at array index kparent(k)
  ?k/2?

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HEAPIFY

• Build the sub-array Ap .. Aq into a heap
• void heapify(int p,int q) int c 2 p if(c
  lt q) if(c1 lt q Ac lt Ac1)
  c if(Ap lt Ac) swap(c,p)
  heapify(c,q)

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Nonrecursive Heapify

• void heapify(int p, int q) int c for(
  c2p, c lt q pc) if(c1 lt q
  Ac lt Ac1) c if(Ap gt Ac)break
  exchange Ac and Ap

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BUILD HEAP

• Initially, we build the entire array A1 .. An
  into a heap.
• void buildheap(void) int j for(jn/2 j gt
  1 --j) heapify(j,n)

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HEAPSORT

• Then we do the exchange to put the max element
  at the end of the array.
• void heapifyexchange(void) int j for(jn j
  gt 2 --j) heapify(1,j) exchange(1,j)

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Final Heapsort Code

• We put everything together to get heapsort
• main() buildheap() heapify_exchange()

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Running Time of Heapsort

• Buildheap() runs in O(n lg(n)) time.heapify_excha
  nge() runs in O(n lg(n)) time.
• Thus heapsort main() runs in O(n lg(n)) time.