Post on 25-Feb-2016
description
Heap And Heap Sort
Ethan Coulter4/22/2004
CS 146 – Dr. Lee
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.
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.
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 storedin the root
Max Heap Example
Properties Of The Heap
The parent of node v, stored in A[i], isstored in A[i/2]• The left child of node v, stored in A[i], is stored in A[2i]
• The right child of node v, stored in A[i], is stored in A[2i+1]
Maintaining The Heap Order
In order to maintain the heap orderthe value of all children of A[i] mustbe less than the value of A[i].
A[i].child.value <= A[i].value
To keep this property we use a function called Heapify()
Heapify()
– Input: an array A and an index i.
– Assumption:• the binary trees rooted at left and rightchildren of A[i] are heaps• A[i] may violate the heap order property
– Purpose: Push the value at A[i] downthe heap until the tree rooted at A[i] isa heap.
The Algorithm
Heapify(A, i)l Left_child(i)r Right_child(i)if (l ≤ heap-size[A] and A [l] > A [i] thenLargest lelseLargest iif (r ≤ heap-size[A] and A[r] > A [largest] thenLargest rif (largest = i) thenswap(A [i], A [largest]) A [i] A [largest]Heapify(A, largest)
4 < 14 so we need to swap
Now Check 4’s New Children and Swap
The Heap Property Is Now Maintained
Heap Sort
1. Build_heap(A)2. For i length[A] down to 2 do3. Swap(A[1], A[i])4. heap-size[A] heap-size[A]-15. Heapify(A,1)
Time Cost Analysis
Line 1 takes O(n) time
There are n-1 calls to Heapify each callrequires O(log n) time.
Total cost O(n log n).
Example
19
12 16
41 7
19 12 16 1 4 7
Array A
Example
19
12 16
41 7
1912 16 1 4 7
Array ASorted:
Take out biggest
Move the last elementto the root
Example
12 16
41
7
1912 16 1 47
Array ASorted:
HEAPIFY()swap
Example
12
16
41
7
191216 1 47
Array ASorted:
Example
12
16
41
7
1912 161 47
Array ASorted:
Take out biggest
Move the last elementto the root
Example
12
4
1
7
1912 1614 7
Array ASorted:
Example
12
4
1
7
1912 1614 7
Array ASorted:
HEAPIFY()
swap
Example
12
4
1
7
1912 1614 7
Array ASorted:
Example
12
4
1
7
1912 1614 7
Array ASorted:
Take out biggest
Move the lastelement to the root
Example
4
1
7
1912 161 4 7
Array ASorted:
HEAPIFY()swap
Example
4 1
7
1912 16147
Array ASorted:
Example
4 1
7
1912 161 4 7
Array ASorted:
Take out biggest
Move the lastelement to the root
Example
4
1
1912 1614 7
Array ASorted:
HEAPIFY()
swap
Example
4
1
1912 161 4 7
Array ASorted:
Take out biggestMove the lastelement to the root
Example
1
1912 161 4 7
Array ASorted:
Take out biggest
Example
1912 161 4 7
Sorted: