Heap Sort sorting algorithm analysis of algorithm.pptx

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• Priority Queues
• Heaps
• Heapsort

Priority Queue
A data structure implementing a set S of elements, each associated with a
key, supporting the following operations:
increase_key(S, x, k) :
insert element x into set S
return element of S with largest key return
element of S with largest key and remove it
from S
increase the value of element x’ s key to new
value k
(assumed to be as large as current value)
insert(S, x) :
max(S) :
extract_max(S) :

Heap
• Implementation of a priority queue
• An array, visualized as a nearly complete binary tree
•Max Heap Property: The key of a node is≥ than the keys of its children
(Min Heap defined analogously)
161410 8 7 9 3 2 4 1
1 2 3 4 5 6 7 8 9 10
10
8 9
2 4
1
16
1
2
14
3
4
5
7
6 7
3
8 9 10

Heap as a Tree
root of tree. :first element in the array, corresponding to i = 1
parent(i) =i/2: returns index of node's parent
left(i)=2i. : returns index of node's left child
right(i)=2i+1: returns index of node's right child
10
8 9
16
1
2
14
161410 8 7 9 3 2 4 1
1 2 3 4 5 6 7 8 9 10
3
4
5
7
6 7
3
8 9 10
2 4 1

Heap Operations
insert, extract_max, heapsort
produce a max-heap from an unordered
array
correct a single violation of the
heap property in a subtree at its root
build_max_heap :
max_heapify :

Max_heapify (Example)
Time=? O(log n)

Max_Heapify Pseudocode
Max_Heapify(A,i):
l = left(i)
r = right(i)
if (l <= heap-size(A) and A[l] > A[i])
then largest = l
else largest = i
if (r <= heap-size(A) and A[r] > A[largest])
then largest = r
if largest != i
then exchange A[i] and A[largest]
Max_Heapify(A, largest)

Build_Max_Heap(A)
• Converts A[1…n] to a max heap
• Build_Max_Heap(A):
• heap-size(A) = length(A)
for i=n/2 downto 1
do Max_Heapify(A, i)
• Why start at n/2?
• Because elements A[n/2 + 1 … n] are all leaves of the tree 2i > n, for i > n/2 +
1
Time= O(n log n) via simple analysis

Heap-Sort
Sorting Strategy:
1. Build Max Heap from unordered array;
2. Find maximum element A[1];
3. Swap elements A[n] and A[1]:
now max element is at the end of the array!
4. Discard node n from heap
(by decrementing heap-size variable)
5.New root may violate max heap property, but its children are max
heaps. Run max_heapify to fix this.
6. Go to Step 2 unless heap is empty.

Heap-Sort Demo
Swap A[10] and
A[1]
Max_heapify(A,1)

Heap-Sort Demo
Swap A[9] and
A[1]

Heap-Sort Demo
Swap A[8] and
A[1]

Heap-Sort
• Running time:
• after n iterations the Heap is empty
• every iteration involves a swap and a max_heapify operation; hence it
takes O(log n) time
Overall O(n log n)

Heap Sort sorting algorithm analysis of algorithm.pptx

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