HeapAndHeapsort PDF

7/29/2019 HeapAndHeapsort PDF

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Heaps and Heapsort

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Heaps are used to efficiently implement twooperations:

Insert

Find Max

Definition: A Heap is a complete binary tree witheach node satisfying the heap property:

The key stored in the node is greater than or equalto both keys stored in its children, if any.

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Heap example



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The heap data structure supports the followingoperations

delete_max[H]: returns and deletes max.

insert[H,x]: insert x into H.

delete[H,i]: delete x from H.

makeheap[A]: Transform an array into a heap.

We restrict our discussion to max-heaps.

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Heap Operations



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In a max-heap, if the value at a node becomes lessthan the key of any of its children, the heapproperty can be restored by swapping the current

node and the child with maximum key value,repeating this process if necessary until

the key at the node is greater than or equal to

the keys of both children.

we reach a leaf.

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Sift Down Operation



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Procedure Sift-DownInput: H[1..n], i where 1 in.Output: H[i] is percolated down, if needed, so

that its not smaller than its children.

done := false;if (2*i) > nthen exit; { is a leaf node }repeat i = 2*i; if (i+1) n and key(H[i+1]) > key(H[i]) then i := i+1; if key(H[i/2]) < key(H[i]) then

swap(H[i],H[i/2]); else done := true; end if;until 2*i > n or done;

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Algorithm for sift down



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Example of sift down

In Heap example, change 17 to 3. Then 3 will be sifted down.



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Work from high end of array to low end. Call SiftDown for each item. Dont need to callSiftDown on leaf nodes.

AlgorithmMake-Heap

Input: Array A[1..n] of n elements

Output: Max-heap A[1..n]

fori= n/2downto 1

SiftDown(A,i);

endfor;

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Make-heap operation

(Transform a given array into a heap)

Time complexity to make a heap is O(n).Friday, September 30, 11


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Example of make-heap



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AlgorithmHeapsort

Input: Array A[1..n] of n elements

Output:A[1..n]sorted.

Make-Heap[A]

forj= ndownto 2

Interchange A[1] and A[j]

SiftDown(A[1..j-1],i);

endfor;

Heapsort



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There are (n-1) iterations. Each iteration executes thesift-down operation.

The cost of one sift-down is O(log n).

Hence, the time complexity of heapsort is O(n log n).

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Time complexity of Heapsort



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Example of Heapsort

Sort the array: 3 2 5 4 1


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