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Applications of Heap Data Structure
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Time Complexity of building a heap

Last Updated : 17 Nov, 2024
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Consider the following algorithm for building a Heap of an input array A. 

A quick look over the above implementation suggests that the running time is [Tex]O(n * lg(n))  [/Tex] since each call to Heapify costs [Tex]O(lg(n))  [/Tex]and Build-Heap makes [Tex]O(n)  [/Tex]such calls. 

This upper bound, though correct, is not asymptotically tight. 

We can derive a tighter bound by observing that the running time of Heapify depends on the height of the tree ‘h’ (which is equal to log n, where n is a number of nodes) and the heights of most sub-trees are small. The height ’h’ increases as we move upwards along the tree. Line-3 of Build-Heap runs a loop from the index of the last internal node (heapsize/2) with height=1, to the index of root(1) with height = lg(n). Hence, Heapify takes a different time for each node, which is:

For finding the Time Complexity of building a heap, we must know the number of nodes having height h. For this we use the fact that, A heap of size n has at most [Tex]\left \lceil \frac{n}{2^{h+1}} \right \rceil  [/Tex]nodes with height h. 

a  to derive the time complexity, we express the total cost of Build-Heap as-

 [Tex]T(n) = \sum_{h = 0}^{lg(n)}\left \lceil \frac{n}{2^{h+1}} \right \rceil * O(h)= O(n * \sum_{h = 0}^{lg(n)}\frac{h}{2^{h}})= O(n * \sum_{h = 0}^{\infty}\frac{h}{2^{h}})[/Tex]

Step 2 uses the properties of the Big-Oh notation to ignore the ceiling function and the constant 2([Tex]2^{h+1} = 2.2^h  [/Tex]). Similarly in Step three, the upper limit of the summation can be increased to infinity since we are using Big-Oh notation. Sum of infinite G.P. (x < 1)

 [Tex]\sum_{n = 0}^{\infty}{x}^{n} = \frac{1}{1-x}[/Tex]

On differentiating both sides and multiplying by x, we get

 [Tex]\sum_{n = 0}^{\infty}n{x}^{n} = \frac{x}{(1-x)^{2}}[/Tex]

Putting the result obtained in (3) back in our derivation (1), we get

 [Tex]= O(n * \frac{\frac{1}{2}}{(1 – \frac{1}{2})^2})= O(n * 2)= O(n)[/Tex]

Hence the Time complexity for Building a Binary Heap is [Tex]O(n)[/Tex]. 


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Applications of Heap Data Structure

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Chirag Manwani
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Article Tags :
  • Analysis of Algorithms
  • DSA
  • Heap
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  • Heap

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