Learn Programming online

Started Two months Online Programming Classes for XI, XII (all boards), B.Sc.(CS/IT), BCA, M.Sc.(CS/IT), MCA.

Programming Languages

1. C

2. C++

3. Python Basics

4. Python for Data Science

5. R programming

Web Designing

1. HTML

2. CSS

3. PHP

Theory Papers

1. DBMS

2. OS

3. Data Structure using C/C++

New Batches starting from 25 May. Limited Seats. 

Comment Below your contact details(name, email-id and mobile number) if your are interested.

Doubts / Queries

Comment below your queries, doubts or you want to understand any concept related with Data Structure again. 

Heap Sort Algorithm

What is heap?

A heap is defined as a complete binary tree with n nodes such that

1. Value of each node is less than or equal to the root. When root of the binary tree is the biggest number in the heap. This type of heap is known as max Heap  or descending Heap.


2. Value of each node is greater than or equal to the root. When root of the binary tree is the smallest number in the heap. This type of heap is known as min Heap  or ascending Heap.

Each node in the heap corresponds to an element of the array, with the root node corresponding to the element with index 0 in the array. Considering a node corresponding to index i, then its left child has index (2*i + 1) and its right child has index (2*i + 2).

Heap can be used to sort the data in ascending and descending order. The method is known as Heap Sort. Heap Sort involves two basic step

1. Build Heap

•      Create max heap to sort the data in ascending order
•      Create min Heap to sort data in descending order

2. Heapify


ALGORITHM 
    Heapify(A, i){
        l <- left(i)
        r <- right(i)
        if l <= heapsize[A] and A[l] > A[i]
            then largest <- l
            else largest <- i
        if r <= heapsize[A] and A[r] > A[largest]
            then largest <- r
        if largest != i
            then swap A[i] <-> A[largest]
                Heapify(A, largest)
    }

    Buildheap(A){
        heapsize[A] <- length[A]
        for i <- |length[A]/2| downto 1
            do Heapify(A, i)
    }

    Heapsort(A){
        Buildheap(A)
        for i <- length[A] downto 2
            do swap A[1] <-> A[i]
            heapsize[A] <- heapsize[A] - 1
            Heapify(A, 1)
    }

Kruskal Algorithm

DEFINITION - Minimum Cost Spanning Tree

Let G=(V,E,W) be any weighted graph. Then a spanning tree whose cost is minimum is called minimum spanning tree. The cost of the spanning tree is defined as the sum of the costs of the edges in that tree.

Kruskal’s Algorithm

Kruskal’s Algorithm uses greedy approach to select an edge with a minimum cost that doesn't result in cycle when added to already selected edges. Time complexity of Kruskal’s algorithm is o(n+eloge).
In this method all the edges are arranged in the increasing order of the cost. Now select one by one the edges with minimum cost but edges are accepted if and only if there is no cycle. All the other edges having high cost or cycle is formed are rejected.

The steps involved are

Step 1: Construct a min heap of edges on the basis of ascending order of cost of edges.

Step 2: Delete edge from heap and add to minimum cost spanning tree T if no cycle

Step 3: Repeat until e=n-1 or heap is empty

EXAMPLE

Prim's Method

DEFINITION - Minimum Cost Spanning Tree

Let G=(V,E,W) be any weighted graph. Then a spanning tree whose cost is minimum is called minimum spanning tree. The cost of the spanning tree is defined as the sum of the costs of the edges in that tree.

Prim's Algorithm 

In Prim’s algorithm, greedy criterion is used to determine the next edge that have minimum cost. It has a time complexity of O(n²). The steps to find minimum cost spanning tree using Prim’s algorithm is

The steps involved are

Step 1: Select the minimum cost edge and include in minimum spanning tree T

Step 2: Select adjacent edges to T which is minimum in cost and add to T

Step 3: Repeat Step 1 and 2 for n-1 edges and no cycle

EXAMPLE