Category: Design & Analysis of Algorithms

Insertion Sort Algorithm | Example | Time Complexity

Insertion Sort-

 

  • Insertion sort is an in-place sorting algorithm.
  • It uses no auxiliary data structures while sorting.
  • It is inspired from the way in which we sort playing cards.

 

How Insertion Sort Works?

 

Consider the following elements are to be sorted in ascending order-

6, 2, 11, 7, 5

 

Insertion sort works as-

 

Firstly,

  • It selects the second element (2).
  • It checks whether it is smaller than any of the elements before it.
  • Since 2 < 6, so it shifts 6 towards right and places 2 before it.
  • The resulting list is 2, 6, 11, 7, 5.

 

Secondly,

  • It selects the third element (11).
  • It checks whether it is smaller than any of the elements before it.
  • Since 11 > (2, 6), so no shifting takes place.
  • The resulting list remains the same.

 

Thirdly,

  • It selects the fourth element (7).
  • It checks whether it is smaller than any of the elements before it.
  • Since 7 < 11, so it shifts 11 towards right and places 7 before it.
  • The resulting list is 2, 6, 7, 11, 5.

 

Fourthly,

  • It selects the fifth element (5).
  • It checks whether it is smaller than any of the elements before it.
  • Since 5 < (6, 7, 11), so it shifts (6, 7, 11) towards right and places 5 before them.
  • The resulting list is 2, 5, 6, 7, 11.

 

As a result, sorted elements in ascending order are-

2, 5, 6, 7, 11

 

Also Read- Selection Sort

 

Insertion Sort Algorithm-

 

Let A be an array with n elements. The insertion sort algorithm used for sorting is as follows-

 

for (i = 1 ; i < n ; i++)

{

   key = A [ i ];

   j = i - 1;

   while(j > 0 && A [ j ] > key)

   {

       A [ j+1 ] = A [ j ];

       j--;

    }

    A [ j+1 ] = key;

}

 

Here,

  • i = variable to traverse the array A
  • key = variable to store the new number to be inserted into the sorted sub-array
  • j = variable to traverse the sorted sub-array

 

Insertion Sort Example-

 

Consider the following elements are to be sorted in ascending order-

6, 2, 11, 7, 5

 

The above insertion sort algorithm works as illustrated below-

 

Step-01: For i = 1

 

 

Step-02: For i = 2

 

 

Step-03: For i = 3

 

 

251176For j = 2; 11 > 7 so A[3] = 11
2511116For j = 1; 5 < 7 so loop stops and A[2] = 7
257116After inner loop ends

 

Working of inner loop when i = 3

 

Step-04: For i = 4

 

 

Loop gets terminated as ‘i’ becomes 5. The state of array after the loops are finished-

 

 

With each loop cycle,

  • One element is placed at the correct location in the sorted sub-array until array A is completely sorted.

 

Time Complexity Analysis-

 

  • Selection sort algorithm consists of two nested loops.
  • Owing to the two nested loops, it has O(n2) time complexity.

 

Time Complexity
Best Casen
Average Casen2
Worst Casen2

 

Space Complexity Analysis-

 

  • Selection sort is an in-place algorithm.
  • It performs all computation in the original array and no other array is used.
  • Hence, the space complexity works out to be O(1).

 

Important Notes-

 

  • Insertion sort is not a very efficient algorithm when data sets are large.
  • This is indicated by the average and worst case complexities.
  • Insertion sort is adaptive and number of comparisons are less if array is partially sorted.

 

To gain better understanding about Insertion Sort Algorithm,

Watch this Video Lecture

 

Next Article- Quick Sort

 

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Selection Sort Algorithm | Example | Time Complexity

Selection Sort-

 

  • Selection sort is one of the easiest approaches to sorting.
  • It is inspired from the way in which we sort things out in day to day life.
  • It is an in-place sorting algorithm because it uses no auxiliary data structures while sorting.

 

How Selection Sort Works?

 

Consider the following elements are to be sorted in ascending order using selection sort-

6, 2, 11, 7, 5

 

Selection sort works as-

  • It finds the first smallest element (2).
  • It swaps it with the first element of the unordered list.
  • It finds the second smallest element (5).
  • It swaps it with the second element of the unordered list.
  • Similarly, it continues to sort the given elements.

 

As a result, sorted elements in ascending order are-

2, 5, 6, 7, 11

 

Selection Sort Algorithm-

 

Let A be an array with n elements. Then, selection sort algorithm used for sorting is as follows-

 

for (i = 0 ; i < n-1 ; i++)

{

   index = i;

   for(j = i+1 ; j < n ; j++)

   {

      if(A[j] < A[index])

      index = j;

   }

   temp = A[i];

   A[i] = A[index];

   A[index] = temp;

}

 

Here,

  • i = variable to traverse the array A
  • index = variable to store the index of minimum element
  • j = variable to traverse the unsorted sub-array
  • temp = temporary variable used for swapping

 

Selection Sort Example-

 

Consider the following elements are to be sorted in ascending order-

6, 2, 11, 7, 5

 

The above selection sort algorithm works as illustrated below-

 

Step-01: For i = 0

 

 

Step-02: For i = 1

 

 

Step-03: For i = 2

 

 

Step-04: For i = 3

 

 

Step-05: For i = 4

 

Loop gets terminated as ‘i’ becomes 4.

 

The state of array after the loops are finished is as shown-

 

 

With each loop cycle,

  • The minimum element in unsorted sub-array is selected.
  • It is then placed at the correct location in the sorted sub-array until array A is completely sorted.

 

Time Complexity Analysis-

 

  • Selection sort algorithm consists of two nested loops.
  • Owing to the two nested loops, it has O(n2) time complexity.

 

Time Complexity
Best Casen2
Average Casen2
Worst Casen2

 

Space Complexity Analysis-

 

  • Selection sort is an in-place algorithm.
  • It performs all computation in the original array and no other array is used.
  • Hence, the space complexity works out to be O(1).

 

Important Notes-

 

  • Selection sort is not a very efficient algorithm when data sets are large.
  • This is indicated by the average and worst case complexities.
  • Selection sort uses minimum number of swap operations O(n) among all the sorting algorithms.

 

To gain better understanding about Selection Sort Algorithm,

Watch this Video Lecture

 

Next Article- Insertion Sort

 

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Difference Between Prim’s and Kruskal’s Algorithm

Prim’s and Kruskal’s Algorithms-

 

Before you go through this article, make sure that you have gone through the previous articles on Prim’s Algorithm & Kruskal’s Algorithm.

 

We have discussed-

  • Prim’s and Kruskal’s Algorithm are the famous greedy algorithms.
  • They are used for finding the Minimum Spanning Tree (MST) of a given graph.
  • To apply these algorithms, the given graph must be weighted, connected and undirected.

 

Some important concepts based on them are-

 

Concept-01:

 

If all the edge weights are distinct, then both the algorithms are guaranteed to find the same MST.

 

Example-

 

Consider the following example-

 

 

Here, both the algorithms on the above given graph produces the same MST as shown.

 

Concept-02:

 

  • If all the edge weights are not distinct, then both the algorithms may not always produce the same MST.
  • However, cost of both the MSTs would always be same in both the cases.

 

Example-

 

Consider the following example-

 

 

Here, both the algorithms on the above given graph produces different MSTs as shown but the cost is same in both the cases.

 

Concept-03:

 

Kruskal’s Algorithm is preferred when-

  • The graph is sparse.
  • There are less number of edges in the graph like E = O(V)
  • The edges are already sorted or can be sorted in linear time.

 

Prim’s Algorithm is preferred when-

  • The graph is dense.
  • There are large number of edges in the graph like E = O(V2).

 

Concept-04:

 

Difference between Prim’s Algorithm and Kruskal’s Algorithm-

 

Prim’s AlgorithmKruskal’s Algorithm
The tree that we are making or growing always remains connected.The tree that we are making or growing usually remains disconnected.
Prim’s Algorithm grows a solution from a random vertex by adding the next cheapest vertex to the existing tree.Kruskal’s Algorithm grows a solution from the cheapest edge by adding the next cheapest edge to the existing tree / forest.
Prim’s Algorithm is faster for dense graphs.Kruskal’s Algorithm is faster for sparse graphs.

 

To gain better understanding about Difference between Prim’s and Kruskal’s Algorithm,

Watch this Video Lecture

 

Next Article- Linear Search

 

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Kruskal’s Algorithm | Kruskal’s Algorithm Example | Problems

Kruskal’s Algorithm-

 

  • Kruskal’s Algorithm is a famous greedy algorithm.
  • It is used for finding the Minimum Spanning Tree (MST) of a given graph.
  • To apply Kruskal’s algorithm, the given graph must be weighted, connected and undirected.

 

Kruskal’s Algorithm Implementation-

 

The implementation of Kruskal’s Algorithm is explained in the following steps-

 

Step-01:

 

  • Sort all the edges from low weight to high weight.

 

Step-02:

 

  • Take the edge with the lowest weight and use it to connect the vertices of graph.
  • If adding an edge creates a cycle, then reject that edge and go for the next least weight edge.

 

Step-03:

 

  • Keep adding edges until all the vertices are connected and a Minimum Spanning Tree (MST) is obtained.

 

Thumb Rule to Remember

 

The above steps may be reduced to the following thumb rule-

  • Simply draw all the vertices on the paper.
  • Connect these vertices using edges with minimum weights such that no cycle gets formed.

 

Kruskal’s Algorithm Time Complexity-

 

Worst case time complexity of Kruskal’s Algorithm

= O(ElogV) or O(ElogE)

 

Analysis-

 

  • The edges are maintained as min heap.
  • The next edge can be obtained in O(logE) time if graph has E edges.
  • Reconstruction of heap takes O(E) time.
  • So, Kruskal’s Algorithm takes O(ElogE) time.
  • The value of E can be at most O(V2).
  • So, O(logV) and O(logE) are same.

 

Special Case-

 

  • If the edges are already sorted, then there is no need to construct min heap.
  • So, deletion from min heap time is saved.
  • In this case, time complexity of Kruskal’s Algorithm = O(E + V)

 

Also Read- Prim’s Algorithm

 

PRACTICE PROBLEMS BASED ON KRUSKAL’S ALGORITHM-

 

Problem-01:

 

Construct the minimum spanning tree (MST) for the given graph using Kruskal’s Algorithm-

 

 

Solution-

 

To construct MST using Kruskal’s Algorithm,

  • Simply draw all the vertices on the paper.
  • Connect these vertices using edges with minimum weights such that no cycle gets formed.

 

Step-01:

 

 

Step-02:

 

 

Step-03:

 

 

Step-04:

 

 

Step-05:

 

 

Step-06:

 

 

Step-07:

 

 

Since all the vertices have been connected / included in the MST, so we stop.

Weight of the  MST

= Sum of all edge weights

= 10 + 25 + 22 + 12 + 16 + 14

= 99 units

 

To gain better understanding about Kruskal’s Algorithm,

Watch this Video Lecture

 

To practice previous years GATE problems based on Kruskal’s Algorithm,

Watch this Video Lecture

 

Next Article- Prim’s Algorithm Vs Kruskal’s Algorithm

 

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Prim’s Algorithm | Prim’s Algorithm Example | Problems

Prim’s Algorithm-

 

  • Prim’s Algorithm is a famous greedy algorithm.
  • It is used for finding the Minimum Spanning Tree (MST) of a given graph.
  • To apply Prim’s algorithm, the given graph must be weighted, connected and undirected.

 

Prim’s Algorithm Implementation-

 

The implementation of Prim’s Algorithm is explained in the following steps-

 

Step-01:

 

  • Randomly choose any vertex.
  • The vertex connecting to the edge having least weight is usually selected.

 

Step-02:

 

  • Find all the edges that connect the tree to new vertices.
  • Find the least weight edge among those edges and include it in the existing tree.
  • If including that edge creates a cycle, then reject that edge and look for the next least weight edge.

 

Step-03:

 

  • Keep repeating step-02 until all the vertices are included and Minimum Spanning Tree (MST) is obtained.

 

Prim’s Algorithm Time Complexity-

 

Worst case time complexity of Prim’s Algorithm is-

  • O(ElogV) using binary heap
  • O(E + VlogV) using Fibonacci heap

 

Time Complexity Analysis

 

  • If adjacency list is used to represent the graph, then using breadth first search, all the vertices can be traversed in O(V + E) time.
  • We traverse all the vertices of graph using breadth first search and use a min heap for storing the vertices not yet included in the MST.
  • To get the minimum weight edge, we use min heap as a priority queue.
  • Min heap operations like extracting minimum element and decreasing key value takes O(logV) time.

 

So, overall time complexity

= O(E + V) x O(logV)

= O((E + V)logV)

= O(ElogV)

 

This time complexity can be improved and reduced to O(E + VlogV) using Fibonacci heap.

 

PRACTICE PROBLEMS BASED ON PRIM’S ALGORITHM-

 

Problem-01:

 

Construct the minimum spanning tree (MST) for the given graph using Prim’s Algorithm-

 

 

Solution-

 

The above discussed steps are followed to find the minimum cost spanning tree using Prim’s Algorithm-

 

Step-01:

 

 

Step-02:

 

 

Step-03:

 

 

Step-04:

 

 

Step-05:

 

 

Step-06:

 

 

Since all the vertices have been included in the MST, so we stop.

 

Now, Cost of Minimum Spanning Tree

= Sum of all edge weights

= 10 + 25 + 22 + 12 + 16 + 14

= 99 units

 

Problem-02:

 

Using Prim’s Algorithm, find the cost of minimum spanning tree (MST) of the given graph-

 

 

Solution-

 

The minimum spanning tree obtained by the application of Prim’s Algorithm on the given graph is as shown below-

 

 

Now, Cost of Minimum Spanning Tree

= Sum of all edge weights

= 1 + 4 + 2 + 6 + 3 + 10

= 26 units

 

To gain better understanding about Prim’s Algorithm,

Watch this Video Lecture

 

To practice previous years GATE problems based on Prim’s Algorithm,

Watch this Video Lecture

 

Next Article- Kruskal’s Algorithm

 

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