Tag: Kruskal’s Algorithm Questions

Difference Between Prim’s and Kruskal’s Algorithm

Design & Analysis of Algorithms

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 Algorithm Kruskal’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

Design & Analysis of Algorithms

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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