# Which One Is A Better Approach Between Prim’s Algorithm And Kruskal’s Algorithm?

When it comes to finding the shortest path in a graph, two algorithms stand out: Prim’s algorithm and Kruskal’s algorithm. Both are great choices for finding the shortest path, but which one is better?

To answer this question, we must first understand what each algorithm does.

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## Prim’s Algorithm-

Prim’s algorithm is a greedy algorithm that finds a minimum spanning tree for a weighted undirected graph. The algorithm starts with a random vertex and adds the cheapest edge from that vertex to the tree. It then adds the cheapest edge from the tree to a new vertex and continues until all vertices are in the tree.

The algorithm is named after its discoverer, Czech mathematician Václav Matoušek, who found it in 1930 while working on a problem in electronics engineering.

It is similar to Kruskal’s algorithm, but it uses a different rule for choosing the next edge to add to the tree. Kruskal’s algorithm chooses the edge with the lowest weight, while Prim’s algorithm chooses the edge with the lowest weight that connects a vertex to the tree.

Prim’s algorithm can be used to find the minimum spanning tree for a graph with arbitrary weights. However, it is not guaranteed to find the optimal solution, as there may be other minimum spanning trees with different weights.

**Time complexity**: O(n^{2}) [where n is the number of vertices.]

**Space complexity**: O(n)

## Advantages of Prim’s Algorithm:

This algorithm is a greedy algorithm that finds a minimum spanning tree for a weighted undirected graph. This means it finds a subset of the edges that forms a tree that includes every vertex, where the total weight of all the edges in the tree is minimized.

The algorithm begins with a random vertex, v_{0}, and adds the cheapest edge from v0 to some other vertex, v_{1}. It then finds the cheapest edge from v_{1} to some other vertex, v_{2}, and so on until it reaches v_{n-1}. At this point, it has found a minimum spanning tree.

There are several advantages to using Prim’s algorithm. First, it is relatively simple to implement. Second, it is guaranteed to find a minimum spanning tree. Third, it can be used to find both the minimum spanning tree and the maximum spanning tree. Finally, it can be parallelized, so it can be used to find minimum spanning trees for very large graphs.

## Disadvantages of Prim’s Algorithm:

The main disadvantage of Prim’s algorithm is that it is not guaranteed to find the optimal solution. The algorithm can get trapped in local minima, meaning that it will find a solution that is not necessarily the best possible solution.

Another disadvantage of this algorithm is that it is not very efficient. The algorithm has a time complexity of O(n^{2}), meaning that it is not suitable for large graphs.

Overall, Prim’s algorithm is a decent algorithm for finding the minimum cost spanning tree of a graph. However, it has its disadvantages and should only be used if other more efficient algorithms are not available.

## Kruskal’s Algorithm-

Kruskal’s algorithm is a minimum-spanning-tree algorithm that takes a graph as input and finds the subset of the edges of that graph which form a tree that includes every vertex, where the total weight of all the edges in the tree is minimized. The algorithm is greedy, meaning that it finds a locally optimal solution at each stage while hoping that this will lead to a globally optimal solution.

The algorithm was first described by Joseph Kruskal in 1956 and is one of the most famous algorithms in graph theory. It is used in a wide variety of applications, including finding the shortest path between two points, clustering data points, and constructing the minimum spanning tree of a graph.

The algorithm works as follows:

- Sort the edges of the graph by weight.
- Start with the edge with the smallest weight and add it to the spanning tree.
- Repeat step 2 until all the vertices are in the spanning tree.

The Union-Find algorithm is a data structure that is used to keep track of which subsets of a set of elements are connected. It can be used to efficiently find the connected components of a graph.

The algorithm works as follows:

- Initialize the Union-Find data structure with the set of elements.
- For each edge in the graph, find the subset that contains the two vertices connected by the edge.
- If the two subsets are not already connected, then connect them and update the Union-Find data structure.
- Repeat step 2 until all the edges have been processed.

The time complexity of the Union-Find algorithm is **O(log n)** for each edge in the graph.

The time complexity of Kruskal’s algorithm is **O(m log n)**, where m is the number of edges and n is the number of vertices in the graph. This is because the edges must be sorted before the algorithm can begin, and this sorting step takes O(m log n) time.

Kruskal’s algorithm is a simple and elegant solution to the minimum-spanning-tree problem, but it is not without its drawbacks. One issue is that it is not easy to parallelize, as the edges must be sorted before the algorithm can begin. Another issue is that the algorithm can give different results on different graphs with the same weights, due to the way it breaks ties.

## Advantages of Kruskal’s Algorithm:

The algorithm works by first sorting the edges of the graph by weight from smallest to largest. It then begins building the spanning tree by adding edges one at a time, starting with the smallest weight edge. Each time an edge is added, the algorithm checks to see if doing so would create a cycle. If adding the edge would create a cycle, the edge is not added to the spanning tree. Otherwise, the edge is added and the algorithm continues until all vertices are in the spanning tree.

One of the main advantages of Kruskal’s algorithm is that it is very simple to implement and understand. Additionally, the algorithm can be used on any type of graph, including both **undirected and directed** graphs.

Another advantage of Kruskal’s algorithm is that it is relatively efficient. In particular, the algorithm runs in time O(E log V), where E is the number of edges in the graph and V is the number of vertices. This is a significant improvement over the naive algorithm for finding a minimum spanning tree, which runs in time O(E V).

Finally, Kruskal’s algorithm has the useful property of being able to handle graphs with negative edge weights. While the algorithm will still find a minimum spanning tree in this case, it is important to note that the edge weights must all be negative for the algorithm to work correctly.

## Disadvantages of Kruskal’s Algorithm:

There are several disadvantages of Kruskal’s algorithm. One disadvantage is that it is not guaranteed to find the shortest path between two nodes. Another disadvantage is that it can be slow on some types of graphs. Finally, it can be difficult to implement on some types of graphs.

So which one is better?

Let’s see-

## Difference between Prim’s Algorithm And Kruskal’s Algorithm:

Prim’s and Kruskal’s algorithms are two different algorithms used for finding the minimum spanning tree (MST) of a given graph. Both algorithms have the same goal, but they differ in the way they go about finding the MST.

Prim’s Algorithm | Kruskal’s Algorithm |
---|---|

It starts with a single vertex and expands outward until it reaches all vertices in the graph. It does this by always adding the next cheapest edge that does not create a cycle. | It starts with all edges in the graph and sorts them by weight. It then adds edges to the MST one at a time, making sure that each edge does not create a cycle. |

It traverses one node multiple times to get the minimum distance. | It traverses one node only once. |

The time complexity of this algorithm is O(n^{2}) [where n is the number of vertices.] | Its time complexity is O(m log n). [m is the number of edges] |

This algorithm is typically better for dense graphs. | This algorithm is typically better for sparse graphs. |

Generates the minimum spanning tree starting from the root vertex. | It generates the minimum spanning tree starting from the shortest edge. |

Both algorithms are guaranteed to find the MST, but Kruskal’s algorithm is generally faster. This is because Prim’s algorithm needs to maintain a list of all the vertices it has reached, which can take up a lot of memory. Kruskal’s algorithm only needs to keep track of the edges in the MST, which takes up much less memory.

## Conclusion:

Both Prim’s algorithm and Kruskal’s algorithm are popular methods for finding the minimum spanning tree of a graph. Each algorithm has its benefits and drawbacks, so it’s important to choose the right algorithm for the task at hand. In general, Prim’s algorithm is better for dense graphs and Kruskal’s algorithm is better for sparse graphs.

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