# Difference between Prim’s and Kruskal’s Algorithm

In computer science, Prim’s and Kruskal’s algorithm is a basic building block for solving certain kinds of problems. Kruskal’s algorithm is a more powerful version of Prim’s algorithm that can solve a wider range of problems than the original algorithm can. In this article, We will learn the difference between these two algorithms along with an explanation.

## What is Prim’s Algorithm?

Prim’s Algorithm is a relatively simple algorithm used to solve optimization problems. It is named after Russian mathematician Grigory Prim, who developed the algorithm in the 1930s. The basic objective of the algorithm is to find the minimum value of a function subject to a given set of constraints.

Prim’s approach is frequently used to discover the graph’s smallest spanning tree. A minimal spanning tree is one that connects all of the vertices of a graph with the fewest number of edges feasible. That is a tree that connects all of the vertices in the graph without including any edges that are longer than necessary.

Prim’s algorithm is a greedy algorithm that runs in O(n log n) time where n is the number of vertices in the graph. It is able to find the shortest path between two points in a graph, as long as there are no cycles in that graph. It can find the minimum spanning tree in a graph in O(n2) time. This means that it can run fast enough to solve real-world problems where the graphs are very large.

### Steps of Prim’s Algorithm

Let’s look at the steps involved in finding an MST using Prim’s algorithm.

- Start with the root node of the graph and select a parent of this node that is not in MST. Parent nodes are all the vertices adjacent to the root vertex. This step is important since it will select the non-MST parent that will be used in step 3.
- Select a child of each selected parent node. This step selects all of the remaining child nodes that were not previously selected as parents.
- Add each child that was selected in the previous step to the set of selected children to form a new set. These children are then sorted according to the weight assigned to them by their parent(s), and the new set formed contains the minimum-weight subset of these sorted children.
- Repeat steps 2 and 3 until every child in the graph has been selected as a parent or added to a new set. This will result in the set of minimum-weight connected subsets of the graph, which comprise the MST.

### Features of Prim’s Algorithm

- Prim’s algorithm is recursive in nature, which means it needs another function to calculate the minimum value when it is applied to a different instance of the optimization problem.
- It runs faster than many other algorithms because it can solve complex problems very efficiently.
- It requires less memory than other algorithms because it is initialized to only include part of the search space for a particular problem. In other words, it only uses a subset of the data required to solve the problem.
- It is possible to produce multiple solutions to the same problem using Prim’s algorithm, which increases the computational speed further and improves the quality of results produced by the algorithm.

### Advantages of Prim’s Algorithm

- Prim’s algorithm is a radix tree search algorithm. This means that it uses a tree structure to help it find solutions more quickly. The structure of this tree allows it to look for solutions in a variety of different ways, so it can find the optimal solution quickly without getting bogged down in unnecessary computations.
- It is more efficient because it does not have to waste time on needlessly complex calculations to find an optimal solution. Because it only looks at one variable at a time, it does not waste time calculating the value of that variable multiple times.
- It is also able to solve problems that cannot be solved using other algorithms because it doesn’t use a fixed set of instructions to find the optimal solution every time it is run. Instead, it looks at the problem from different angles and uses different strategies for solving it each time.
- It doesn’t require much memory or computing power to complete because it uses a tree structure and not an algorithm graph as some other algorithms do. This makes it quick and simpler to use than other algorithms.
- It can perform operations in parallel, which means it can run multiple parts of the algorithm at the same time without having to wait for them to finish before moving on to the next part. This results in faster total computation time, which can drastically reduce the time required to accomplish complex problems.

### Disadvantages of Prim’s Algorithm

- The running time is quadratic in the size of the data set.
- The complexity of the algorithm is O(n).
- The algorithm does not take into account the order of the items in the data set.
- It is inefficient in the case of large arrays.

## What is Kruskal’s Algorithm?

Kruskal’s algorithm is a search algorithm that can be used to find the minimum value of a set of elements in a sorted list. It is named after J.S. Kruskal, who developed the algorithm in 1956. It performs better than the standard sort algorithms when there are multiple minimal elements in the list.

This algorithm is used to solve a problem that consists of finding the lowest number in a series of numbers. These could be the coordinates of a vehicle or the number of workers required to build a building. It can also be used to solve a problem that involves determining the number of steps to walk from one point to another.

### Steps of Kruskal’s Algorithm

- Initialize a graph with a set of vertices or nodes and a weight assigned to each edge in the graph. We assign the weights by considering the distance from a given node to other nodes of the graph.
- Sort the graph into two sets of adjacent vertices: `A` consists of all vertices that are not incident to any edge of the graph, and `B` consists of all the vertices that are incident to at least one edge of the graph. We use 0 as the default initial weight of any edge.
- For each node in `A`, choose a vertex from `B` with minimum weight and add a directed edge between the chosen vertex and the node in `A`.
- Repeat step three until there is no vertex left in `B` that can be added to the set of nodes of `A` without adding any edges to the edges already present in `A`.
- Delete any edge with a weight equal to zero. Once the edges are deleted, we are left with the shortest path between each pair of vertices of the graph. The set of nodes incident to the unique shortest path is called a minimum spanning tree for the graph.

### Advantages of Kruskal’s Algorithm

- Kruskal’s algorithm is simple to implement and can be used in a variety of scenarios, including the calculation of minimum spanning trees.
- Kruskal’s algorithm maintains all the vertices in the graph in the optimal solution. This helps to reduce the amount of backtracking required to solve the problem, making it ideal for situations where a large amount of computation is required or the solution must be determined quickly.
- Kruskal’s algorithm is efficient and reliable when implemented correctly, which makes it a popular choice for real-world applications.

### Disadvantages of Kruskal’s Algorithm

- It is not guaranteed to converge to a local minimum, whereas it is with Prim’s algorithm.
- It does not produce the same number of possible paths as the other algorithms.
- It requires a quadratic number of function evaluations.

### Features of Kruskal’s Algorithm

- It finds a minimum in the set of points.
- It is an algorithm that runs in polynomial time, meaning it can be solved quickly by a computer or processor.
- It can be solved by mathematical formulas.
- It allows the algorithm to access all points in the entire data set without wasting time on unnecessary data, allowing for a faster processing time.
- It works by sorting the data points into different piles according to the value of the points.

## Example of Prim’s algorithm

Here is an example of how Prim’s algorithm works.

## Example of Kruskal’s algorithm

Here is an example of how Kruskal’s algorithm works.

## Key Difference Prim’s and Kruskal’s Algorithm

Prim’s Algorithm | Kruskal’s Algorithm |

Prim’s algorithm is fast and easy to implement. | Kruskal’s algorithm is more efficient but takes longer to calculate. |

Requires less storage and less computational power than Kruskal’s algorithm. | More storage space and more computational power than Prim’s algorithm |

Has no limit on the number of vertices in a graph. | May have limits on the number of vertices depending on the algorithm used. |

Prim’s algorithm is used to find the minimum cost path in a graph | Kruskal’s algorithm is used to find the shortest path in a graph. |

## Comparison chart Prim’s and Kruskal’s Algorithm

Basis For Comparison | Prim’s Algorithm | Kruskal’s Algorithm |

Basic | Prim’s algorithm is used to find the minimum cost path in a graph | Kruskal’s algorithm is used to find the shortest path in a graph. |

The route is chosen based on | Vertices | Edges |

Speed | Better for dense graph | Good for sparse graph |

Time Complexity | O(V^2) | O(log V) |

Initiates with | A node | An edge |

## Conclusion

The distinctions between Prim’s and Kruskal’s algorithms were briefly discussed in this article. Both techniques are used to determine the minimal spanning tree in a graph, but they do it in different ways. Prim’s algorithm starts with a single vertex and adds edges until it reaches the desired tree, while Kruskal’s algorithm starts with all edges and removes the ones that are not part of the desired tree. Thanks for reading!!

## Frequently Asked Questions (FAQs)

### What is the difference between Prim’s algorithm and Kruskal’s algorithm?

Prim’s algorithm is the shortest path algorithm that calculates the length of the shortest path between two points in a graph. Kruskal’s algorithm is the longest-increasing subsequence algorithm that finds the longest-increasing subsequence of a set of items in a sequence.

### Which algorithm is better Kruskal’s and Prim’s algorithm?

Prim’s algorithm performs better when the number of variables is smaller, while Kruskal’s algorithm works best for larger datasets.

### What is the difference between Prim’s algorithm and Dijkstra’s algorithm?

Prim’s algorithm is a simpler algorithm than Dijkstra’s, and it can be more efficient in some cases, such as when the search space is very large, or if the data is sparse.

Dijkstra’s algorithm does a better job of calculating the shortest path between two points if the search space is not too large.

### What is the faster program, Prim or Kruskal?

Prim’s algorithm is a faster way to solve the optimization problem than Kruskal’s algorithm. Prim’s algorithm runs in polynomial time, while Kruskal’s algorithm takes linear time to solve the same problem.

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