## Introduction

In the field of computer science, graph algorithms are vital tools that are put to use for a wide variety of purposes, including pathfinding, network analysis, and optimization. In this post, we will discuss the most widely used graph algorithms as well as the applications of these algorithms.

## What exactly is a graph?

A mathematical structure known as a graph is constructed from vertices (also known as nodes) and edges. The vertices stand in for the things that are being investigated, and the edges illustrate the connections that can be made between them. In a directed graph, the edges all point in the same direction, but in an undirected graph, the edges might point in any direction. You can also give each edge in a graph a weight that represents its cost or value. This type of graph is called a weighted graph.

Algorithms for Graphs That Are Used Most Often

## Searching the Whole Domain at Once (BFS)

BFS is an algorithm for traversing graphs that begins at a particular vertex and investigates all of the vertices that are adjacent to it before proceeding to the next level. Finding the path in an unweighted graph that is the shortest distance between two vertices is a popular use of the BFS algorithm. The temporal complexity of BFS can be expressed as O(V+E), where V represents the number of vertices and E represents the number of edges.

Searching From the Surface Down (DFS)

DFS is another algorithm for traversing graphs that tries to traverse as far as it can down each branch before turning around. DFS is put to work in a wide variety of contexts, including the detection of cycles, the topological sorting of data, and the creation of mazes. The temporal complexity of DFS can be expressed as O(V+E), where V represents the number of vertices and E represents the number of edges.

Dijkstra’s Algorithm

Dijkstra’s Algorithm is a shortest path algorithm that locates the shortest path from a source vertex to all other vertices in a weighted graph. It does this by comparing the distances between each pair of vertices. In order for Dijkstra’s Algorithm to function, it must first create a priority queue of vertices and then, at each step, choose the vertex that has the shortest distance. The amount of time required to complete Dijkstra’s Algorithm is expressed as O((V+E)logV), where V and E stand for the number of vertices and edges, respectively.

Algorithm of Bellman and Ford

In contrast to Dijkstra’s Algorithm, the Bellman-Ford Algorithm permits the use of edges having a negative weight when calculating the shortest path through a graph. In addition to this, it locates the shortest path from a source vertex to each of the other vertices in a weighted network, but it is also able to identify and manage negative cycles. The time required to complete the Bellman-Ford Algorithm is expressed as O(VE), where V represents the number of vertices and E represents the number of edges.

An Example of the Floyd-Warshall Algorithm

The Floyd-Warshall Algorithm is a type of method known as an all-pairs shortest path algorithm. Its purpose is to locate the weighted network vertex pair that has the shortest path between them. In order for it to function properly, a distance matrix must be routinely updated with the shortest path that exists between each pair of vertices. In terms of its time requirements, the Floyd-Warshall Algorithm has a complexity of O(V3), where V is the total number of vertices.

## Kruskal’s Algorithm

The Kruskal Algorithm is a minimum spanning tree algorithm that locates the tree with the least amount of weight that yet manages to link all of the vertices in a weighted graph. In order for Kruskal’s Algorithm to function properly, it must first remove cycles before adding edges in ascending order of weight. The amount of time required to complete Kruskal’s Algorithm is denoted by the notation O(ElogE), where E is the total number of edges.

## Prim’s Algorithm

Prim’s Algorithm is another example of a minimum spanning tree algorithm. Its purpose is to locate the tree with the least amount of weight that nevertheless manages to connect all of the vertices in a weighted graph. Prim’s Algorithm accomplishes its task by adding vertices to the existing tree in a manner that prioritizes those that are furtherthest away from it. The amount of time needed to complete Prim’s Algorithm is expressed as O(ElogV), where V represents the number of vertices and E represents the number of edges.

Algorithm with an A*

The A* Algorithm is a search technique that use a heuristic function to determine which path in a weighted graph provides the shortest distance between two vertices. The A* Algorithm accomplishes its tasks by preserving a priority queue of vertices ordered according to their proximity to the algorithm’s beginning point and an estimate of how far they are from the program’s conclusion. The temporal complexity of the A* Algorithm is given as O(bd), where b is the branching factor (the average number of successors for each node), and d is the depth of the solution.

## Topological Sort

Vertices are placed in a linear ordering by the Topological Sort algorithm, which is used for directed acyclic graphs (DAGs). This approach ensures that for any directed edge (u,v), vertex u appears before vertex v. Topological Sort is an algorithm that is used for directed acyclic graphs (DAGs). Task scheduling and the creation of dependencies are two typical applications of the Topological Sort. The amount of time required to complete Topological Sort is denoted by the notation O(V+E), where V represents the number of vertices and E represents the number of edges.

Maximum Flow

The Maximum Flow algorithm is used to determine the maximum flow that can be carried through a network while taking into account the capacities of the network’s edges. Maximum Flow is a method that is frequently utilized in the transportation industry, as well as in the construction of networks and the distribution of resources. Algorithms such as the Ford-Fulkerson Algorithm and the Edmonds-Karp Algorithm are only two examples of the many that may be used to locate the maximum flow. The amount of time required to complete Maximum Flow is denoted by the notation O(VE2), where V represents the number of vertices and E represents the number of edges.

## Conclusion

Graph algorithms are an essential component of computer science, and their use can be seen in a wide variety of contexts. In this piece, we will examine the ten most popular graph algorithms as well as the applications that make use of them. It is vital to have a solid understanding of these methods and the difficulties they present in order to solve graph-related problems effectively.