Graph theory introduction: Part 2
Tags: graphtheory, apacheage
Published: 2023-05-27 11:17:41

## Introduction
In part one of our blog series on graph theory, we explored the basic concepts of graphs, their applications, and different types of graphs. We also mentioned some popular graph-related algorithms that play a crucial role in solving various problems efficiently. In this second part, we will delve deeper into these algorithms and understand their working principles. Let's explore the common graph algorithms that every aspiring computer scientist or mathematician should be familiar with.

## Common graph algorithms

- Breadth First Search: -
Breadth-First Search is a fundamental graph traversal algorithm used to explore or search through the vertices of a graph in a breadth-ward motion. It starts at a given source vertex and visits all the vertices in the graph that are reachable from the source in a breadth-first manner. BFS employs a queue data structure to keep track of the vertices to be explored, ensuring that all vertices at a particular depth level are visited before moving to the next level. This algorithm is often used to find the shortest path between two vertices and to determine the connectivity of a graph.

- Depth First Search: -
Depth-First Search is another graph traversal algorithm that explores or searches through the vertices of a graph in a depth-ward motion. Starting from a given source vertex, DFS visits the adjacent unvisited vertices recursively until it reaches a vertex with no unvisited neighbors, and then backtracks. This algorithm employs a stack data structure or recursion to maintain the order of visiting vertices. DFS is particularly useful for applications such as topological sorting, detecting cycles in a graph, and solving puzzles with a graph representation.

- Dijkstra's Algorithm
Dijkstra's algorithm is a widely used algorithm for finding the shortest path between two vertices in a weighted graph with non-negative edge weights. It starts at a source vertex and iteratively selects the vertex with the minimum distance from the source. Dijkstra's algorithm maintains a priority queue or a min-heap to efficiently determine the next vertex to explore. It updates the distances of adjacent vertices as it progresses, ensuring that the shortest path to each vertex is always considered. This algorithm is highly efficient and commonly applied in routing protocols, network analysis, and GPS navigation systems.

- Bellman-Ford Algorithm
The Bellman-Ford algorithm is another algorithm for finding the shortest path in a graph that allows negative edge weights. It is more versatile than Dijkstra's algorithm in handling graphs with negative weight cycles. Bellman-Ford algorithm works by relaxing the edges iteratively, gradually refining the estimates of the shortest path until the optimal solution is achieved. This algorithm is commonly used in network routing, distributed systems, and detecting negative cycles in financial applications.

- Kruskal's Algorithm
Kruskal's algorithm is a greedy algorithm used to find the minimum spanning tree (MST) of a connected, undirected graph. An MST is a subset of the graph's edges that connects all vertices while minimizing the total edge weight. Kruskal's algorithm starts by sorting the edges in ascending order based on their weights. It then considers each edge in the sorted order and adds it to the MST if it does not create a cycle. This process continues until all vertices are included in the MST. Kruskal's algorithm is widely used in network design, clustering, and approximate algorithms.

- Prim's Algorithm
Similar to Kruskal's algorithm, Prim's algorithm is used to find the minimum spanning tree of a connected, undirected graph. Prim's algorithm operates in a different manner by starting from an arbitrary vertex and greedily adding the edge with the minimum weight that connects the already included vertices to the remaining vertices. This process continues until all vertices are included in the MST. Prim's algorithm is efficient for dense graphs and has applications in network design, clustering, and data analysis.

In this second part of our blog series, we have explored some common graph algorithms that form the backbone of graph theory applications. These algorithms play a crucial role in solving problems related to connectivity, shortest paths, spanning trees, and optimization. By understanding their working principles, computer scientists and mathematicians can leverage these algorithms to develop efficient solutions for a wide range of real-world problems. Stay tuned for the next part of our blog series, where we will delve into more advanced graph algorithms and their applications.

To use graph as databases you can use PostgreSQL's extension Apache AGE: -
More about apache age here: https://age.apache.org/
Github here: https://github.com/apache/age/