Graph algorithms are a fundamental concept in computer science, particularly in the field of data structures. They deal with problems on graphs, which are collections of nodes (also called vertices) and edges connecting these nodes.

Common Graph Algorithms

  • Breadth-First Search (BFS): An algorithm used to traverse or search a graph in a breadthward motion, i.e., exploring all the neighbors of a given node before moving on to the next level.
  • Depth-First Search (DFS): A method for traversing or searching tree or graph data structures. It starts at the root and explores as far as possible along each branch before backtracking.
  • Dijkstra’s Algorithm: Used for finding the shortest path between two nodes in a graph with non-negative edge weights.
  • Floyd-Warshall Algorithm: An algorithm for finding shortest paths between all pairs of vertices in a weighted graph.
  • Prim’s Algorithm: A greedy algorithm that finds a minimum spanning tree for a connected weighted undirected graph.

Example: Dijkstra’s Algorithm

Dijkstra’s algorithm is a popular algorithm used in graph theory for finding the shortest paths between nodes in a graph. It works well for graphs with non-negative edge weights.

Step-by-Step Explanation

  1. Create a set S of all nodes in the graph.
  2. Create a map dist that maps each node to its distance from the source node. Initially, the distance from the source node to itself is 0, and the distance from the source node to all other nodes is infinity.
  3. While S is not empty: a. Choose a node u from S that has the smallest distance in dist. b. Remove u from S. c. For each neighbor v of u: i. Calculate the distance from the source node to v through u. ii. If this distance is less than the current distance in dist for v, update the distance in dist.
  4. The distances in dist now represent the shortest path from the source node to every other node in the graph.

Further Reading

For more detailed information and examples, you can visit the following resources:

Graph Theory Diagram