Graph Algorithms: Concepts and Applications

Graph Algorithms
Graphs are a fundamental structure in computer science used to model
relationships between objects. They consist of vertices (nodes) and edges
(connections), and graph algorithms are used to solve various problems such as
traversal, shortest paths, network flow, and more.



Key Concepts in Graphs
1. Directed vs. Undirected Graphs:
o Directed: Edges have a direction (e.g., A→BA \to BA→B).
o Undirected: Edges have no direction (e.g., A−BA - BA−B).
2. Weighted vs. Unweighted Graphs:
o Weighted: Edges have a weight/cost associated with them.
o Unweighted: All edges have the same weight or no weight.
3. Representation of Graphs:
o Adjacency Matrix: A 2D array where matrix[i][j]matrix[i][j]matrix[i][j]
indicates if there is an edge between vertex iii and jjj.
o Adjacency List: A list of lists, where each vertex stores a list of its
neighbors.



Important Graph Algorithms
1. Graph Traversal Algorithms
1. Depth-First Search (DFS):
o Description: Explores as far as possible along each branch before
backtracking.
o Time Complexity: O(V+E)O(V + E)O(V+E) (where VVV is the number
of vertices, EEE is the number of edges).
o Applications:
 Detecting cycles in a graph.
 Topological sorting in Directed Acyclic Graphs (DAGs).

,  Solving maze problems.
2. Breadth-First Search (BFS):
o Description: Explores all neighbors of a node before moving to the
next level.
o Time Complexity: O(V+E)O(V + E)O(V+E).
o Applications:
 Finding the shortest path in unweighted graphs.
 Solving puzzles like finding the shortest path in a maze.




2. Shortest Path Algorithms
1. Dijkstra’s Algorithm:
o Description: Finds the shortest path from a source node to all other
nodes in a weighted graph (non-negative weights).
o Time Complexity: O((V+E)log⁡V)O((V + E) \log V)O((V+E)logV) (using a
priority queue).
o Applications:
 GPS navigation systems.
 Network routing protocols.
2. Bellman-Ford Algorithm:
o Description: Finds the shortest path from a source node to all other
nodes, even with negative edge weights.
o Time Complexity: O(V⋅E)O(V \cdot E)O(V⋅E).
o Applications:
 Detecting negative weight cycles.
 Financial arbitrage problems.
3. Floyd-Warshall Algorithm:
o Description: Computes shortest paths between all pairs of nodes.
o Time Complexity: O(V3)O(V^3)O(V3).
o Applications:
 All-pairs shortest path problems.
 Transitive closure of graphs.
4. A Algorithm*:
o Description: Combines features of Dijkstra's Algorithm and a
heuristic to optimize pathfinding.