Graph algorithms are essential tools for solving complex problems in computer science. They help us navigate through interconnected data structures, finding optimal paths and uncovering hidden patterns. From search techniques to shortest path calculations, these algorithms form the backbone of many real-world applications.

This section dives into various graph algorithms, including depth-first and breadth-first search, topological sorting, and shortest path algorithms. We'll explore their implementations, time complexities, and practical applications, giving you a solid foundation in graph theory and its algorithmic solutions.

Graph Traversal Algorithms

Depth-First and Breadth-First Search

• Depth-First Search (DFS) explores graph by going as deep as possible before backtracking
  • Uses stack data structure to keep track of vertices
  • Implemented recursively or iteratively
  • Time complexity: $O(V + E)$ where V is number of vertices and E is number of edges
  • Applications include maze solving, detecting cycles, and topological sorting
• Breadth-First Search (BFS) explores graph by visiting all neighbors before moving to next level
  • Uses queue data structure to keep track of vertices
  • Always implemented iteratively
  • Time complexity: $O(V + E)$
  • Applications include finding shortest path in unweighted graphs and social network analysis
• Comparison between DFS and BFS:
  • DFS uses less memory for deep graphs, BFS uses less for wide graphs
  • DFS may not find shortest path, BFS always finds shortest path in unweighted graphs
  • DFS better for decision trees, BFS better for finding nearby nodes

Advanced Graph Algorithms

• Topological sorting orders vertices in directed acyclic graph (DAG) such that for every edge (u, v), u comes before v
  • Uses modified DFS algorithm
  • Time complexity: $O(V + E)$
  • Applications include scheduling tasks, build systems, and resolving dependencies
• Strongly Connected Components (SCC) are maximal subgraphs where every vertex is reachable from every other vertex
  • Kosaraju's algorithm finds SCCs using two DFS passes
  • Time complexity: $O(V + E)$
  • Applications include analyzing social networks and simplifying complex graphs

Shortest Path Algorithms

Single-Source Shortest Path

• Dijkstra's algorithm finds shortest path from source to all other vertices in weighted graph with non-negative edges
  • Uses greedy approach and priority queue
  • Time complexity: $O((V + E) \log V)$ with binary heap implementation
  • Cannot handle negative edge weights
  • Applications include GPS navigation and network routing protocols
• Bellman-Ford algorithm finds shortest path from source to all other vertices, even with negative edge weights
  • Uses dynamic programming approach
  • Time complexity: $O(VE)$
  • Can detect negative cycles
  • Applications include currency exchange and arbitrage detection in financial markets

All-Pairs Shortest Path

• Floyd-Warshall algorithm finds shortest paths between all pairs of vertices in weighted graph
  • Uses dynamic programming approach
  • Time complexity: $O(V^3)$
  • Can handle negative edge weights but not negative cycles
  • Applications include finding transitive closure of a graph and solving all-pairs shortest path problem in small graphs

Minimum Spanning Tree Algorithms

Fundamentals and Greedy Approaches

• Minimum Spanning Tree (MST) connects all vertices in weighted, undirected graph with minimum total edge weight

  • Properties include:
    • Contains exactly V-1 edges for V vertices
    • No cycles in the tree
    • Unique if all edge weights are distinct
  • Applications include network design, clustering algorithms, and image segmentation

• Prim's algorithm builds MST by iteratively adding cheapest edge that connects tree to a new vertex

  • Uses priority queue to select cheapest edge
  • Time complexity: $O((V + E) \log V)$ with binary heap implementation
  • Efficient for dense graphs
  • Step-by-step process:
    1. Start with arbitrary vertex
    2. Add cheapest edge connecting tree to new vertex
    3. Repeat until all vertices are in tree

• Kruskal's algorithm builds MST by iteratively adding cheapest edge that doesn't create a cycle

  • Uses disjoint-set data structure to detect cycles
  • Time complexity: $O(E \log E)$ or $O(E \log V)$ (since $E \leq V^2$)
  • Efficient for sparse graphs
  • Step-by-step process:
    1. Sort all edges by weight
    2. Add cheapest edge that doesn't create cycle
    3. Repeat until V-1 edges are added