Graph Theory

  1. Shortest Path Algorithms
    1. Single-Source Shortest Paths
      1. Problem Definition
        1. Source Vertex
          1. Distance Function
            1. Shortest Path Tree
            2. Unweighted Graphs
              1. BFS for Shortest Paths
                1. Level-by-Level Exploration
                  1. Optimality Proof
                  2. Weighted Graphs with Non-Negative Edges
                    1. Dijkstra's Algorithm
                      1. Greedy Approach
                        1. Priority Queue Implementation
                          1. Relaxation Process
                            1. Correctness Proof
                              1. Complexity Analysis
                                1. Limitations with Negative Weights
                              2. Weighted Graphs with Negative Edges
                                1. Bellman-Ford Algorithm
                                  1. Dynamic Programming Approach
                                    1. Edge Relaxation
                                      1. Negative Cycle Detection
                                        1. Complexity Analysis
                                    2. All-Pairs Shortest Paths
                                      1. Problem Definition
                                        1. Distance Matrix
                                          1. Path Reconstruction
                                          2. Floyd-Warshall Algorithm
                                            1. Dynamic Programming Formulation
                                              1. Intermediate Vertex Approach
                                                1. Implementation Details
                                                  1. Negative Cycle Detection
                                                    1. Time and Space Complexity
                                                    2. Johnson's Algorithm
                                                      1. Reweighting Technique
                                                        1. Combination with Dijkstra's Algorithm
                                                          1. Efficiency for Sparse Graphs
                                                            1. Complexity Analysis

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                                                        8. Network Flow