Final Review

1. Review for Test 1
2. Review for Test 2
3. New Material:
   • Graphs
     • Representation of Graphs:
       • Adjacency Matrix
       • Adjacency Lists
     • Graph Algorithms
       • Graph Traversal: Assignment of timestamp depends on order of vertices in adjacency lists or matrix, for example: if vertices are alphabetically ordered then if a has b,c and d in its list then b will be visited/discovered first from a, not c or d (this concept also applies for adjacency matrix).
         • Breadth First Search:
           • Runtime: O(|V|+|E|) [if adjacency lists are given]
           • If the graph is given in an adjacency matrix, the time complexity of BFS_iter(G) will change. Why?
• Depth First Search:
  • Runtime: O(|V|+|E|) [if adjacency lists are given]
  • DFS edge classification
  • Directed Acyclic Gragh(DAG) and Topological Sort
  • If the graph is given in an adjacency matrix, the time complexity of DFS_rec(G) will change. Why?

• Minimum Spanning Tree(MST)
  • Prim
    • Runtime: Θ(|V|).T EXTRACT-MIN + Θ(|E|).TDECREASE-KEY
    • Runtimes using different variants of graph representations and priority queue(linked list, min-heap or array)
  • Kruskal
    • Runtime: O(|E|log|E|) which comes from sorting all the edges.
• Shortest Path Algorithms
  • Single Source Shortest Paths
    • Dijkstra: It works only for non-negative edge weights.
      • Runtime: Θ(|V|).T EXTRACT-MIN + Θ(|E|).TDECREASE-KEY
      • Runtimes using different variants of graph representations and priority queue(linked list, min-heap or array)
    • Bellman-Ford: It works for any edge weights and can detect negative weight cycle.
      • Runtime: O(|V||E|)
      • For a DAG, run Topological Sort and then one round of Bellman-Ford. runtime O(|V|+|E|)
  • All Pairs Shortest Paths
    • Floyd-Warshall:
      • Runtime: O(|V|^3) [when adjacency matrix is given]
    • Dijkstra: Run single source Dijkstra once for each vertex as source.
    • Bellman-Ford: Run single source Bellman-Ford once for each vertex as source.
      • Runtime: O(|V|^2|E|)
• P and NP
  • Definition of P, NP, NP-hard, NP-Complete
  • Reductions, NP-complete problems (SAT, Clique, TSP, HC, Vertex Cover, Independent Set)