Final Review

1. Review for midterm 1: html, pdf,
   
2. Review for midterm 2
3. New Material:
   • Graphs
     • Representation of Graphs:
       • Adjacency Matrix
       • Adjacency Lists
       • Sample Questions
         • Come up with an algorithm to convert given adjacency lists to matrix or vice versa.
     • 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 adjacency matrix is given 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 adjacency matrix is given complexity of DFS_rec(G) will change. Why?
• Sample Questions:
  1. For a given graph give discover/finish time using DFS
  2. For a given graph give visit time using BFS
  3. Runtimes of BFS and DFS when different varients of graph representations(lists or matrix) are used.
• 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.
  • Sample Questions:
    • For a given graph run Prim's algorithm and show all different steps(vertex weights, priority queue, predecessor array)
    • For a given graph run Kruskal's algorithm and show the subsets of different steps.
• 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|)
    • Sample Questions:
      • For a given graph run Dijkstra's/Bellman-Ford algorithm and show all different steps(vertex weights, priority queue, predecessor array)
  • 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|)
    • Sample Questions
      • Analyze the runtime of Floyd-Warshall if adjacency lists are given instead of adjacency matrix.