Lecture Notes on 15 Nov 2023

* Directed Acyclic Graphs (DAGs)

https://www.cs.utexas.edu/users/mitra/csFall2023/cs313/notes/DAG.pdf

Topological Sort (Topo Sort)

Works on directed graphs that do not have cycles (DAGs)

0. Determine the in_degree for all vertices. The in_degree is
   the number of edges that are incident on that vertex.

1. Remove the vertices that have an in_degree of 0 to a list and
   remove the out going edges from those vertices. Sort the list
   in a given order. Enqueue the vertices into a Queue and then 
   update the in_degree of all remaining vertices.

2. Repeat step 1 until there are no more vertices in the Graph.

3. Dequeue the vertices and print.

* Minimum Spanning Tree (MST) - Kruskal's Algorithm and Prim's
  Algorithm.

* Kruskal's Algorithm
  - Order the edges by increasing weight.
  - Start with an empty graph having only the vertices but no
    edges. Add an edge to this graph as long as it does not form 
    a cycle.
  - When all the vertices are connected you are done.

* Prim's Algorithm
  - Start with an empty graph having only the vertices.
  - Start with any vertex and add it to the list of visited vertices.
  - Choose the edge with the smallest weight to an unvisited vertex
    as long as it does not form a cycle. Add that vertex to the list
    of visited vertices.
  - Keep adding the edges with the smallest weight from any of the
    vertices in the list of visited vertices as long as those edges
    do not form a cycle.
  - When all the vertices have been visited then you are done.


* Dijkstra's Single Source Shortest Path Algorithm
  - Choose a starting vertex.
  - Create a table with all the other vertices in the graph.
  - From the starting vertex fill the table with the cost of
    visiting all the other vertices.
  - Go to the vertex with the lowest cost and update the cost
    of visiting all the other vertices from the starting vertex.
  - Go to all the other vertices in order of increasing cost and
    update the costs of visiting the other vertices from the 
    starting vertex.
  - When you have visited all the vertices that you can visit 
    from the starting vertex then you are done.

* Worked out example using Dijkstra's Algorithm
https://www.cs.utexas.edu/users/mitra/csFall2023/cs313/notes/Dijkstra.pdf