Lecture Notes on 25 Apr 2022

* 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/csSpring2022/cs313/notes/Dijkstra.pdf

* Let us look at some fun puzzles that you can use graphs to solve:
https://www.cs.utexas.edu/users/mitra/csSpring2022/cs313/notes/Graph_Puzzles.pdf

* Here is one solution:
https://www.cs.utexas.edu/users/mitra/csSpring2022/cs313/notes/Graph_Images.pdf

* Some additional terms:
https://www.cs.utexas.edu/users/mitra/csSpring2022/cs313/notes/Graph_Terms.pdf

Dynamic Programming is a computational problem solving paradigm that
relies on two factors - overlapping sub-problems and optimal substructure.

In dynamic programming you break a problem into sub-problems and use
the optimal solution of the sub-problems to solve the main problem.

In dynamic programming you store the results of the sub-problems so that
you do not repeat the same computations (otherwise known as memoization).

A great example of overlapping sub-problems is the computation of terms
in the Fibonacci series. The general term in the Fibonacci series can
be written as 

fib (n) = fib (n - 1) + fib (n - 2)

fib (n - 1) = fib (n - 2) + fib (n - 3)
fib (n - 2) = fib (n - 3) + fin (n - 4)

A problem must have optimal substructure property if it can be solved
by dynamic programming. Consider the shortest path algorithm that we
did last class. If a vertex q lies on the shortest path from p to r
then the shortest path from p to r is the sum of the shortest path
from p to q and the shortest path from q to r.