Lecture Notes on 29 Apr 2022

* Longest Increasing Subsequence:
  Given a sequence of numbers find the length of the longest
  increasing subsequence. The subsequence contains elements
  that are not necessarily contiguous. Here is a sequence of
  numbers:
  3, 1, 4, 5, 9, 2, 6, 8
  What is the length of the longest increasing subsequence?
  The length of the longest increasing subsequence is 5. There
  are two increasing subsequences of length 5.
  3, 4, 5, 6, 8
  1, 4, 5, 6, 8

* Here are the worksheets for the Longest Increasing Subsequence:
https://www.cs.utexas.edu/users/mitra/csSpring2022/cs313/notes/Longest_Increasing_Subsequence.pdf

https://www.cs.utexas.edu/users/mitra/csSpring2022/cs313/notes/Longest_Increasing_Subsequence.docx

* Coin Change problem: 
  You have a large number of coins of the following denominations 
  [1, 7, 13, 23]. What is the minimum number of coins that you can 
  use to hand out change for 37?

  Number of coins: 
  C[37] = min {(C[37 - 1], C[37 - 7], C[37 -13], C[37 -23]} + 1
        = min {C[36], C[30], C[24], C[14]} + 1

* Here are the worksheets for the Coin Change Problem:
https://www.cs.utexas.edu/users/mitra/csSpring2022/cs313/notes/Coin_Change_Problem.pdf

https://www.cs.utexas.edu/users/mitra/csSpring2022/cs313/notes/Coin_Change_Problem.docx

* Maximum Weighted Independent Set: 
  You have a set of bottle with different volumes filled with your 
  favorite beverage. You may drink as much as you want with the 
  constraint that you cannot drink from two adjacent bottles. How 
  can you maximize your intake?

  The volumes are {6, 2, 9, 5, 1, 4, 7, 8, 3}

  Let us say you have n bottles. Then you chose the last bottle or you do
  not chose the last bottle. If you do not chose the last bottle then you
  want to find the best solution from among the first n - 1 bottles. 

  If V[i] is the volume of the ith bottle and S[i] is the largest sum so
  far, then the solution is of the form:
  
  S[i] = max {(V[i] + S[i - 2]), S[i - 1]}

* Here are the worksheets for the Maximum Weighted Independent Set:
https://www.cs.utexas.edu/users/mitra/csSpring2022/cs313/notes/Maximum_Weighted_Independent_Set.pdf

https://www.cs.utexas.edu/users/mitra/csSpring2022/cs313/notes/Maximum_Weighted_Independent_Set.docx