Asymptotics II Study Guide
Author: Josh Hug and Kartik Kapur

Live QA #

Linked here.

Check-in Exercise #

Linked here.

Overview #

Runtime Analysis. Understanding the runtime of code involves deep thought. It amounts to asking: “How long does it take to do stuff?”, where stuff can be any conceivable computational process whatsoever. It simply cannot be done mechanically, at least for non-trivial problems. As an example, a pair of nested for loops does NOT mean $\Theta(N^2)$ runtime as we saw in lecture.

Cost Model. As an anchor for your thinking, recall the idea of a “cost model” from last lecture. Pick an operation and count them. You want the one whose count has the highest order of growth as a function of the input size.

Important Sums. This is not a math class so we’ll be a bit sloppy, but the two key sums that you should know are that:

  • $1 + 2 + 3 + … + N \in \Theta(N^2)$
  • $1 + 2 + 4 + 8 + … + N \in \Theta(N)$

Practice. The only way to learn this is through plenty of practice. Make sure to work through the problems in lecture and below when you have some time.

C level #

  1. Prove that $O(N + \frac{N}{2} + \frac{N}{4} +…. 2 + 1)= O(N)$ (hand wavy proof is okay as long as you gain the intuition)

  2. What would the runtime of modified_fib be. Assume that values is an array of size n. If a value in an int array is not initialized to a number, it is automatically set to 0.

     public void modified_fib(int n, int[] values){
       if(n <= 1){
         values[n] = n;
         return n;
       }
       else{
         int val = values[n];
         if(val == 0){
           val = modified_fib(n-1, values) + modified_fib(n-2, values);
           values[n] = val;
         }
         return val;
       }
     }  
    
  3. Prove to yourself that $\Theta(log_2(n)) = \Theta(log_3(n)) $

B level #

  1. Find the runtime of running print_fib with for arbitrary large n.

     public void print_fib(int n){
       for(int i = 0; i < n; i++i){
           System.out.println(fib(i));
       }
     }
    
     public int fib(int n){
       if(n <= 0){
         return 0;
       }
       elif(n == 1){
         return 1;
       }
       else{
         return fib(n-1) + fib(n-2);
       }
     }
    
  2. Do problem 5 again, but change the body of the for loop in print_fib to be

     System.out.println(fib(n));
    
  3. Find the runtime of this function

     public void melo(int N){
       for(int i = 0; i < N*N; i++){
         System.out.println("Gelo is fruit pudding");
       }
       for(int i = 0; i < N*N*N; i++){
         System.out.println("Zo Two the Warriors");
       }
     }
    
  4. Find the runtime of this function

     public void grigobreath(int N){
         if(N==0){
           return;
         }
         for(int i  = 0; i < N; i++){
           System.out.println("Gul-great")
         }
         grigobreath(N * 1/2);
         grigobreath(N * 1/4);
         grigobreath(N * 1/4);
     }
    
  5. Problem 8 from Spring 2018 midterm #2

  6. Problem 4 from Spring 2017 midterm #2

Last built: 2022-10-01 06:07 UTC