HTW Berlin Fachbereich 4 Internationaler Bachelor Studiengang Internationale Medieninformatik (Bachelor) Info 2: Informatik II Summer Term 2023

Finger Exercises

Please have these completed before coming to the lab. BTW, I like questions like this on the exam...

1. Programs A and B are analyzed and are found to have worst-case running times no greater than 150 N log N and N² , respectively. Answer the following questions, if possible:

  1. Which program has the better guarantee on the running time for large values of N (N > 10 000)?
  2. Which program has the better guarantee on the running time for small values of N (N < 100)?
  3. Which program will run faster on average for N = 1000?
  4.  Is it possible that program B will run faster than program A on all possible inputs?

2. An algorithm takes 0.5 ms for input size 100. How long will it take for input size 500 if the running time is the following:

  1. linear
  2. O (N log N)
  3. quadratic
  4. cubic

3. An algorithm takes 0.5 ms for input size 100. How large a problem can be solved in 1 min if the running time is the following:

  1. linear
  2. O (N log N)
  3. quadratic
  4. cubic

4. Order the following functions by growth rate, and indicate which, if any, grow at the same rate.:
N, square root of N, N^1.5, N² , N log N, N log log N, N log² N, N log (N²), 2/N, 2N, 2N/2, 37,N³, N² log N

Lab exercises:

These are the required exercises for this week. Work in groups of two, doing pair programming, and turn in the same report for each member of the pair. Remember to put your names on the report. You can continue to use BlueJ, or move to Eclipse, if you wish.

1. For each of the following eight program fragments, do the following:
   1. Give your own Big-Oh analysis of the running time for each fragment.
   2. If you would like to, ask ChatGPT for an analysis of the Big-Oh for each fragment.
   3. Implement the code in a simple main class and run it for these values of n: 10, 100, 1000, 10.000, 100.000, and 1.000.000
   4. Compare your analysis (and ChatGPT's) with the actual number of steps (i.e. the value of sum after the loop) for your report.

// Fragment #1 for ( int i = 0; i < n; i ++)     sum++; // Fragment #2 for ( int i = 0; i < n; i ++)     for ( int j = 0; j < n; j ++)         sum++; // Fragment #3 for ( int i = 0; i < n; i ++)     for ( int j = i; j < n; j ++)         sum++; // Fragment #4 for ( int i = 0; i < n; i ++)     sum ++;     for ( int j = 0; j < n; j ++)         sum ++; // Fragment #5 for ( int i = 0; i < n; i ++)     for ( int j = 0; j < n*n; j ++)     sum++; // Fragment #6 for ( int i = 0; i < n; i ++)     for ( int j = 0; j < i; j ++)         sum++; // Fragment #7 for ( int i = 1; i < n; i ++)     for ( int j = 0; j < n*n; j ++)         if (j % i == 0)            for (int k = 0; k < j; k++)                sum++; // Fragment #8 int i = n; while (i > 1) {     i = i / 2;     sum++; }

2. A prime number has no factors besides 1 and itself. Do the following:
1. Write a simple method
   public static bool isPrime (int n) {...}
   to determine if a positive integer N is prime.
2. In terms of N, what is the worst-case running time of your program?
3. Let B equal the number of bits in the binary representation of N. What is the relationship between B and N?
4. In terms of B, what is the worst-case running time of your program?
5. Compare the running times needed to determine if a 20-bit number and a 40-bit number are prime by running 100 random examples of each through your program. Report on the results in your lab report. You can use a spread sheet to make some diagrams if you wish.

For the bored:

3. Try the sequence that is generated by the following method:
        
    public int sequence (int n)
    {
        int sum = 0;
        int i = n;
        while (i > 1) {
            if (i%2 == 0) {  
               i = i/2;
            } else {  
               i = 3*i + 1;
            }
            sum++;
        }
        return sum;
    } 

   Can you determine the complexity of the number of steps needed for the loop to terminate for various values of n? Can you find out more about this sequence?
   
   
4. The Sieve of Eratosthenes is a method used to compute all primes less than N. Begin by making a table of integers 2 to N.
   Find the smallest integer i that is not crossed out. Print i (it is prime!) and cross out i , 2i , 3i , .... Terminate when i is greater than the square root of N. The running time has been shown to be O (N log log N). Write a program to implement the Sieve and verify that the running time is as claimed. If you are really bored, animate this with a GUI like on the Wikipedia!

Your report is due by 10.00 pm the night before your next lab! As in Informatics 1, We are more interested in process than in product, although we are now getting more interested in products as well. Your report should include any collaborators, summarize what you learned, and note the time you invested in this exercise. How many lines of code did you write for each exercise? Record this in your report.