Worksheet:  Analysis of Recursive and Non-recursive Algorithms

Instructions:

• Collaboration is allowed on these problems.
• As with all work for this course, each problem submitted must include a complete list of all sources used (e.g., from the textbook, Web materials, conversations with other students, etc.) References must be specific, indicating publication information (e.g., publisher, date, Web site URL) or individual(s) consulted.
• Grading will be based on the insight and comprehension shown in the student work. Thus, copying an answer from another source may yield few or no points.

Reference

Many Web sites provide summation formulas for sums of the form:

for various constants a (e.g., a = 1, 2, 3, ...).

Once such reference is "Summation formulas" from Tripod

Problems

1. Consider the following code segment:

       int sum = 0;
   
       for (int i=1 ; i <= n * n; i+= n)
         for (int j=1 ; j <= 10 ; j++)
           for (int k=0; k < i ; k++ )
               sum++;
   

   1. Develop an algebraic expression for the function f(n) that describes the time required for processing, given n. (If the time depends upon specific details of input, consider a "worst-case" analysis.)

   2. Determine if the resulting code has O(n), O(n²), O(n³), O(n⁴), and O(n⁵). In each case, give a careful argument justifying your conclusions.

   3. Determine if the resulting code has Θ(n), Θ(n²), Θ(n³), Θ(n⁴), and Θ(n⁵). In each case, give a careful argument justifying your conclusions.

   4. Naturally, the running time of this code segment will depend upon the particular computer and compiler used. On a particular machine and compiler, suppose the running time for this code is 8 seconds when n is 1000. Based on your answers earlier in this problem, estimate the likely time for the running of this code when n is 2000 and when n is 5000. Justify your answer briefly.

   Notes

   • For an initial analysis, use constants a, b, c, ..., to represent the amount of time required for a type of machine code. For example,
     • a might be the time for an assignment (e.g., sum = 0
     • b might be the time for a comparison (e.g., i < j)
     • c might be the time for adding 1 (e.g., i++)
     • etc.

2. Consider the C program simple-loop-analysis.c.

   1. Give a micro-analysis of the iter_compute procedure to describe its run time.

   2. On the basis of your answer to part a, determine if the resulting code has Θ(n), Θ(n²), Θ(n³), Θ(n⁴), and Θ(n⁵). In each case, give a careful argument justifying your conclusions.

   3. Develop a recurrence relation that describes the run time of rec_compute procedure. (For this problem, you need not solve the recurrence relation — but wait until later in the semester.)

   4. On a particular machine and compiler, suppose the running time for this code is 5 seconds when n is 800. Based on your answers earlier in this problem, estimate the likely time for the running of this code when n is 1600 and when n is 8000. Justify your answer briefly.

3. Review the recurrence examples from Appendix B in the text, and study the discussion of the Master Theorem that begins in page 490.

   1. Find two examples that illustrate Case 1 of the Master Theorem .
   2. Find two examples that illustrate Case 2 of the Master Theorem .
   3. Find two examples that illustrate Case 3 of the Master Theorem .

   Notes:

   • For each example, indicate which case applies and solve the recurrence relation.
   • If the examples from the text do not include two examples of each case, develop additional examples of your own. (The examples should not repeat examples done in the book or in class.)

created December, 2021 revised December-January 2021
For more information, please contact Henry M. Walker at walker@cs.grinnell.edu.