Homework 3, Homework 3, Homework 3, ....

Short Description:

This assignment is worth 15 points.

Goals

• Have written recursive functions.
• Have discovered that recursive functions can be inefficient.

Formal Description

1. [5 points] Starting with the code in fibonacci.C.
   • Add a parameter,long & count to the recursive version. Each time Fib is called, increment count by 1. How many times is the Fib function called for each input value? Add this to the output line. Change the return type to be a long.
   • Write a non-recursive Fibonacci function. Test this function to make sure it accurately computes values for n between 1 and 20. How many times does the non-recursive version loop to compute each value? Add this to the output line. Make sure the return type is a long.

     Example Output

     Recursive version
     Fib(1) = 1 calls 1
     Fib(2) = 1 calls 1
     Fib(3) = 2 calls 3
     ...
     Non-recursive version
     Fib(1) = 1 loops 0
     Fib(2) = 1 loops 0
     Fib(3) = 2 loops 1
     ...
            

   • Using each function, compute the 50^th Fibonacci number, Fib(50), and observe the time to compute. To compute the time, use the system call clock().

     #include <time.h>
     ...
     const int US_TO_SEC = 1000000;
     ...
     clock_t start, stop, time;
     ...
     start = clock();
     Fib(50,count);
     stop = clock();
     time = stop-start
     cout << "That took " << time  << "microseconds or " << time/US_TO_SEC << " seconds." << endl;
     

   • In the README file, describe the difference in ease of implementation and efficiency. Make sure that this is clearly labeled.

2. [5 points] Implement Ackermann's Function.. This function is historically important in that it demonstrated a new class of functions. Don't worry about understanding the function, just identify the base case and the recursive cases and implement the code. Demonstrate that your implementation is correct by printing a table similar to this table. Please limit 0 ≤ m ≤ 3 and 0 ≤ n ≤ 4.

   Example Output

    m\n |     0    1    2    3    4
   -----+--------------------------
      0 |     1    2    3    4    5
      1 |     2    3    4    5    6
      2 |     3    5    7    9   11
      3 |     5   13   29   61  125
   

   setw and setfill might be very helpful producing this table.

3. [5 points] Implement a recursive function that checks to see if a string is a perfect palindrome. Write a main routine to test your function. This routine should open a file, read this file one line at a time, print the line in quotes and "YES!" if the line represents a perfect palindrome and "no." otherwise. One answer per line. The program should prompt for the file name and produce errors consistent with those in assignment two.

   Example Output

   Enter the file name => example.in
   
   "racecar" YES!
   "god saw I was dog" YES!
   "I did, did I?" no.
   "Able was I ere I saw Elba." no.
   "A man, A plan, A canal: Panama!" no.
   "Madam, I'm Adam." no.
   "Hello World!" no.
   "spam" no.
   

   Discussion

   • All code should involve recursive functions.
   • Each sub-portion should be implemented as a separate program.
   • All programs should compile with make.

   Required Files

   All work should be contained in a single directory, with a Makefile, source code and README file.

   Submission

   Please create a tar file your solution directory and submit it to your instructor as an attachment to an email message.