Assignment 1

• Developed a testbed for examining the empirical efficiency of GCD algorithms. (And others as well)
• Implemented several GCD algorithms.
• Experienced collecting and interpreting data related to the empirical performance of algorithms.

1. Implement a non recursive Euclid's GCD algorithm.
2. Implement the Binary GCD algorithm
   1. gcd(0, v) = v, because everything divides zero, and v is the largest number that divides v. Similarly, gcd(u, 0) = u. gcd(0, 0) is not typically defined, but it is convenient to set gcd(0, 0) = 0.
   2. If u and v are both even, then gcd(u, v) = 2*gcd(u/2, v/2), because 2 is a common divisor. Don't test for even with divide or mod, us a logical and. Don't divide either, shift right one.
   3. If u is even and v is odd, then gcd(u, v) = gcd(u/2, v), because 2 is not a common divisor. Similarly, if u is odd and v is even, then gcd(u, v) = gcd(u, v/2).
   4. If u and v are both odd, and u >= v, then gcd(u, v) = gcd((u - v)/2, v). If both are odd and u < v, then gcd(u, v) = gcd((v - u)/2, u). These are combinations of one step of the simple Euclidean algorithm, which uses subtraction at each step, and an application of step 3 above. The division by 2 results in an integer because the difference of two odd numbers is even.
   5. Repeat steps 2-4 until u = v, or (one more step) until u = 0. In either case, the GCD is 2kv, where k is the number of common factors of 2 found in step 2.
3. Add a second test driver which will execute a given routine a given number of times, generating new random input each time. The driver should return the average run time for the routine.
4. Use the driver described above to test each algorithm at least 10 times for different random values with 6 bits, 14 bits, 22 bits and 30 bits. (Do not test the School routine with this driver, but do test your two new routines)
5. Use the data that you collected, along with your favorite graphing software (ie excel or oocalc) to draw a graph comparing the speeds of the different implementations. You may need to present the data in different graphs to provide meaningful data.
6. Write a summary describing what you have discovered in your empirical investigation of GCD algorithms.