UMBC CMSC202, Computer Science II, Fall 1998, Sections 0101, 0102, 0103, 0104

12. Recursion, again

Tuesday October 13, 1998

Assigned Reading:  none

Handouts (available on-line):

Programs from this lecture.

Topics Covered:

• The greatest common divisor (gcd) of two numbers m and n is the largest integer that divides both m and n. We show how recursion is used to find a fast algorithm to compute gcd.

  • First, we implement a naive algorithm to compute the gcd of two numbers. This program simply tries every number between 2 and the smaller of m and n. The largest divisor is returned. The sample run shows that this program is quite slow. It took 22 seconds in one case to compute the gcd of two large-ish numbers.

  • Euclid, an ancient Greek mathematician, showed that

    gcd(m,n) = gcd(n, m % n).

    Using Euclid's recursive algorithm, we can produce a faster program to compute the gcd of two numbers. The sample run shows a running time of less than 0.1 seconds for the same input that took 22 seconds previously. Note that the worst case input to Euclid's gcd algorithm is two consecutive Fibonacci numbers.

  • We can also implement Euclid's algorithm using a while loop instead of recursion. (Program and sample run.) With modern optimizing compilers, this program isn't necessarily faster than the recursive version. One disadvantage of not using recursion is that Euclid's original formulation is lost.
   

• Another classic example of problem solving by recursion is the Towers of Hanoi problem. Program and sample run. Again, the emphasis here is how to think recursively. Unravelling the recursive calls for this problem gets pretty confusing.

• Next, we look at an example of mutually recursion where two functions call each other. In our program, the function count_alpha() does not call itself directly, but does call the function count_non_alphas(). Since count_non_alphas() also calls count_alpha(), the execution of count_alpha() does not return before another call to count_alpha() is issued. Hence we must take great care to ensure that the recursion terminates properly. In this program, we rely on the null character at the end of the string to trigger the base case of the recursion. See sample run.

• We went over a few questions on recursion from past exams.