Recursive Algorithms

Copyright © 1996-97 Kenneth J. Goldman

When we defined the Rational class, we didn't reduce the fractions, so we ended up with numbers like 10/8 and 4/6. To avoid this, we could reduce the fractions in the constructor by dividing both the numerator and the denominator by the GCD of the two.

The mathematical definition is:

The greatest common divisor (GCD) of two integers m and n is the greatest integer that divides both m and n with no remainder.
So, we'd like a procedure

Thus, the problem is:

Given integers m and n such that m >= n > 0, find the GCD of m and n.
To solve this problem, the mathematical definition isn't enough. It tells us what we want, but not how to get it. Before we can write a procedure, we need to know how to compute the value.

We need an algorithm: a method for computing process.
step by step description of the activity the idea of how to perform the activity activity
(The same algorithm can be described in many different ways.)

So far, the procedures we have written contained only simple formulae and occasional conditional statements. We have essentially translated the specifications directly into code. Now we need to come up with an algorithm, a way to compute the results that does not fall out immediately from the statement of the problem.

Before considering possible GCD algorithms, let's design algorithms for some simpler problems.

Example: factorial

Recall that n! = n*(n-1)*(n-2)*...*2*1, and that 0! = 1. In other words,

We want to build a procedure

Let's try to devise an algorithm straight from the mathematical definition.

int factorial(int n) {
   if (n == 0)
      return 1;
      return (n * factorial(n-1));
Will this work? Let's try using the substitution model.
=> 1

3 * factorial(2)
3 * 2 * factorial(1)
3 * 2 * 1 * factorial(0)
3 * 2 * 1 * 1
=> 6
This looks good. We've done a reduction of the factorial problem to a smaller instance of the same problem. This idea of reducing a problem to itself is known as recursion.

Let's pick apart the code:

int factorial(int n) {
   if (n == 0)       // Termination condition (base case)
      return 1;      // Result for base case
      return (n   *    factorial(n-1));
//                ^    --------------
//                |           ^ 
//                |           |___ Recursive call
//                | 
//                |___ Combining step (used to combine other values with
//                     results of the recursive call)
Hints in writing recursive procedures: Another way to think about the execution of a recursive procedure is with the "actors" model (or dataflow model). In this simple model, we draw a box for each procedure invocation and show the flow of data into and out of each box. To do this, we put the output arrow on the left with the input.

For example,

This corresponds very closely to what actually happens on the execution stack in the computer's memory.

Example: add

Let's do another example. Suppose that addition were not built into Java, and all you had to use was the increment operator, ++, and decrement operator, --. (++j increments j by 1.) Can you build an addition procedure?

Let's assume i >= 0. Then we can rewrite i+j as (i-1)+(j+1). This way, we bring i closer and closer to 0 until it reaches 0. Then we can return j.

Algorithm idea: At each step, subtract one from i and add one to j until i is 0. Then return j. More formally,

int add(int i, int j) {
   if (i == 0)
      return j;
      return add(--i, ++j);
Using the substitution model,
add(3, 7)
add(2, 8)
add(1, 9)
add(0, 10)
=> 10
Notice that there is no combining. The result of the recursive call is the final result. This is known as tail recursion.

Example: Primality Tester

Recall: an integer n is prime iff n >= 2 and n's only factors are 1 and itself. We want:

Algorithm idea: BRUTE FORCE -- test every integer from 2 up to n-1 to see if any divides n with no remainder.

In order to try values in the range 2 to n-1, we'll need a procedure that tests for factors of n in a range, say k to n-1.

boolean factorInRange(int k, int n) {
   if (k >= n)                         // is the range empty?
      return false;                    // the range is empty so there are
                                       // no factors in the range
   else if ((n % k) == 0)              // is n divisible by k?
      return true;                     // yes, we found a factor, namely k
   else                                // k is not a factor
      return factorInRange(k+1, n);    // so let's see if there's a factor
                                       // among the values in the rest
                                       // of the range
Given this, we can write:
boolean isPrime(int n) {
   return ((n > 1) && !factorInRange(2, n));

(16 > 1) && !factorInRange(2, 16)
true && !true
=> false

true && !factorInRange(2, 5)
         factorInRange(3, 5)
         factorInRange(4, 5)
         factorInRange(5, 5)
true && !false
=> true

true && !factorInRange(2, 2)
true && !false
=> true

true && !factorInRange(2,35)
true && !true

false  (doesn't even call factorInRange)

Example: Fibonacci Numbers

Here is another example, this time of purely academic interest. The fibonacci series is defined as follows:

The series is 0 1 1 2 3 5 8 13 21 ...

Let's write a recursive procedure for Fibonacci numbers directly from the definition:

int fib(int n) {
   if (n <= 1)
      return n;
      return (fib(n-1) + fib(n-2));
This combines results from 2 different recursive calls. This is sometimes known as "deep" recursion, or in other cases "divide and conquer."

Let's consider all the recursive calls for fib(5).

This returns the correct answer, but it takes a long time, since there are many calls. This is because there's a lot of repeated work. For example, the call to fib(4) repeats the calculation of fib(3) (see the circled regions of the tree). In general, when n increases by 1, we roughly double the work; that makes about 2n calls!

Let's try to think of another algorithm that is less wasteful. When we compute the series on paper, what do we do?

0 1 1 2 3 5 8 13 21 34 ...
We look at the previous two numbers and add them to get the next number. We don't recompute the previous two, we just write them down (or remember them).

Let's use this algorithm. We'll start the implementation by writing a helper procedure whose parameters are n, a number k, and the values of fib(k) and fib(k-1). Think of k as the place we've come so far in writing out the series. When k reaches n, we're done.

int helpFib(int n, int k, int fibk, int fibk1) {
   if (n == k)
      return fibk;
      return helpFib(n, k+1, fibk+fibk1, fibk);
Now we can write (assuming n>=1):
int fib(int n) {
   return helpFib(n, 1, 1, 0);

helpFib(6, 1, 1, 0)
helpFib(6, 2, 1, 1)
helpFib(6, 3, 2, 1)
helpFib(6, 4, 3, 2)
helpFib(6, 5, 5, 3)
helpFib(6, 6, 8, 5)
=> 8
This is much better -- this tail recursive procedure needs n iterations to compute fib(n). That is much better than 2n.

The lesson here is that being clever about the algorithm can yield significant savings.

Example: GCD

Now let's return to the problem of computing GCD's.

(Here, we assume m >= n > 0.) Recall: The greatest common divisor (GCD) of m and n is the largest integer that divides both m and n with no remainder.

GCD Algorithm 1: Brute Force

The idea is to try all integers from n down until finding one that divides m and n evenly.

First, define tryDivisor that takes in m, n, and a guess. If the guess works, then it returns the guess. Otherwise, it tries a smaller guess.

int tryDivisor(int m, int n, int g) {
   if (((m % g) == 0) && ((n % g) == 0))
      return g;
      return tryDivisor(m, n, g - 1);
Now we can reduce GCD to tryDivisor:
int gcd(int m, int n) {
   return tryDivisor(m, n, n);   // use n as our first guess

gcd(6, 4)
tryDivisor(6, 4, 4)
tryDivisor(6, 4, 3)
tryDivisor(6, 4, 2)
=> 2
This works, but for large numbers, this could take a while. Let's consider another algorithm.

GCD Algorithm 2: Euclid's Algorithm

(This algorithm dates from c. 200 B.C.!)

The basis of the algorithm is the following fact:

Why is this true? We can rewrite m as follows:

Now any divisor d common to m and n must divide the first term with no remainder, since it is the product of n and an integer. Therefore, d must also divide the second term since d divides m and m is the sum of the two terms.

Since any divisor common to m and n must divide the remainder of m/n, we know that, in particular, the gcd does, since it is a common divisor. It just happens to be the greatest such divisor.

So by taking the GCD(n, remainder of m/n), we can "close in quickly" on the GCD of m and n. (This is extremely clever -- you wouldn't be expected to come up with something like this in an algorithm question on a CS101 exam.)

Now we can write:

int gcd(int m, int n) {
   if ((m % n) == 0)
      return n;
      return gcd(n, m % n);

gcd(468, 24)
gcd(24, 12)
=> 12

gcd(135, 19)
gcd(19, 2)
gcd(2, 1)
=> 1
Euclid's GCD algorithm is very fast, but divison (taking remainders) is a more time-consuming operation than simple addition and subtraction. Can we devise a GCD algorithm that doesn't use division?

GCD Algorithm 3: Dijkstra's Algorithm

(Dutch mathematician / computer scientist)

The idea: If m>n, GCD(m,n) is the same as GCD(m-n,n).

Why? If m/d and n/d both leave no remainder, then (m-n)/d leaves no remainder. (Again, this is clever. Most graduate students probably couldn't come up with this if they haven't already seen it.)

This leads to the following algorithm:

int gcd(int m, int n) {
   if(m == n)
      return m;
   else if (m > n)
      return gcd(m-n, n);
      return gcd(m, n-m);
An example using the substitution model:
gcd(468, 24)
gcd(444, 24)
gcd(420, 24)
gcd(36, 24)
gcd(12, 24)     (Now n is bigger)
gcd(12, 12)     (Same)
=> 12
This does accomplish the calculation with no division.

Kenneth J. Goldman
Last modified: Thu Feb 13 23:35:13 CST