Recursion

A recursive function is a function that is defined in terms of itself.

Consider this recursive factorial function:

def factorial(n):
    """Return the factorial of N, a positive integer."""
    if n == 1:
        return 1
    else:
        return n * factorial(n - 1)

Inside of the body of factorial, we are able to call factorial itself, since the function body is not evaluated until the function is called.

When n is 1, we can directly return the factorial of 1, which is 1. This is known as the base case of this recursive function, which is the case where we can return from the function call directly, without having to first recurse (i.e. call factorial) and then returning. The base case is what prevents factorial from recursing infinitely.

Since we know that our base case factorial(1) will return, we can compute factorial(2) in terms of factorial(1), then factorial(3) in terms of factorial(2), and so on.

There are three main steps in a recursive definition:

1. Base case. You can think of the base case as the case of the simplest function input, or as the stopping condition for the recursion.

   In our example, factorial(1) is our base case for the factorial function.

2. Recursive call on a smaller problem. You can think of this step as calling the function on a smaller problem that our current problem depends on. We assume that a recursive call on this smaller problem will give us the expected result; we call this idea the "recursive leap of faith".

   In our example, factorial(n) depends on the smaller problem of factorial(n-1).

3. Solve the larger problem. In step 2, we found the result of a smaller problem. We want to now use that result to figure out what the result of our current problem should be, which is what we want to return from our current function call.

   In our example, we can compute factorial(n) by multiplying the result of our smaller problem factorial(n-1) (which represents (n-1)!) by n (the reasoning being that n! = n * (n-1)!).

Q1: Warm Up: Recursive Multiplication

These exercises are meant to help refresh your memory of the topics covered in lecture.

Write a function that takes two numbers m and n and returns their product. Assume m and n are positive integers. Use recursion, not mul or *.

Hint: 5 * 3 = 5 + (5 * 2) = 5 + 5 + (5 * 1).

For the base case, what is the simplest possible input for multiply?

For the recursive case, what does calling multiply(m - 1, n) do? What does calling multiply(m, n - 1) do? Do we prefer one over the other?

def multiply(m, n):
    """ Takes two positive integers and returns their product using recursion.
    >>> multiply(5, 3)
    15
    """
    if n == 1:
        return m
    else:
        return m + multiply(m, n - 1)

Q2: Recursion Environment Diagram

Draw an environment diagram for the following code:

Note: If you can't move elements around, make sure you're logged in!

def rec(x, y):
    if y > 0:
        return x * rec(x, y - 1)
    return 1

rec(3, 2)

def rec(x, y):
    if y > 0:
        return x * rec(x, y - 1)
    return 1

rec(3, 2)

Note: This problem is meant to help you understand what really goes on when we make the "recursive leap of faith". However, when approaching or debugging recursive functions, you should avoid visualizing them in this way for large or complicated inputs, since the large number of frames can be quite unwieldy and confusing. Instead, think in terms of the three steps: base case, recursive call, and solving the full problem.

Q3: Find the Bug

Find the bug with this recursive function.

def skip_mul(n):
    """Return the product of n * (n - 2) * (n - 4) * ...

    >>> skip_mul(5) # 5 * 3 * 1
    15
    >>> skip_mul(8) # 8 * 6 * 4 * 2
    384
    """
    if n == 2:
        return 2
    else:
        return n * skip_mul(n - 2)

def skip_mul(n):
    if n == 1:
        return 1
    elif n == 2:
        return 2
    else:
        return n * skip_mul(n - 2)

Q4: Recursive Hailstone

Recall the hailstone function from Homework 2. First, pick a positive integer n as the start. If n is even, divide it by 2. If n is odd, multiply it by 3 and add 1. Repeat this process until n is 1. Write a recursive version of hailstone that prints out the values of the sequence and returns the number of steps.

Hint: When taking the recursive leap of faith, consider both the return value and side effect of this function.

def hailstone(n):
    """Print out the hailstone sequence starting at n, and return the number of elements in the sequence.
    >>> a = hailstone(10)
    10
    5
    16
    8
    4
    2
    1
    >>> a
    7
    >>> b = hailstone(1)
    1
    >>> b
    1
    """
    print(n)
    if n == 1:
        return 1
    elif n % 2 == 0:
        return 1 + hailstone(n // 2)
    else:
        return 1 + hailstone(3 * n + 1)

Q5: Merge Numbers

Write a procedure merge(n1, n2) which takes numbers with digits in decreasing order and returns a single number with all of the digits of the two, in decreasing order. Any number merged with 0 will be that number (treat 0 as having no digits). Use recursion.

Hint: If you can figure out which number has the smallest digit out of both, then we know that the resulting number will have that smallest digit, followed by the merge of the two numbers with the smallest digit removed.

def merge(n1, n2):
    """ Merges two numbers by digit in decreasing order
    >>> merge(31, 42)
    4321
    >>> merge(21, 0)
    21
    >>> merge (21, 31) 
    3211
    """
    if n1 == 0:
        return n2
    elif n2 == 0:
        return n1
    elif n1 % 10 < n2 % 10:
        return merge(n1 // 10, n2) * 10 + n1 % 10
    else:
        return merge(n1, n2 // 10) * 10 + n2 % 10

Q6: Is Prime

Write a function is_prime that takes a single argument n and returns True if n is a prime number and False otherwise. Assume n > 1. We implemented this in Discussion 1 iteratively, now time to do it recursively!

Hint: You will need a helper function! Remember helper functions are nested functions that are useful if you need to keep track of more variables than the given parameters, or if you need to change the value of the input.

def is_prime(n):
    """Returns True if n is a prime number and False otherwise.

    >>> is_prime(2)
    True
    >>> is_prime(16)
    False
    >>> is_prime(521)
    True
    """
    def helper(i):
        if i > (n ** 0.5): # Could replace with i == n
            return True
        elif n % i == 0:
            return False
        return helper(i + 1)
    return helper(2)