Overview

The call stack is what a program uses to keep track of function calls. The call stack is made up of stack frames—one for each function call.

For instance, say we called a function that rolled two dice and printed the sum.

  def roll_die():
    return random.randint(1, 6)

def roll_two_and_sum():
    total = 0
    total += roll_die()
    total += roll_die()
    print total

roll_two_and_sum()

First, our program calls rollTwoAndSum(). It goes on the call stack:

rollTwoAndSum()

That function calls rollDie(), which gets pushed on to the top of the call stack:

rollDie()

rollTwoAndSum()

Inside of rollDie(), we call random.randint(). Here's what our call stack looks like then:

random.randint()

rollDie()

rollTwoAndSum()

When random.randint() finishes, we return back to rollDie() by removing ("popping") random.randint()'s stack frame.

rollDie()

rollTwoAndSum()

Same thing when rollDie() returns:

rollTwoAndSum()

We're not done yet! rollTwoAndSum() calls rollDie() again:

rollDie()

rollTwoAndSum()

Which calls random.randint() again:

random.randint()

rollDie()

rollTwoAndSum()

random.randint() returns, then rollDie() returns, putting us back in rollTwoAndSum():

rollTwoAndSum()

Which calls print():

print()

rollTwoAndSum()

The Space Cost of Stack Frames

Each function call creates its own stack frame, taking up space on the call stack. That's important because it can impact the space complexity of an algorithm. Especially when we use recursion.

For example, if we wanted to multiply all the numbers between $1$ and $n$, we could use this recursive approach:

  int product1ToN(int n)
{
    // we assume n >= 1
    return (n > 1) ? (n * product1ToN(n - 1)) : 1;
}

What would the call stack look like when n = 10?

First, product1ToN() gets called with n = 10:

    product1ToN()    n = 10

This calls product1ToN() with n = 9.

    product1ToN()    n = 9

    product1ToN()    n = 10

Which calls product1ToN() with n = 8.

    product1ToN()    n = 8

    product1ToN()    n = 9

    product1ToN()    n = 10

And so on until we get to n = 1.

    product1ToN()    n = 1

    product1ToN()    n = 2

    product1ToN()    n = 3

    product1ToN()    n = 4

    product1ToN()    n = 5

    product1ToN()    n = 6

    product1ToN()    n = 7

    product1ToN()    n = 8

    product1ToN()    n = 9

    product1ToN()    n = 10

Look at the size of all those stack frames! The entire call stack takes up $O(n)$ space. That's right—we have an $O(n)$ space cost even though our function itself doesn't create any data structures!

What if we'd used an iterative approach instead of a recursive one?

  int product1ToN(int n)
{
    // we assume n >= 1
    int num;
    int result = 1;

    for (num = 1; num <= n; num++) {
        result *= num;
    }

    return result;
}

This version takes a constant amount of space. At the beginning of the loop, the call stack looks like this:

    product1ToN()    n = 10, result = 1, num = 1

As we iterate through the loop, the local variables change, but we stay in the same stack frame because we don't call any other functions.

    product1ToN()    n = 10, result = 2, num = 2

    product1ToN()    n = 10, result = 6, num = 3

    product1ToN()    n = 10, result = 24, num = 4

In general, even though the compiler or interpreter will take care of managing the call stack for you, it's important to consider the depth of the call stack when analyzing the space complexity of an algorithm.

Be especially careful with recursive functions! They can end up building huge call stacks.