WARNING! You should not read this section unless you are already
comfortable with implementing recursive
functions.
One of the biggest hang-ups for students learning recursion is too
much focus on the recursive “process”.
The right way to think about recursion is to just think about the
return value that the recursive call gives back.
Thinking about how that answer is computed just gets in the way of
understanding.
There are good reasons to understand how recursion is implemented,
but helping you to write recursive functions is not one of them.
Perhaps the most common computer application that uses
stacks is not even visible to its users.
This is the implementation of subroutine calls in most programming
language runtime environments.
A subroutine call is normally implemented by pushing
necessary information about the subroutine (including the return
address, parameters, and local variables) onto a stack.
This information is called an activation record.
Further subroutine calls add to the stack.
Each return from a subroutine pops the top activation
record off the stack.
As an example, here is a recursive implementation for the factorial
function.
// Recursively compute and return n!staticlongrfact(intn){// fact(20) is the largest value that fits in a longif((n<0)||(n>20)){return-1;}if(n<=1){return1;}// Base case: return base solutionreturnn*rfact(n-1);// Recursive call for n > 1}
Here is an illustration for how the internal processing works.
\(\beta\) values indicate the address of the program instruction
to return to after completing the current function call.
On each recursive function call to fact, both the return
address and the current value of n must be saved.
Each return from fact pops the top activation record off the
stack.
Todo
type: Slideshow
The figure above and the following text should all be rolled into
a slideshow.
Consider what happens when we call fact with the value 4.
We use \(\beta\) to indicate the address of the program
instruction where the call to fact is made.
Thus, the stack must first store the address \(\beta\), and the
value 4 is passed to fact.
Next, a recursive call to fact is made, this time with value 3.
We will name the program address from which the call is
made \(\beta_1\).
The address \(\beta_1\), along with the current value for
\(n\) (which is 4), is saved on the stack.
Function fact is invoked with input parameter 3.
In similar manner, another recursive call is made with input
parameter 2, requiring that the address from which the call is made
(say \(\beta_2\)) and the current value for \(n\) (which is 3)
are stored on the stack.
A final recursive call with input parameter 1 is made, requiring that
the stack store the calling address (say \(\beta_3\)) and current
value (which is 2).
At this point, we have reached the base case for fact, and so
the recursion begins to unwind.
Each return from fact involves popping the stored value for
\(n\) from the stack, along with the return address from the
function call.
The return value for fact is multiplied by the restored value
for \(n\), and the result is returned.
Because an activation record must be created and placed onto the stack
for each subroutine call, making subroutine calls is a relatively
expensive operation.
While recursion is often used to make implementation easy and clear,
sometimes you might want to eliminate the overhead imposed by the
recursive function calls.
In some cases, such as the factorial function above,
recursion can easily be replaced by iteration.
An iterative form of the factorial function is both
simpler and faster than the version shown in the example.
But it is not always possible to replace recursion with iteration.
Recursion, or some imitation of it, is necessary when implementing
algorithms that require multiple branching such as in the Towers of
Hanoi algorithm, or when
traversing a binary tree.
The Mergesort and
Quicksort sorting algorithms
also require recursion.
Fortunately, it is always possible to imitate recursion with a stack.
Let us now turn to a non-recursive version of the Towers of
Hanoi function, which cannot be done iteratively.
Recursive algorithms lend themselves to efficient implementation with
a stack when the amount of information needed to describe a
sub-problem is small.
For example, Quicksort can effectively
use a stack to replace its recursion since only bounds information for
the subarray to be processed needs to be saved.