R supports recursive functions, but does not optimize tail recursive functions the way some other languages do. Fortunately, with a mechanism known as a trampoline, the R programmer can implement something like the optimization manually and with very little code.
To understand trampolines, one must first understand the mechanics of function calls and recursion.
The Call Stack
As in many other languages, functions in R may call themselves. For example, here is a recursive function that decrements its argument until 0 is reached:
This function has no problem with small values of
n is big enough, an error is raised:
Error: C stack usage 7969236 is too close to the limit
The problem here is that the top-most invocation of the
countdown function, the one we called with
countdown(10000), can’t return until
countdown(9999) returned, which can’t return until
countdown(9998) returned, and so on.
R keeps track of all of these calls in a data structure called the call stack or sometimes just the stack. The stack contains references to all outstanding function calls, recursive or not.
Since the stack is stored in memory, and since computers only have so much memory, the number of nested calls that can occur in a program is limited.
If we want to decrement a number 10000 or more times and print something when we’re done, we have to do it a different way. That way is to use a loop.
countdown function using a loop instead of recursion:
It doesn’t overflow the stack:
countdown contains the same essential pieces as the recursive version: the
n > 0 test, decrementing
n, and returning
"done" at the end. The pieces are just slightly differently arranged so that
countdown doesn’t need to call itself.
If it does what we want, and looks only slightly different than the recursive version… why did we care about recursion again?
Well, maybe we don’t. The choice to use recursion is a stylistic one with arguable benefits. People with a mathematical bent seem to enjoy it. So do I, usually.
Forgoing a debate of the merits of recursive style, let’s assume we want it, and continue on to trampolines: a means to stack-friendly recursive functions.
A trampoline is a function or set of functions that together give us the tools we need to write code in a recursive style, in a way that doesn’t overflow the stack. Here’s an awesome trampoline by Jim Hester:
countdown now looks like this:
It’s very close stylistically to the original recursive version, but has no stack issues:
The trampoline works because it’s thin veneer over a regular loop. Compared to our direct loop version of
countdown, the body of the trampoline’s
while is parameterized by the
f function instead of being hard-coded.
The only new requirement of this re-arrangement is that the body, or the
f function, return
recur instead of calling itself if it wants to keep going.
In languages that perform this optimization automatically, applicable cases are recognized by the compiler and the recursive code is rewritten as a loop. Compilers that do this are said to perform TCO, where TCO stands for tail-call optimization.
Tail call conversion
Trampolines only apply to singly-recursive functions that call themselves in tail position, but many algorithms commonly expressed do not meet these criteria. For example, here’s a recursive
factorial function in R that can’t immediately be trampolined:
Note: It probably wouldn’t make sense to trampoline this function without other modifications first, because
nis coerced to the
numericclass if it wasn’t already. For medium-sized
n, the numeric (double precision)
Infbefore the stack overflows. The gmp library might be a way to produce the necessary large integers. I’ll use
factorialanyway because it’s a compact function and the transformation is clear.
The “tail” of
factorial is the expression
n*factorial(n-1), which places a call to
factorial on the stack before returning. This is exactly the operation that eventually leads to stack overflow and that we need to eliminate.
The way forward is to introduce an accumulator, or a variable to store intermediate state between calls. It’s a step towards an explicit loop, but with R’s named and default argument support, can be done in a decidedly un-loopy way:
Instead of relying on a recursive call for the number to multiply
n by, we store it explicitly in the
prod argument and pass it along. In this way the running product is maintained across invocations and the stack doesn’t need to grow.
Of course, in R, the stack does grow, but now we’ve refactored the function sufficiently enough to apply
trampoline. Let’s do that:
Jim’s trampoline is really efficient, but can’t handle interdependent, mutually recursive functions. These are functions that call one another.
Arrangements like this don’t come up much in my experience, and require a different kind of trampoline, and so I generally prefer solutions like Jim’s. One type of program where mutual recursion seems to come up is in parsers.
But just for completeness, here’s a
trampoline function and two mutually recursive functions from SICP:
Thanks for reading, I hope you enjoyed! In summary:
- A certain kind of recursive function, tail recursive, can be mechanically transformed into a loop and so not consume stack space.
- In languages that don’t perform the transformation automatically, it can be applied manually by the programmer using a trampoline.
- Some recursive functions can be transformed to tail recursive functions with the introduction of accumulator variables, which are facilitated by R’s support for named arguments.
- Mutually-recursive functions can also be trampolined.