将递归转换为“尾递归”

Question

I have a question about how to convert 'recursion' to 'tail recursion'. this is not a homework, just a question pop up when I tried to polish the recursion theorem from algorithm book. I am familiar with the 2 typical examples of using recursion(factorial and Fibonacci sequence), and also know how to implement them in recursive way and tail-recursive way. My code is as below(I use Perl just to make it simple, but can be easily convert to C/Java/C++)

#this is the recursive function
sub recP    {
    my ($n) = @_;
    if ($n==0 or $n==1 or $n==2)    {
        return 1;
    } else {
        return (recP($n-3)*recP($n-1))+1;
    }

}
for (my $k=1;$k<10;$k++) {
    print "*"x10,"\n";
    print "recP($k)=", recP($k), "\n";
}

When running the code, the output as below:

recP(1)=1 
recP(2)=1 
recP(3)=2 
recP(4)=3 
recP(5)=4 
recP(6)=9 
recP(7)=28 
recP(8)=113 
recP(9)=1018 

the recursive function invoke itself twice with different parameter before return; I tried several ways to convert this to a tail recursive way but all turns out wrong.

Can anybody take a look at the code and show me the correct way to make it tail-recursive? especially I believe there is a routine for the conversion for this tree recursion(invoke recursive function multiple times before return), can any shed some light on this? So I can use the same logic to handle different questions later. thanks in advance.

Answer 1

Although you often see the following as an example of converting factorial to tail-call:

int factorial(int n, int acc=1) {
  if (n <= 1) return acc;
  else        return factorial(n-1, n*acc);
}

it's not quite correct, since it requires multiplication to be both associative and commutative. (Multiplication is associative and commutative, but the above doesn't serve as a model for other operations which don't satisfy those constraints.) A better solution might be:

int factorial(int n, int k=1, int acc=1) {
  if (n == 0) return acc;
  else        return factorial(n-1, k+1, acc*k);
}

This also serves as a model for the fibonacci transform:

int fibonacci(int n, int a=1, int b=0) {
  if (n == 0) return a;
  else        return fibonacci(n-1, a+b, a);
}

Note that these compute the sequence starting at the beginning, as opposed to queueing pending continuations in a call stack. So they are structurally more like the iterative solution than the recursive solution. Unlike the iterative program, though, they never modify any variable; all bindings are constant. This is an interesting and useful property; in these simple cases it doesn't make much difference, but writing code without reassignments makes some compiler optimizations easier.

Anyway, the last one does provide a model for your recursive function; like the fibonacci sequence, we need to keep the relevant past values, but we need three of them instead of two:

int mouse(int n, int a=1, int b=1, int c=1) {
  if (n <=2 ) return a;
  else        return mouse(n-1, a*c+1, a, b);
}

Addenda

In comments, two questions were raised. I'll try to answer them (and one more) here.

First, it should be clear (from a consideration of the underlying machine architecture which has no concept of function calling) that any function call can be rephrased as a goto (possibly with non-bounded intermediate storage); furthermore, any goto can be expressed as a tail-call. So it is possible (but not necessarily pretty) to rewrite any recursion as tail-recursion.

The usual mechanism is "continuation-passing style" which is a fancy way of saying that every time you want to call a function, you instead package the rest of the current function as a new function (the "continuation"), and pass that continuation to the called function. Since every function then receives a continuation as an argument, it has to finish any continuation it creates with a call to the continuation it received.

That's probably enough to make your head spin, so I'll put it another way: instead of pushing arguments and a return location onto the stack and calling a function (which will later return), you push arguments and a continuation location onto the stack and goto a function, which will later goto the continuation location. In short, you simply make the stack an explicit parameter, and then you never need to return. This style of programming is common in event-driven code (see Python Twisted), and it's a real pain to write (and read). So I strongly recommend letting compilers do this transformation for you, if you can find one which will do that.

@xxmouse suggested that I pulled the recursion equation out of a hat, and asked how it was derived. It's simply the original recursion, but reformulated as a transformation of a single tuple:

f n = f n-1 *f n-3 + 1
=>
F n = <F n-1 1 *F n-1 3 +1, F n-1 1 , F n-1 2 >

I don't know if that's any clearer, but it's the best I can do. Look at the fibonacci example for a slightly simpler case.

@j_random_hacker asks what the limits on this transformation are. It works for a recursive sequence where each element can be expressed by some formula of the previous k elements, where k is a constant. There are other ways to produce a tail-call recursion. For example:

// For didactic purposes only
bool is_odd(int n) { return n%2 == 1; }

int power(int x, int n, int acc=1) {
  if (n == 0)         return acc;
  else if (is_odd(n)) return power(x, n-1, acc*x);
  else                return power(x*x, n/2, acc);
}

The above is not the same as the usual non-tail-call recursion, which does a different (but equivalent and equally long) sequence of multiplications.

int squared(n) { return n * n; }

int power(int x, int n) {
  if (n == 0)         return 1;
  else if (is_odd(n)) return x * power(x, n-1));
  else                return squared(power(x, n/2));
}

---

Thanks to Alexey Frunze for the following test: Output ( ideone ):

mouse(0) = 1
mouse(1) = 1
mouse(2) = 1
mouse(3) = 2
mouse(4) = 3
mouse(5) = 4
mouse(6) = 9
mouse(7) = 28
mouse(8) = 113
mouse(9) = 1018

Answer 2

Using google, I found this page that describes Tail Recursion . Basically, you need to split the function into at least two other functions: one that does the work, keeping an "accumulation" of the current value, and another that is a driver for your workhouse function. The factorial example in C is below:

/* not tail recursive */
unsigned int
factorial1(unsigned int n)
{
    if(n == 0)
        return 1;
    return n * factorial1(n-1);
}

/* tail recursive version */
unsigned int 
factorial_driver(unsigned int n, unsigned int acc)
{
    if(n == 0)
        return acc;

    /* notice that the multiplication happens in the function call */
    return factorial_driver(n - 1, n * acc);
}

/* driver function for tail recursive factorial */
unsigned int
factorial2(unsigned int n)
{
    return factorial_driver(n, 1);
}

Answer 3

@Alexey Frunze's answer is okay but not precisely right. It is indeed possible to convert any program into one where all recursion is tail recursion by transforming it into Continuation Passing Style .

I don't have time right now but will try to re-implement your program in CPS if I get some minutes.

Answer 4

You could do something like this:

#include <stdio.h>

void fr(int n, int a[])
{
  int tmp;

  if (n == 0)
    return;

  tmp = a[0] * a[2] + 1;
  a[2] = a[1];
  a[1] = a[0];
  a[0] = tmp;

  fr(n - 1, a);
}

int f(int n)
{
  int a[3] = { 1, 1, 1 };

  if (n <= 2)
    return 1;

  fr(n - 2, a);

  return a[0];
}

int main(void)
{
  int k;
  for (k = 0; k < 10; k++)
    printf("f(%d) = %d\n", k, f(k));
  return 0;
}

Output ( ideone ):

f(0) = 1
f(1) = 1
f(2) = 1
f(3) = 2
f(4) = 3
f(5) = 4
f(6) = 9
f(7) = 28
f(8) = 113
f(9) = 1018

The compiler may transform fr() into something like this:

void fr(int n, int a[])
{
  int tmp;

label:    

  if (n == 0)
    return;

  tmp = a[0] * a[2] + 1;
  a[2] = a[1];
  a[1] = a[0];
  a[0] = tmp;

  n--;

  goto label;
}

And that would be tail-call optimization.

Answer 5

The problem is that the last operation is not one of the recursive calls, but the addition of 1 to the multiplication. Your function in C:

unsigned faa (int n)  // Ordinary recursion
{
    return n<3 ? 1 :
                 faa(n-3)*faa(n-1) + 1;  // Call, call, multiply, add
}

If you change the order in which the values are requested, you can turn one of the calls into a loop:

unsigned foo (int n)  // Similar to tail recursion
{                     // (reverse order)
    int i;
    unsigned f;

    for (i=3, f=1; i<=n; i++)
        f = f*foo(i-3) + 1;

    return f;
}

The key is thinking about the order in which values are actually computed in the original function, instead of the order in which they are requested.

Note that I am assuming that you want to remove one recursive call. If you want to write the recursive call at the end of the function expecting the compiler to optimize it away for you, see the other answers.

Though, "The Right Thing(TM)" to do here is to use dynamic programming to avoid computing the same values many times:

unsigned fuu (int n)  // Dynamic programming
{
    int i;
    unsigned A[4]={1,1,1,1};

    for (i=3; i<=n; i++)
    {
        memmove (A+1, A, 3*sizeof(int));
        A[0] = A[1]*A[3] + 1;
    }

    return A[0];
}

The array A contains a sliding window of the sequence: A[0]==f(i), A[1]==f(i-1), A[2]==f(i-2) and so on.

The memmove might have been written as:

        A[3] = A[2];
        A[2] = A[1];
        A[1] = A[0];