Question

Consider a function f(x,y):

f(x,0) = x*x;
f(0,y) = y*(y + 1);
f(x,y) = f(x,y-1) + f(x-1,y);

If one tries to implement that recursively in some language like C++ he will encounter a problem.

Suppose the function is first called with x = x0 and y = y0. Then for any pair (x,y) where 0 <= x < x0 and 0 <= y < y0 the intermediate values will be computed multiple times – recursive calls will form a huge tree in which multiple leaves will in fact contain the same pairs (x,y). For pairs (x,y) where x and y are both close to 0 values will be computed numerous times.

For instance, I tested a similar function implemented in C++ – for x=20 and y=20 its computation takes about 4 hours (yes, four Earth hours!).

Obviously the implementation can be rewritten in such way that repeated computation doesn’t occur – either iteratively or with a cache table.

The question is: will functional languages perform any better and avoid repeated computations when implementing a function like above recursively?

Answer 1

The term you’re looking for here is Memoization:

In computing, memoization is an
optimization technique used primarily
to speed up computer programs by
having function calls avoid repeating
the calculation of results for
previously-processed inputs.

No, functional language do not automatically implement memoization. You can implement it in them, but in C++ as well. It is true, however, that when you write purely functional code (i.e. with no side effects), then memoization is easier. Some dynamic languages (Perl, for instance) have auto-memoization modules that can easily memoize any function. There’s a discussion of this subject in the Automatic memoization section of the Wikipedia article.

For example here’s a naive C++ Fibonacci:

long fib_rec(long index)
{
    if (index < 2)
        return index;
    else
        return fib_rec(index - 1) + fib_rec(index - 2);
}

And here’s a memoized version:

long fib_memoized_aux(vector<long>& cache, long index)
{
    if (cache[index] >= 0)
    return cache[index];

    cache[index] = fib_memoized_aux(cache, index - 1) + fib_memoized_aux(cache, index - 2);
    return cache[index];
}


long fib_memoized(long index)
{
    vector<long> cache(index + 1, -1);
    cache[0] = 0;
    cache[1] = 1;

    return fib_memoized_aux(cache, index);
}