When we presented the Y combinator, we said that it was very aesthetic but not so practical.

Today, we are going to show a real life application of the Y combinator: the memoization of a recursive function. # The problem

Did you ever try to memoize a recursive function?

At first glance it seems easy, especially in `clojure` with the memoize function:

``````(defn factorial [n]
(print n)
(if (zero? n)
1
(* n (factorial (dec n)))))

(def factorial-memo (memoize factorial))
``````

And indeed subsequent calls to `factorial-memo` are cached:

``````(def factorial-memo (memoize factorial))

(with-out-str
(factorial-memo 6)
(factorial-memo 6))
``````

The numbers are only printed once.

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But what happens to subsequent calls with smaller numbers? We’d like them to be cached also. But they are not.

Here is the proof:

``````(def factorial-memo (memoize factorial))

(with-out-str
(factorial-memo 6)
(factorial-memo 5))
``````

The reason is that the code of `factorial-memo` uses `factorial` and not `factorial-memo`.

In `clojure`, we could modify the code of `factorial` so that it calls `factorial-memo`, but it is very very ugly: the code of the recursive function has to be aware of its memoizer!!!

``````(defn factorial-ugly [n]
(print n)
(if (zero? n)
1
(* n (factorial-memo-ugly (dec n)))))

(def factorial-memo-ugly (memoize factorial-ugly))

(with-out-str
(factorial-memo-ugly 6)
(factorial-memo-ugly 5))
``````

With the Y combinator we can solve this issue with elegance.

# The Y combinator for recursive memoization

As we explained here, the Y combinator allows us to generate recursive functions without using any names.

As envisioned by Bruce McAdam in his paper Y in Practical Programs and exposed here by Viksit Gaur, we are going to tweak the code of the Y combinator, so that it receives a wrapper function and apply it before executing the original function. Something like this:

``````(def Ywrap
(fn [wrapper-func f]
((fn [x]
(x x))
(fn [x]
(f (wrapper-func (fn [y]
((x x) y))))))))
``````

And here is the code for a memo wrapper generator:

``````(defn memo-wrapper-generator []
(let [hist (atom {})]
(fn [f]
(fn [y]
(if (find @hist y)
(@hist y)
(let [res (f y)]
(swap! hist assoc y res)
res))))))
``````

It is almost the same code as the clojure memoize.

And now, we are going to build a Y combinator for memoization:

``````(def Ymemo
(fn [f]
(Ywrap (memo-wrapper-generator) f)))
``````

And here is how we get a memoized recursive factorial function:

``````(def factorial-gen
(fn [func]
(fn [n]
(println n)
(if (zero? n)
1
(* n (func (dec n)))))))
(def factorial-memo (Ymemo factorial-gen))
``````

And here is the proof that it is memoized properly:

``````(with-out-str
(factorial-memo 6)
(factorial-memo 5))
``````

Isn’t it elegant?

# Fibonacci without exponential complexity

The worst effective implementation (exponential complexity) of the Fibonacci function is the recursive one:

``````(defn fib [n]
(if (< n 2) 1
(+ (fib (- n 1))
(fib (- n 2)))))
``````

There are a couple of effective implementations for the Fibonacci sequence without using recursion.

Using our `Ymemo` combinator, one can write an effective recursive implementation if the Fibonnaci sequence:

``````(defn fib-gen [f]
(fn [n]
(if (< n 2) 1
(+ (f (- n 1))
(f (- n 2))))))

(def fib-recursive-memo (Ymemo fib-gen))
``````

Let’s compare the performances of the naive recursive version and the memoized recursive:

(We have to redefine `fib-recursive-memo`, in order to reset the cache each time we re-run the code snippet.)

``````(def fib-recursive-memo (Ymemo fib-gen))
(def n 35)
(with-out-str
(time (fib n))
(time (fib-recursive-memo n)))
``````

On my computer, the memoized one is around 300 times faster!

Please share your thoughts about this really exciting topic…