Fast Factorial

Factorial function is simple enough. But there is still some fun about it. In Stirling's Approximation article we've seen how to calculate good factorial approximation faster than exact value. By the way, there is an algorithm to calculate exact value of factorial faster than "by definition".

Factorials

Most of programming languages tutorials shows the following approach to calculate factorial (transformed to clojure):

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

Simple recursive solution.

Note: Since clojure allows using ! character in function name, it is good name for factorial.

(! 10) works, (! 20) works, (! 30)...

> ArithmeticException integer overflow

Oops.

Default clojure integer type is 64 bit (Long). It does not fit well for such large input, so we rewrite function.

(defn ! [n]
  (if (zero? n) 1
      (*' n (! (dec n)))))

Just one character added, * became *'. It is automatic type promotion. If type can not hold the value without losing precision, it automatically extends. In this case, Long becames BigInteger and all works fine.

(! 30) works, (! 300) works, (! 3000) works, (! 30000)...

> StackOverflowError

Yet another overflow.

Well, stack size is limited. We don't need to consume stackframes and will use loop/recur approach.

(defn ! [n]
  (loop [cur n acc 1]
    (if (zero? cur) acc
        (recur (dec cur) (*' cur acc)))))

It is also recursive approach, but instead of saving so far calculated value in stack, we pass it to the function as a parameter. Clojure optimize tail recursion to iterative version and does not consume stack. It succesfully calculates factorial for 30000, and for any other value. Great!

If we look at the factorial as a calculation process, not definition, it means "product of all numbers between 1 and n". It is very concise for functional programming:

(defn ! [n]
  (reduce *' (range 1 (inc n))))

Does not consume stack, no number overflow, but works ~2 times slower than loop/recur version.

I don't know the reason, but it seems time wasted by reduce for creating intermediate results. So we choose previous loop/recur function as baseline factorial.

Fast Factorial

Here is another way to calculate factorial.

Let's consider example 12!.

Factorial is a product of numbers 12! = 1 * 2 * 3 * ... * 11 * 12. Perform prime factorization of each number. For example, prime factorization of 12 is 2 * 2 * 3.

After that we get a factorization of the 12!, which contains only prime numbers. To find factorial value we can just multiply them all. But we do clever trick instead.

Group every prime number and its power. E.g. if we have factorization 2 * 2 * 2 * 3 * 5 it becomes vector of pairs [[2 3] [3 1] [5 1]].

To calculate power we have simple power implementation:

(defn power [x n]
  (reduce *' (repeat n x)))

Trick is using exponentiation by squaring, which reduce exponentiation complexity from O(n) to O(log n).

(defn power [x n]
  (cond (= 0 n) 1
        (= 1 n) x
        (even? n) (power* (*' x x) (/ n 2))
        (odd? n) (*' x (power* (*' x x) (/ (dec n) 2)))))

It is not tail-recursive solution and theoretically may cause stackoverflow, but its not critical. It works well for large numbers.

Actually, we do not perform factorization. We just know that factorization of n! contains all prime numbers below or equal n and every number have some multiplicity.

The function calculates how many times prime number k occurs in factorial factorization for n

(defn- find-power [n k]
  (loop [total n sum 0]
    (let [i (int (/ total k))]
      (if (zero? i) sum
          (recur i (+ sum i))))))

Binding all together:

(defn !! [n]
  (loop [[h & t] 
         (map #(power % (find-power n %))
              (take-while #(<= % n) (primes)))
         acc 1]
    (if h (recur t (*' h acc)) acc)))

Code available here

Note: primes is a function generates lazy-sequence of prime numbers. It can be taken from clojure.contrib.lazy-seqs.

Testing

Anyway, why do you think it is faster? You performing more calculations than just multiplying numbers.

Let's see.

(do (println "== 10 ==")
    (time (! 10))
    (time (!! 10))
    (println "== 100 ==")
    (time (! 100))
    (time (!! 100))
    (println "== 1000 ==")
    (time (! 1000))
    (time (!! 1000))
    (println "== 10000 ==")
    (time (! 10000))
    (time (!! 10000))
    (println "== 100000 ==")
    (time (! 100000))
    (time (!! 100000))
    (println "== 1000000 ==")
    (time (! 1000000))
    (time (!! 1000000)) nil)
== 10 ==
"Elapsed time: 0.051054 msecs"
"Elapsed time: 0.137587 msecs"
== 100 ==
"Elapsed time: 0.081365 msecs"
"Elapsed time: 0.376653 msecs"
== 1000 ==
"Elapsed time: 1.59252 msecs"
"Elapsed time: 3.212842 msecs"
== 10000 ==
"Elapsed time: 161.856965 msecs"
"Elapsed time: 76.452529 msecs"
== 100000 ==
"Elapsed time: 19403.375319 msecs"
"Elapsed time: 6372.35266 msecs"
== 1000000 ==
"Elapsed time: 2863893.471718 msecs"
"Elapsed time: 1079632.255919 msecs"
nil

For small factorials (< 1000) improved version works ~2 times slower. Around the thousand it has the same performance as standard version. And, finally, some win (up to 3 times) for larger numbers.

Theoretically, you can implement generalized factorial with these two algorithms and switch between them, depending on input. But who really need it?

Question: What the complexity of this algorithm?

P.S. Enterprise lovers would say factorial complexity is O(1). Just precompute all values and save them to database.

mishadoff 07 June 2013
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