calculate time to process combinations javascript

Question

I have this function from : https://rosettacode.org/wiki/Combinations#ES6

In my environementt console.log(show(comb(3,15))); (same as this snippet bellow) take aprox. 4 seconds to process

If I try console.log(show(comb(3,16))); that take aprox. 16 seconds

If I try console.log(show(comb(3,17))); that take aprox. 90 seconds

But If I tryed : console.log(show(comb(3,20))); After one hour of process are not yet finish and I have stopped it.

The question is:

How to calculate beforehand the time to process comb(3,50) or comb(3,80) ?

(() => {
    'use strict';

 
    // COMBINATIONS -----------------------------------------------------------
 
    // comb :: Int -> Int -> [[Int]]
    const comb = (m, n) => combinations(m, enumFromTo(0, n - 1));
 
    // combinations :: Int -> [a] -> [[a]]
    const combinations = (k, xs) =>
        sort(filter(xs => k === xs.length, subsequences(xs)));
 
 
    // GENERIC FUNCTIONS -----------------------------------------------------
 
    // cons :: a -> [a] -> [a]
    const cons = (x, xs) => [x].concat(xs);
 
    // enumFromTo :: Int -> Int -> [Int]
    const enumFromTo = (m, n) =>
        Array.from({
            length: Math.floor(n - m) + 1
        }, (_, i) => m + i);
 
    // filter :: (a -> Bool) -> [a] -> [a]
    const filter = (f, xs) => xs.filter(f);
 
    // foldr (a -> b -> b) -> b -> [a] -> b
    const foldr = (f, a, xs) => xs.reduceRight(f, a);
 
    // isNull :: [a] -> Bool
    const isNull = xs => (xs instanceof Array) ? xs.length < 1 : undefined;
 
    // show :: a -> String
    const show = x => JSON.stringify(x) //, null, 2);
 
    // sort :: Ord a => [a] -> [a]
    const sort = xs => xs.sort();
 
    // stringChars :: String -> [Char]
    const stringChars = s => s.split('');
 
    // subsequences :: [a] -> [[a]]
    const subsequences = xs => {
 
        // nonEmptySubsequences :: [a] -> [[a]]
        const nonEmptySubsequences = xxs => {
            if (isNull(xxs)) return [];
            const [x, xs] = uncons(xxs);
            const f = (r, ys) => cons(ys, cons(cons(x, ys), r));
 
            return cons([x], foldr(f, [], nonEmptySubsequences(xs)));
        };
 
        return nonEmptySubsequences(
            (typeof xs === 'string' ? stringChars(xs) : xs)
        );
    };
 
    // uncons :: [a] -> Maybe (a, [a])
    const uncons = xs => xs.length ? [xs[0], xs.slice(1)] : undefined;
 
 
    // TEST -------------------------------------------------------------------
    // return show(
        // comb(3, 5)
    // );
    
    console.log(show(comb(3,15)));
})();

Answer 1

Use binomial coefficients. The time to process comb(3,n) is n choose 3 which works out to n*(n-1)*(n-2)/6 hence is O(n^3). For example, increasing n by a factor of 10 should increase the runtime by a factor of roughly 1000.

20 choose 3 is only 1140, so if it takes over an hour to generate them, the algorithm in question isn't particularly good. Furthermore, the gap between 20 choose 3 and 17 choose 3 is not so large that it really explains the time difference. Thus, the asymptotic analysis is only suggestive of what is happening. The actual runtime seems to be much worse.