Finding all possible combinations of numbers to reach a given sum

Question

How would you go about testing all possible combinations of additions from a given set N of numbers so they add up to a given final number?

A brief exam

Answer 1

Deduce 0 in the first place. Zero is an identiy for addition so it is useless by the monoid laws in this particular case. Also deduce negative numbers as well if you want to climb up to a positive number. Otherwise you would also need subtraction operation.

So... the fastest algorithm you can get on this particular job is as follows given in JS.

function items2T([n,...ns],t){
    var c = ~~(t/n);
    return ns.length ? Array(c+1).fill()
                                 .reduce((r,_,i) => r.concat(items2T(ns, t-n*i).map(s => Array(i).fill(n).concat(s))),[])
                     : t % n ? []
                             : [Array(c).fill(n)];
};

var data = [3, 9, 8, 4, 5, 7, 10],
    result;

console.time("combos");
result = items2T(data, 15);
console.timeEnd("combos");
console.log(JSON.stringify(result));

This is a very fast algorithm but if you sort the data array descending it will be even faster. Using .sort() is insignificant since the algorithm will end up with much less recursive invocations.