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

There are a lot of solutions so far, but all are of the form generate then filter. Which means that they potentially spend a lot of time working on recursive paths that do not lead to a solution.

Here is a solution that is O(size_of_array * (number_of_sums + number_of_solutions)). In other words it uses dynamic programming to avoid enumerating possible solutions that will never match.

For giggles and grins I made this work with numbers that are both positive and negative, and made it an iterator. It will work for Python 2.3+.

def subset_sum_iter(array, target):
    sign = 1
    array = sorted(array)
    if target < 0:
        array = reversed(array)
        sign = -1

    last_index = {0: [-1]}
    for i in range(len(array)):
        for s in list(last_index.keys()):
            new_s = s + array[i]
            if 0 < (new_s - target) * sign:
                pass # Cannot lead to target
            elif new_s in last_index:
                last_index[new_s].append(i)
            else:
                last_index[new_s] = [i]

    # Now yield up the answers.
    def recur (new_target, max_i):
        for i in last_index[new_target]:
            if i == -1:
                yield [] # Empty sum.
            elif max_i <= i:
                break # Not our solution.
            else:
                for answer in recur(new_target - array[i], i):
                    answer.append(array[i])
                    yield answer

    for answer in recur(target, len(array)):
        yield answer

And here is an example of it being used with an array and target where the filtering approach used in other solutions would effectively never finish.

def is_prime (n):
    for i in range(2, n):
        if 0 == n%i:
            return False
        elif n < i*i:
            return True
    if n == 2:
        return True
    else:
        return False

def primes (limit):
    n = 2
    while True:
        if is_prime(n):
            yield(n)
        n = n+1
        if limit < n:
            break

for answer in subset_sum_iter(primes(1000), 76000):
    print(answer)

This prints all 522 answers in under 2 seconds. The previous approaches would be lucky to find any answers in the current lifetime of the universe. (The full space has 2^168 = 3.74144419156711e+50 possible combinations to run through. That...takes a while.)