Getting all possible sums that add up to a given number

Question

I'm making an math app for the android. In one of these fields the user can enter an int (no digits and above 0). The idea is to get all possible sums that make this int, without doubles (4+1 == 1+4 in this case). The only thing known is this one int.

For example:

Say the user enters 4, I would like the app to return:

• 4
• 3+1
• 2+2
• 2+1+1
• 1+1+1+1

Obviously 4 == 4 so that should be added too. Any suggestions as to how i should go about doing this?

Answer 1

Here's a simple algorithm that purports to do that

from : http://introcs.cs.princeton.edu/java/23recursion/Partition.java.html

 public class Partition { public static void partition(int n) { partition(n, n, ""); } public static void partition(int n, int max, String prefix) { if (n == 0) { StdOut.println(prefix); return; } for (int i = Math.min(max, n); i >= 1; i--) { partition(ni, i, prefix + " " + i); } } public static void main(String[] args) { int N = Integer.parseInt(args[0]); partition(N); } } 

Answer 2

There are short and elegant recursive solution to generate them, but the following may be easier to use and implement in existing code:

import java.util.*;

public class SumIterator implements Iterator<List<Integer>>, Iterable<List<Integer>> {

  // keeps track of all sums that have been generated already
  private Set<List<Integer>> generated;

  // holds all sums that haven't been returned by `next()`
  private Stack<List<Integer>> sums;

  public SumIterator(int n) {

    // first a sanity check...
    if(n < 1) {
      throw new RuntimeException("'n' must be >= 1");
    }

    generated = new HashSet<List<Integer>>();
    sums = new Stack<List<Integer>>();

    // create and add the "last" sum of size `n`: [1, 1, 1, ... , 1]
    List<Integer> last = new ArrayList<Integer>();
    for(int i = 0; i < n; i++) {
      last.add(1);
    }
    add(last);

    // add the first sum of size 1: [n]
    add(Arrays.asList(n));
  }

  private void add(List<Integer> sum) {
    if(generated.add(sum)) {
      // only push the sum on the stack if it hasn't been generated before
      sums.push(sum);
    }
  }

  @Override
  public boolean hasNext() {
    return !sums.isEmpty();
  }

  @Override
  public Iterator<List<Integer>> iterator() {
    return this;
  }

  @Override
  public List<Integer> next() {
    List<Integer> sum = sums.pop();                         // get the next sum from the stack
    for(int i = sum.size() - 1; i >= 0; i--) {              // loop from right to left
      int n = sum.get(i);                                   //   get the i-th number
      if(n > 1) {                                           //   if the i-th number is more than 1
        for(int j = n-1; j > n/2; j--) {                    //     if the i-th number is 10, loop from 9 to 5
          List<Integer> copy = new ArrayList<Integer>(sum); //       create a copy of the current sum
          copy.remove(i);                                   //       remove the i-th number
          copy.add(i, j);                                   //       insert `j` where the i-th number was
          copy.add(i + 1, n-j);                             //       insert `n-j` next to `j`
          add(copy);                                        //       add this new sum to the stack
        }                                                   //     
        break;                                              //   stop looping any further
      }                                                     
    }
    return sum;
  }

  @Override
  public void remove() {
    throw new UnsupportedOperationException();
  }
}

You can use it like this:

int n = 10;
for(List<Integer> sum : new SumIterator(n)) {
  System.out.println(n + " = " + sum);
}

which would print:

10 = [10]
10 = [6, 4]
10 = [6, 3, 1]
10 = [6, 2, 1, 1]
10 = [7, 3]
10 = [7, 2, 1]
10 = [8, 2]
10 = [9, 1]
10 = [5, 4, 1]
10 = [5, 3, 1, 1]
10 = [5, 2, 1, 1, 1]
10 = [8, 1, 1]
10 = [7, 1, 1, 1]
10 = [4, 3, 1, 1, 1]
10 = [4, 2, 1, 1, 1, 1]
10 = [6, 1, 1, 1, 1]
10 = [5, 1, 1, 1, 1, 1]
10 = [3, 2, 1, 1, 1, 1, 1]
10 = [4, 1, 1, 1, 1, 1, 1]
10 = [3, 1, 1, 1, 1, 1, 1, 1]
10 = [2, 1, 1, 1, 1, 1, 1, 1, 1]
10 = [1, 1, 1, 1, 1, 1, 1, 1, 1, 1]

Answer 3

This is the mathematical concept known as partitions . In general, it's... difficult, but there are techniques for small numbers. A load of useful stuff linked from the wiki page.

Answer 4

For a number N you know that the max number of terms is N. so, you will start by enumerating all those possibilities.

For each possible number of terms, there are a number of possibilities. The formula eludes me now, but basically, the idea is to start by (N+1-i + 1 + ... + 1) where i is the number of terms, and to move 1s from left to right, second case would be (Ni + 2 + ... + 1) until you cannot do another move without resulting in an unsorted combination.

(Also, why did you tagged this android again?)

Answer 5

This is related to the subset sum problem algorithm.

N = {N*1, (N-1)+1, (N-2)+2, (N-3)+3 .., N-1 = {(N-1), ((N-1)-1)+2, ((N-1)-1)+3..}

etc.

So it's a recursive function involving substitution; whether that makes sense or not when dealing with large numbers, however, is something you'll have to decide for yourself.

Answer 6

All of these solutions seem a little complex. This can be achieved by simply "incrementing" a list initialized to contain 1's=N.

If people don't mind converting from c++, the following algorithm produces the needed output.

bool next(vector<unsigned>& counts) {
    if(counts.size() == 1)
        return false;

    //increment one before the back
    ++counts[counts.size() - 2];

    //spread the back into all ones
    if(counts.back() == 1)
        counts.pop_back();
    else {
        //reset this to 1's
        unsigned ones = counts.back() - 1;
        counts.pop_back();
        counts.resize(counts.size() + ones, 1);
    }
    return true;
}

void print_list(vector<unsigned>& list) {
    cout << "[";
    for(unsigned i = 0; i < list.size(); ++i) {
        cout << list[i];
        if(i < list.size() - 1)
            cout << ", ";
    }
    cout << "]\n";
}

int main() {
    unsigned N = 5;
    vector<unsigned> counts(N, 1);
    do {
        print_list(counts);
    } while(next(counts));
    return 0;
}

for N=5 the algorithm gives the following

[1, 1, 1, 1, 1]
[1, 1, 1, 2]
[1, 1, 2, 1]
[1, 1, 3]
[1, 2, 1, 1]
[1, 2, 2]
[1, 3, 1]
[1, 4]
[2, 1, 1, 1]
[2, 1, 2]
[2, 2, 1]
[2, 3]
[3, 1, 1]
[3, 2]
[4, 1]
[5]