Question

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Algorithm: Extract subset based on property sum

in the simple case we have an array:

{6, 1, 3, 11, 2, 5,12}

and we want to know all combinations the sum of elements contained in that array to getting 12.

in this case we will get four combinations:

• 12
• 1 + 11
• 6 + 5 + 1
• 1 + 3 + 2 + 6

so, how can we do this in BASIC or PHP?. maybe pseudo-code first ;-).

I’ve been looking but there just got a combination with a predetermined number of elements.

Answer 1

The problem is NP-Hard. Even determining if there is ANY subset of the problem that sums to the desired number is NP-Hard (known as the subset sum problem), and there is no known polynomial solution to it.

Thus, you should look for an exponantial solution, such as a backtracking one – generate all possible combinations, and check if they are valid.

You can use trimming to get your search faster (for example, if you generate a partial subset of sum 13, no need to check other subsets which are supersets of this subset, since they will definetly won’t lead to a solution.

pseudo code:

findValidSubsets(sum,arr,idx,currSolution):
   if (sum == 0):
       print currSolution
   if (sum < 0): //trim the search, it won't be succesful
       return
   //one possibility: don't take the current candidate
   findPermutations(sum,arr,idx+1,currSolution) 

   //second poassibility: take the current candidate
   currSolution.add(arr[idx])
   findPermutations(sum-arr[idx],arr,idx+1,currSolution)

   //clean up before returning: 
   currSolution.removeLast()

Complexity is O(2^n) – need to generate at worst case all 2^n possible subsets

Invoke with findValidSubsets(desiredSum,myArray,0,[]), where [] is an empty initial list