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Algorithm to find which number in a list sum up to a certain number

question:

there is a list: [1,2,3,4,6,8,10,12], i want to use these numbers to sum up to a new number 16.

rules:

1) don’t need to use all of the numbers, 6 + 10 will be ok.

2) a number can use multi times, 12+2+1+1 will be ok.

3) order matters, 12 + 6 and 6 + 12 are two different combinations.

i have see algorithm to sum up a list of numbers for all combinations, but this is not the same.

don’t know much about algorithm, if this fit certain algorithm, please let me know, or some python code / pseudo-code will be much appreciated.

Answer 1

First – note that even finding if there is any subset that sums to the desired number is NP-Complete and is known as the subset sum problem, so there is no known polynomial solution for it.

Now, regarding the specific issue, here are some options:

First there is of course the obvious “generate all subsets and check the sum” way. Note that if your elements are all non-negative, you can use branch and bound and terminate a large portion of the possibilities before you actually develop them (If you found a subset X with sum(X) == s, and you are looking for the number n < s – you can be sure any set containing X will NOT find the solution). Something along the lines of:

findSubsets(list,sol,n):
  if (list.empty() and n == 0): #found a feasible subset!
     print sol 
     return
  else if (n < 0): #bounding non feasible solutions
     return 
  else if (list.empty()): #a solution that sums to a smaller number then n
     return
  e <- list.removeAndReturnFirst()
  sol <- sol.add(e)
  findSubsets(list,sol,n-e)
  sol <- sol.removeLast()
  findSubsets(list,sol,n)
  list.addFirst(e) #cleanup, return the list to original state

Invoke with findSubsets(list,[],n) where list is your list of items, n is the desired number and [] is an empty list.

Note that it can be easily parallelized if needed, there is no real syncrhonization needed between two subsets explored.

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Another alternative if the list contains only integers is using Dynamic Programming for solving the subset sum problem. Once you have the matrix you can re-create all the elements from the table by going back in the table. This similar question discusses how to get a list from the knapsack DP solution. The principles of the two problems are pretty much the same.