Subset sum Problem

Question

recently I became interested in the subset-sum problem which is finding a zero-sum subset in a superset. I found some solutions on SO, in addition, I came across a particula

Answer 1

I'm not quite sure if your solution is exact or a PTA (poly-time approximation).

But, as someone pointed out, this problem is indeed NP-Complete.

Meaning, every known (exact) algorithm has an exponential time behavior on the size of the input.

Meaning, if you can process 1 operation in .01 nanosecond then, for a list of 59 elements it'll take:

2^59 ops -->     2^59     seconds -->     2^26      years -->      1 year
            --------------           ---------------
            10.000.000.000           3600 x 24 x 365

You can find heuristics, which give you just a CHANCE of finding an exact solution in polynomial time.

On the other side, if you restrict the problem (to another) using bounds for the values of the numbers in the set, then the problem complexity reduces to polynomial time. But even then the memory space consumed will be a polynomial of VERY High Order.
The memory consumed will be much larger than the few gigabytes you have in memory. And even much larger than the few tera-bytes on your hard drive.

( That's for small values of the bound for the value of the elements in the set )

May be this is the case of your Dynamic programing algorithm.

It seemed to me that you were using a bound of 1000 when building your initialization matrix.

You can try a smaller bound. That is... if your input is consistently consist of small values.

Good Luck!

Answer 2

An interesting article on optimizing python code is available here. Basically the main result is that you should inline your frequent loops, so in your case this would mean instead of calling get_element twice per loop, put the actual code of that function inside the loop in order to avoid the function call overhead.

Hope that helps! Cheers

Answer 3

Someone on Hacker News came up with the following solution to the problem, which I quite liked. It just happens to be in python :):

def subset_summing_to_zero (activities):
  subsets = {0: []}
  for (activity, cost) in activities.iteritems():
      old_subsets = subsets
      subsets = {}
      for (prev_sum, subset) in old_subsets.iteritems():
          subsets[prev_sum] = subset
          new_sum = prev_sum + cost
          new_subset = subset + [activity]
          if 0 == new_sum:
              new_subset.sort()
              return new_subset
          else:
              subsets[new_sum] = new_subset
  return []

I spent a few minutes with it and it worked very well.

Answer 4

The following code works for Python 3.3+ , I have used the itertools module in Python that has some great methods to use.

from itertools import chain, combinations
def powerset(iterable):
    s = list(iterable)
    return chain.from_iterable(combinations(s, r) for r in range(len(s)+1))


nums = input("Enter the Elements").strip().split()
inputSum = int(input("Enter the Sum You want"))

for i, combo in enumerate(powerset(nums), 1):
    sum = 0
    for num in combo:
        sum += int(num)
    if sum == inputSum:
        print(combo)

The Input Output is as Follows:

Enter the Elements 1 2 3 4
Enter the Sum You want 5
('1', '4')
('2', '3')

Answer 5

Just change the values in your set w and correspondingly make an array x as big as the len of w then pass the last value in the subsetsum function as the sum for which u want subsets and you wl bw done (if u want to check by giving your own values).

def subsetsum(cs,k,r,x,w,d):
    x[k]=1
    if(cs+w[k]==d):
        for i in range(0,k+1):

            if x[i]==1:
                print (w[i],end=" ")
        print()

    elif cs+w[k]+w[k+1]<=d :
        subsetsum(cs+w[k],k+1,r-w[k],x,w,d)

    if((cs +r-w[k]>=d) and (cs+w[k]<=d)) :
        x[k]=0
        subsetsum(cs,k+1,r-w[k],x,w,d)
#driver for the above code
w=[2,3,4,5,0]
x=[0,0,0,0,0]

subsetsum(0,0,sum(w),x,w,7)     

Answer 6

, 1st eye catch

def split_sum(A):
  N_list = 0
  P_list = 0
  for x in A:
    if x < 0:
        N_list+=x
    elif x > 0:
        P_list+=x
  return [N_list, P_list]

Some advices:

1. Try to use 1D list and use bitarray to reduce memory footprint at minimum (http://pypi.python.org/pypi/bitarray) so you will just change get / set functon. This should reduce your memory footprint by at lest 64 (integer in list is pointer to integer whit type so it can be factor 3*32)

2. Avoid using try - catch, but figure out proper ranges at beginning, you might found out that you will gain huge speed.