Given a set of n integers, list all possible subsets with k1<= sum <=k2 , k1 and k2 floating numbers

Question

Given an unsorted set of integers in the form of array, find all possible subsets whose sum is greater than k1 and less than k2 ,where k1 ,k2 are two floating constants, eg:- Our set is {2,3,5,8,10} and k1 =10 and k2 = 12.

Possible subsets:-

{2,3,5}
{8,2}
{8,3}
{10}
{10,2}

I can only think of a naive algorithm, is there any better method or similar problems ,please give some suggestions. It is actually an important part of my project work? Is there any dynamic programming method available ?

Answer 1

Yes, dynamic programming method does exist.
Make table (array of lists) with length k2+1 and fill it:
For every value V=A[i] scan array from the end to beginning, and add V to the list in j-th cell, if Table[j - V] is not empty (so we can compose value j adding V to all subsets from Table[j - V] )
At the end check cells with needed sums and restore sequences.

Example for 2,3,5:

 0    1    2   3   4   5   6   7   8   9   10    //index
 [0]                                             //initial state
 [0]      [2]                                    //after 2   
 [0]      [2]  [3]    [3]                        //after 3     
 [0]      [2]  [3]    [3,5]   [5] [5]      [5]   //after 5

To retrieve sets for sum 5 - got to the fifth cell and unwind values to the left. 3 denotes we go to 5-3=2 , then use 2 until zero entry. Second variant - with 5 we go to zero entry. So value 5 might be produced from set {5} or from set {3,2}

Note that storing data for sequences might require a lot of space, and restoring sequences might require up to 2^N time - when there are many good subsets.

Answer 2

Yes it can be actually solved by: subset sum problem

As the set contains all integers, so you can just just type cast K1 and K2 to integers and just check which are the subsets that falls in the given range.

DP[N][sum_of_all_elements];

Now you got one element of your set say DP[N-1][x],ie K1<=x<=K2.

and now you need to move up in the DP[][] array to find the next element in the set and so on.And you need to do this for each subset which has value(K1<= value <=K2)

Answer 3

Reduce this problem to F(k1,k2) = F'(k1)+F'(k1+1)+...+F'(K2) .

Then F'(k) is an easy DP solution.