1 the subset-sum problem is defined as follows

Question

1. The subset-sum problem is defined as follows: given a set B of n positive integers and an integer K, can you find a subset of B whose elements' summation is equal to K? Design an algorithm to solve this problem. Address its correctness and running time.

Input: set B of n positive integers {b₁, b₂,....., b_n} and an integer K.

Output: whether there exist such a subset of B called B' its elements summation is equal to K.

B'= BA, where A = {a_1, a₂,........, a_n} in which AB= b₁a₁ +b₂a₂ +......+ b_na_n. Where a_i is either 0 or 1.

Algorithm:

- For i= 1 to 2ⁿ  (We have 2ⁿ different combinations set to be checked)

1. Create all the possible combinations of Array A and do:

-  Compute Sum =

-  If Sum = K then there is a subset sum to K. This subset B'= {b₁a₁, b₂a₂,     ......, b_na_n}when a_i representing 1.

-  return the subset B'

- Otherwise return there is no subset sum to K.

The run time is O(2ⁿ) since it needs to go through all possible subsets to find the subset that sum to K.

2. Suppose you have a procedure that can partition a set of positive integers into two equal weight subsets. How could you use this procedure to solve the subset-sum problem?

To solve this we use reduction. In which prove that subset-sum problem can be reduce to partition problem and visa versa.

The set B of n positive integers whose element summation is equal to an integer K.

Partition reduces to Subset Sum:

Calculate Sum = , which is the summation of all the given numbers. A partition ≤ Subset Sum if K = Sum/2.

 

Subset Sum reduces to Partition:

Calculate Sum = , which is the summation of all the given numbers.

Calculate some number x= Sum - 2K. Create new set A by add x to the set B {b₁, b₂,....., b_n} U {x}, where the summation now is B+x. it is possible to split the numbers in A into some subsets iff they can summing up to K:

Subset sum of B indicates partition of A means the set that adds up to Sum with x form a partition.

Partition of A indicates subset sum of B means the numbers which are put together with x must add up to K. Therefore, a partition exists iff some numbers in the B add up to K.