26 Jun The subset-sum problem is defined as follows:
Question
9) Thesubset-sumproblem is defined as follows: given a setBofnpositive integers and an integerK,can you find a subset ofBwhose elements’ summation is equal toK?Design an algorithm to solve this problem. Address its correctness and running time.
Input: setBofnpositive integers {b1, b2,….., bn}and an integerK.
Output: whether there exist such a subset of B called B’ its elements summation is equal to K.
B’= BA, where A = {a1,a2,…….., an} in which AB= b1a1 +b2a2 +……+ bnan. Where ai 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 =.png”>
§ If Sum = K then there is a subset sum to K. This subset B’= {b1a1, b2a2, ……, bnan}when ai 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.
10) Suppose you have a procedure that canpartitiona set of positive integers into two equal weight subsets. How could you use this procedure to solve thesubset-sumproblem?
To solve this we use reduction. In which prove thatsubset-sumproblem can be reduce topartition problem and visa versa.
The setBofnpositive integers whose element summation is equal to an integerK.
Partition reduces to Subset Sum:
Calculate Sum =.png”>, which is the summation of all the given numbers. A partition.png”> Subset Sum if K = Sum/2.
Subset Sum reduces to Partition:
Calculate Sum =.png”>, 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,….., bn}.png”> {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.
Thesereference could help you :
1. Bron, C. and Kerbosch, J. “Algorithm 457: Finding All Cliques of an Undirected Graph.” Comm. ACM 16, 48-50, 1973.
2. Tomita, E.; Tanaka, A.; and Takahashi, H. “The Worst-Case Time Complexity for Generating All Maximal Cliques and Computational Experiments.” Theor. Comput. Sci. 363, 28-42, 2006.
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