Sum of subset problem.pptx

Sum of subset problem.pptx

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  SUM OF SUBSETPROBLEM Mrs.G.Chandraprabha., M.Sc.,M.Phil., Assistant Professor, Department of Information Technology, V.V.V.anniaperumal College for Women, Virudhunagar.
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  SUM OF SUBSETPROBLEM  Subset sum problem is to find subset of elements that are selected from a given set whose sum adds up to a given number K. We are considering the set contains non-negative values. It is assumed that the input set is unique (no duplicates are presented).  In the sum of subsets problem, there are n positive integers(weights) Wi and a positive integer W. The goal is to find all subsets of the integers that sum to W.
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  SUM OF SUBSETPROBLEM  Backtracking Algorithm is used for Sum of subset problem.Using exhaustive search we consider all subsets irrespective of whether they satisfy given constraints or not.  Backtracking can be used to make a systematic consideration of the elements to be selected.  Assume given set of 4 elements, say w[1] … w[4]. Tree diagrams can be used to design backtracking algorithms. The following tree diagram depicts approach of generating variable sized tuple.
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  STATE SPACE TREEOF SUM OF SUBSET State Space Tree
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  PROBLEM:  Input: Theproblem of determining such sets is called the sum of subset problem. In the sum of subsets problem there are n positive integers(weights) wi and a positive integer w. The goal is to find all subsets of the integers that sum to W.  Problem: n=5, w=21 and W1=5 w2=6 w3=10 w4=11 w5=16 Because w1+w2+w3=5+6+10=21, w1+w5=5+16=21, w3+w4=10+11=21 The solutions are {w1,w2,w3},{w1,w5} and {w3,w4}
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  EXAMPLE:  Problem: Showsthe pruned state space tree when backtracking is used with n=4, w=13 W1=3, w2=4, w=5,w4=6 The solution is {w1,w2,w4}
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  PRUNED STATE SPACETREE
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  EXPLANATION:  The nonpromising nodes are marked with crosses. The nodes containing 12, 8, 9 are non promising because the adding the next weight (6) would make the value of weight exceed W. The nodes containing 7,3,4 and 0 are non-promising because there is not enough total weight remaining to bring the value of weight up to W.  Notice that a leaf in the state space tree that does not containing solution automatically non promising because there are no weights remaining that could bring up to W.  The leaf containing 7 illustrates this. There are only 15 nodes in the pruned state space tree , whereas the entire state space tree contains 31 nodes.
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  ALGORITHM: THE BACKTRACKING ALGORITHMFOR THE SUM OF SUBSETS PROBLEM  Problem: Given n positive integers (weights) and a positive integer w, determine all combinations of the integers that sum to W.  Input: positive integer n, sorted ( non decreasing order) array of positive integer u indexed from 1 to n, and a positive w.  Problem: Void sum of subsets(index i , int weight , int total) { If (promising(i)) If (weight==w) Cout<<include[1] through include[i]; Else { Include[i+1]=“yes”; Sum_of_subsets (i+1, weight + w[i+1].total- w[i+1]);
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  . Include[i+1]= “no” ; Sum_of_subsets(i+1, weight, total- w[i+1]); } } bool promising(index i); { Return(weight+total>=w)&&(weight==w||weight +w[i+1] }
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  THANK YOU