![18
Dynamic Programming Algorithm
bool SumSub (w , n , m)
{
1. t[0,0] = true; for (j = 1 to m) t[0,j] = false;
2. for (i = 1 to n)
3. for (j = 0 to m)
4. t[i,j] = t[i-1,j];
5. if ((j-wi) >= 0)
6. t[i,j] = t[i-1,j] || t[i-1, j – wi];
7. return t[n,m];
}
i-1,
j - wi
i-1,
j
i , j
Its Complexity???
Time complexity is T(n) = O(m) + O(nm) = O(nm)](https://image.slidesharecdn.com/subsetsumproblem-160609202613/85/Subset-sum-problem-Dynamic-and-Brute-Force-Approch-18-320.jpg)
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Subset sum problem Dynamic and Brute Force Approch
- 1. Willing is not enough, we must do Bruce lee
- 2. Problem Statement: In the subset-sum problem, we are given a finite set S of positive integers and an integer target t > 0. We ask whether there exists a subset S`⊆ S whose elements sum to t.
- 3. TRUE FALSE t = 138457 TRUE How to Design an Algorithm to Solve this Problem??
- 4. Brute Force?? Making all possible Subsets
- 5. A={1,2,4} What will be its Complexity??? All possible subsets??? 1. A’={} 2. A’={1} 3. A’={1,2} 4. A’={1,4} 5. A’={1,2,4} 6. A’={2} 7. A’={2,4} 8. A’={4} SumOFSubset=2n=8
- 6. Brute Force Subset Sum Solution Running time Q(n 2n ) What does this tell us about the time complexity of the subset sum problem? bool SumSub (w, i, j) { if (i == 0) return (j == 0); else if (SumSub (w, i-1, j)) return true; else if ((j - wi) >= 0) return SumSub (w, i-1, j - wi); else return false; }
- 7. Complexity 0( 2n N) All possible subsets and time required to sum them. Brute Force??? No, thanks Why is this Failing???
- 8. • Sometimes, the divide and conquer approach seems appropriate but fails to produce an efficient algorithm. • One of the reasons is that D&Q produces overlapping sub problems.
- 9. Dynamic Approach Divide and conquer with Tabular form Solves a sub-problem by making use of previously stored solutions for all other sub problems. HOW?????
- 10. Lets take an example 0 1 2 3 4 5 6 7 8 9 10 11 2 3 7 8 10 A={2,3,7,8,10} t=11 Any subset of { 2} making sum=0 ???
- 11. Lets take an example 0 1 2 3 4 5 6 7 8 9 10 11 2 T 3 7 8 10 A={2,3,7,8,10} t=11
- 12. Lets take an example 0 1 2 3 4 5 6 7 8 9 10 11 2 T F 3 7 8 10 A={2,3,7,8,10} t=11 Any subset of { 2} making sum=1 ???
- 13. Lets take an example 0 1 2 3 4 5 6 7 8 9 10 11 2 T F T 3 7 8 10 A={2,3,7,8,10} t=11 Any subset of { 2} making sum=3 ???
- 14. Lets take an example 0 1 2 3 4 5 6 7 8 9 10 11 2 T F T F F F F F F F F F 3 T F T T F T F F F F F F 7 T F T T F T F T F T T F 8 T F T T F T F T T T T T 10 T F T T F T F T T T T T A={2,3,7,8,10} t=11 How to get the Answer??
- 15. 8 is one of the Element
- 16. 3 is one of the Element
- 17. As we have reached the Last row.. A’={8,3} is our Answer
- 18. 18 Dynamic Programming Algorithm bool SumSub (w , n , m) { 1. t[0,0] = true; for (j = 1 to m) t[0,j] = false; 2. for (i = 1 to n) 3. for (j = 0 to m) 4. t[i,j] = t[i-1,j]; 5. if ((j-wi) >= 0) 6. t[i,j] = t[i-1,j] || t[i-1, j – wi]; 7. return t[n,m]; } i-1, j - wi i-1, j i , j Its Complexity??? Time complexity is T(n) = O(m) + O(nm) = O(nm)
- 19. Application Traveling Salesperson Problem –Input: a graph of cities and roads with distance connecting them and a minimum total distant –Output: either a path that visits each with a cost less than the minimum, or “no”. • If given a path, easy to check if it visits every city with less than minimum distance travelled.
- 20. Drug Discovery Problem –Input: a set of proteins, a desired 3D shape –Output: a sequence of proteins that produces the shape (or impossible) If given a sequence, easy to check if sequence has the right shape.
- 21. THE END