Pintu ram
- 1. PRESENTATION Topic : “DYNAMIC PROGRAMMING” Prepared By: Pintu Ram Course: B.Sc.(Hons.)C.S. Sem : 2nd Roll no : 2k17/CS/72 Submitted to : Mr. Ashish Jha College Of Vocational Studies
- 2. INDEX WHAT IS DYNNAMIC PROGRAMMMING WHAT IS OPTIMIZATION PROBLEM WHAT IS OPTIMAL SOLUTION PRICIPLE OF OPTIMALITY DYNAMIC PROGRAMMING ALGORITHM ADVANTAGE OF DYNAMIC PROGRAMMING DISADVANTAGE OF DYNAMIC PROGRAMMING EXAMPLE OF DYNAMIC PROGRAMMING
- 3. WHAT IS DYNAMIC PROGARMMING DEFINITION: Dynamic Programming is a general algorithm design technique for solving a complicated problem defined by recurrences with overlapping sub problems. Invented by : “American mathematician” “Richard Bellman” in 1950
- 4. Dp and dcp DP:> Dynamic Programming DCP:> Divide and Conquer Programming DP likes DCP. But, Some difference DCP :applies when sub problems does not overlap. (repeatedly works)/(top down) DP :applies when sub problems overlap. (avoids repeatedly works)(bottom up)
- 5. Optimization problem Dynamic Programming is generally applied for solving optimization problems. OPTIMIZATION PROBLEM: Such problems that can have many possible solutions and each solution has a value and we wish to find a solution with optimal value (minimum and maximum).
- 6. Optimal solution It is a feasible/suitable/appropriate/right solution which is our favorable. Or not a solution but the best solution. It provides the most beneficial result for the specified objective problems. If the objective problem is related to profit then optimal solution has a maximum value While if the objective problem is related to the cost the optimal solution has a minimum value.
- 7. Principle of optimality The Dynamic Programming works on a principle of optimality. “The principle of optimality states that in an optimal sequences of decisions/choices, each sequences must also be optimal.”
- 8. Dynamic programming algo or dpa Steps for designing a DPA: ♠CHRACTERIZE optimal substructure. ♠RECURSIVELY define the. optimal value ♠COMPUTE the optimal value in bottom up. ♠CONSTRUCT an optimal solution.
- 9. ADVANTAGE OF DYNAMIC PROGRAMMING Easier to implement. Require much less computing resources. Much faster to executer. Greedy algorithm is used to solve optimization problems.
- 10. DISADVANTAGE OF DYNAMIC PROGRAMMING It does not always reach to the global solution. Or, Even the global solution is not found, most of the times is lose, and in this case the sub optimal solution is the best solution.
- 11. EXAMPLE OF DYNAIMC PROGRAMMING Multi graph: A multistage graph is a directed weighted graph and the vertices are divided into stages such that the edges are connecting the vertices from one stage to next stage only. First stage and last stage have a single vertex to represent the starting point to sink of a graph.
- 12. Objective of problem We have to selecting the path which is giving us the minimum cost.
- 13. –Diagram of a multistage graph – 1 2 1 2 1 1 1 0 9 5 4 3 8 7 6 9 6 4 2 5 68 2 7 3 21 2 7 11 11 5 4 5 3 Stage 1 Stage 2 Stage 3 Stage 4 Stage 5 4
- 14. Finding the cost of each stages and vertices are following: Cost (5,12)=0 | Cost (4,9)=4 | Cost (4,9)=2 | Cost (4,11)=5 Cost (3,6)=min { cost(6,9)+cost(4,9), cost(6,10)+cost(4,10) } =min{ 6+4 , 5+2 } =7 Similarly: Cost(3,7)=5 | Cost(3,8)=7 | Cost(2,2)=7 | Cost(2,3)=9 | Cost(2,4)=18 | Cost(2,5)=15 | Cost(1,1)=16
- 15. •Table representing minimum cast in multistage graph V 1 2 3 4 5 6 7 8 9 1 0 1 1 12 C 16 7 9 1 8 1 5 7 5 7 4 2 5 0 2/ 7 6 8 8 10 10 11 1 12 12 12
- 16. Formulation for finding the minimum cast in multistage graph. cost(i , j) = min{cost(j , l)+cost(i+1 , l)} (j, l) € edges l € s+1 (next stage) stage vertex next stage
- 17. Thank you