What is dynamic programing


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What is dynamic programing

  1. 1. What is dynamic programing? In mathematics, computer science, economics, and bioinformatics, dynamic programming is a method for solving complex problems by breaking them down into simpler sub problems. It is applicable to problems exhibiting the properties of overlapping sub problems and optimal substructure (described below). When applicable, the method takes far less time than naive methods that don't take advantage of the sub problem overlap (like depth-first search). The idea behind dynamic programming is quite simple. In general, to solve a given problem, we need to solve different parts of the problem (sub problems), then combine the solutions of the sub problems to reach an overall solution. Often when using a more naive method, many of the sub problems are generated and solved many times. The dynamic programming approach seeks to solve each sub problem only once, thus reducing the number of computations: once the solution to a given sub problem has been computed, it is stored or "memo-ized": the next time the same solution is needed, it is simply looked up. This approach is especially useful when the number of repeating sub problems grows exponentially as a function of the size of the input. Dynamic programming algorithms are used for optimization (for example, finding the shortest path between two points, or the fastest way to multiply many matrices). A dynamic programming algorithm will examine all possible ways to solve the problem and will pick the best solution. Therefore, we can roughly think of dynamic programming as an intelligent, brute-force method that enables us to go through all possible solutions to pick the best one. If the scope of the problem is such that going through all possible solutions is possible and fast enough, dynamic programming guarantees finding the optimal solution. The alternatives are many, such as using a greedy algorithm, which picks the best possible choice "at any possible branch in the road". While a greedy algorithm does not guarantee the optimal solution, it is faster. Fortunately, some greedy algorithms (such as minimum spanning trees) are proven to lead to the optimal solution. For example, let's say that you have to get from point A to point B as fast as possible, in a given city, during rush hour. A dynamic programming algorithm will look into the entire traffic report, looking into all possible combinations of roads you might take, and will only then tell you which way is the fastest. Of course, you might have to wait for a while until the algorithm finishes, and only then can you start driving. The path you will take will be the fastest one (assuming that nothing changed in the external environment). On the other hand, a greedy algorithm will start you driving immediately and will pick the road that looks the fastest at every intersection. As you can imagine, this strategy might not lead to the fastest arrival time, since you might take some "easy" streets and then find yourself hopelessly stuck in a traffic jam. Figure 1.
  2. 2. Figure 1. Finding the shortest path in a graph using optimal substructure; a straight line indicates a single edge; a wavy line indicates a shortest path between the two vertices it connects (other nodes on these paths are not shown); the bold line is the overall shortest path from start to goal. • An algorithm design technique (like divide and conquer) • Divide and conquer  Partition the problem into independent sub problems  Solve the sub problems recursively  Combine the solutions to solve the original problem  A divide and conquer approach would repeatedly solve the common sub problems Dynamic programing  Dynamic programming solves every sub problem just once and stores the answer in a table.  Used for optimization problems  A set of choices must be made to get an optimal solution  Find a solution with the optimal value(minimum or maximum)  There may be many solutions that return the optimal vale:an optimal solution Dynamic Programming Algorithm It has four steps: 1. Characterize the structure of an optimal solution 2. Recursively define the value of an optimal solution 3. Compute the value of an optimal solution in a bottom- up fashion 4. Construct an optimal solution from computed information. Step 1-3 are the basis of dynamic programming solution to a problem. o It gives the optimal value only
  3. 3. o Step 4 is used to construct the Optimal Solution 1 2 3 S1,i 7 8 6 S2,i 5 11 13 S3,i 4 9 6 E1 2 - - E2 3 - - E3 4 - - X1 3 - - X2 1 - - X3 5 - - T1,i 7,11 4,9 - T2,i 8,3 3,6 - T3,i 5,7 12,14 - Cost: j 1 2 3 F1[j] 7 8 6 F2[j] 5 11 13 F3[j] 4 9 6 Line:
  4. 4. L1[j] 1 3 2 L2[j] 3 1 2 l3[j] 1 2 1