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• Dynamic_programming abstract "In mathematics, management science, economics, computer science, and bioinformatics, dynamic programming is a method for solving a complex problem by breaking it down into a collection of simpler subproblems, solving each of those subproblems just once, and storing their solutions - ideally, using a memory-based data structure. The next time the same subproblem occurs, instead of recomputing its solution, one simply looks up the previously computed solution, thereby saving computation time at the expense of a (hopefully) modest expenditure in storage space. (Each of the subproblem solutions is indexed in some way, typically based on the values of its input parameters, so as to facilitate its lookup.) The technique of storing solutions to subproblems instead of recomputing them is called \"memoization\". 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 the previously solved subproblems and will combine their solutions to give the best solution for the given problem. The alternatives are many, such as using a greedy algorithm, which picks the locally optimal choice at each branch in the road. The locally optimal choice may be a poor choice for the overall solution. While a greedy algorithm does not guarantee an optimal solution, it is often faster to calculate. Fortunately, some greedy algorithms (such as minimum spanning trees) are proven to lead to the optimal solution.For example, let's say that someone must get from point A to point B as fast as possible, in a given city, during rush hour. A dynamic programming algorithm will look at finding the shortest paths to points close to A, and use those solutions to eventually find the shortest path to B. On the other hand, a greedy algorithm will start them driving immediately and will pick the road that looks the fastest at every intersection. This strategy might not lead to the fastest arrival time, since it might take some \"easy\" streets that lead to a gridlock.Sometimes, applying memoization to a naive basic recursive solution already results in a dynamic programming solution with asymptotically optimal time complexity; however, the optimal solution to some problems requires more sophisticated dynamic programming algorithms. Some of these may be recursive as well but parametrized differently from the naive solution. Others can be more complicated and cannot be implemented as a recursive function with memoization. Examples of these are the two solutions to the Egg Dropping puzzle below.".
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