Dynamic

In mathematics, computer science, economics, and bioinformatics, dynamic programming is a method for solving complex problems by breaking them down into simpler subproblems. It is applicable to problems exhibiting the properties of overlapping subproblems and optimal substructure (described below). When applicable, the method takes far less time than naive methods that don't take advantage of the subproblem 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 (subproblems), then combine the solutions of the subproblems to reach an overall solution. Often when using a more naive method, many of the subproblems are generated and solved many times. The dynamic programming approach seeks to solve each subproblem only once, thus reducing the number of computations: once the solution to a given subproblem 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 subproblems 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.

Sometimes, applying memoization to a naive basic recursive solution already results in an optimal dynamic programming solution; however, many problems require 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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