【DP Album】ACM Dynamic Programming Summary

Source blog link: http://blog.csdn.net/cc_again/article/details/25866971

Dynamic programming ( DP ) is a method used in mathematics , computer science and economics to solve complex problems by decomposing the original problem into relatively simple sub-problems. Dynamic programming is often suitable for problems with overlapping subproblems and optimal substructure properties, and the time consumption of dynamic programming methods is often much less than that of naive solutions.

The basic idea behind dynamic programming is very simple. Roughly speaking, to solve a given problem, we need to solve its different parts (i.e. sub-problems), and then combine the solutions of the sub-problems to arrive at the solution of the original problem. Often many sub-problems are very similar, and for this reason dynamic programming attempts to solve each sub-problem only once, thus reducing the amount of computation: Once the solution to a given sub-problem has been solved, it is memoized so that the same sub-problem is needed next time Check the table directly when solving. This practice is especially useful when the number of repeated subproblems grows exponentially with the size of the input.

Dynamic programming problems satisfy three important properties

Optimal substructure property: If the solution of the subproblems contained in the optimal solution of the problem is also optimal, we say that the problem has the optimal substructure property (that is, it satisfies the optimization principle). The optimal substructure properties provide important clues for dynamic programming algorithms to solve problems.

Sub-problem overlapping property: The sub-problem overlapping property means that when a recursive algorithm is used to solve the problem from top to bottom, the sub-problems generated each time are not always new problems, and some sub-problems will be recalculated many times. The dynamic programming algorithm takes advantage of the overlapping nature of this sub-problem. It only calculates each sub-problem once, and then saves its calculation results in a table. When the calculated sub-problem needs to be calculated again, it is only in the table. Simply look at the results to get high efficiency.

No aftereffect :After the stages are arranged in a certain order, for a given stage state, the state of the previous stages cannot directly affect its future decision-making, but only through the current state. In other words, each state is a complete summary of past history. This is no retrospective, also known as no retrospective.

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There are many division methods for dynamic programming classification, and many online are divided according to the state, divided into one-dimensional, two-dimensional, interval, tree and so on. I think it is better to classify the problem according to the type and difficulty of the problem solved by the function. According to my own understanding and induction, the classification of dynamic programming is as follows:

1. Simple basic dp

This type of dp is mainly because some states are easier to represent, the transition equation is easier to think, and the problems are relatively common. It mainly includes recursion , knapsack , LIS (longest increasing sequence) , LCS (longest common subsequence) . For these types, we recommend some good learning materials and topics.