#### Dynamic Programming and Methods

**Dynamic Programming and Methods**

**Dynamic programming and its methods** are a way of solving complex and multi-process problems by breaking them down into repetitive, sub-problems, solving each sub-problem once and then finding an easier solution. When a complex problem is solved by breaking it down into sub-problems, time is saved. However, more free space is required. In short, it saves time by sacrificing some of the storage space. This method is used to solve most optimization problems. It also saves code costs by eliminating the effort of recalculating the same operations.

Another answer to the question of **what is dynamic programming** is to find the shortest path from one place to another on a line by recursion technique. Problems in various algorithms are easily and quickly solved using dynamic programming methods. This method solves complex problems with inductive and deductive approaches and techniques. In this way, the problem of making a complex problem more understandable is eliminated. Problems are solved without changing their characteristics. In this way, it is ensured that the desired problem is solved completely without deviating from other issues.

In the **dynamic programming method**, the optimal solution is to find the optimal solution for the main problem by successively solving sub-problems that are independent of the initial problem. In other words, the sub-problems are solved first and the other main problem is solved in the light of the data obtained from the sub-problems. In this way, problems are solved much faster and more practically by obtaining a whole from small parts.

**Dynamic Programming Methods and Usage**

**Dynamic programming methods** are used to solve problems using various methods. The method of solving the main problem by finding the solutions to small problems and obtaining metadata is called memoization. In this method, analysis is done by starting from the top data and going down. Going from the lowest parts to the top data and solutions is called tabulation. Thanks to these methods, the work to be done and the problems to be solved become much easier.

Dynamic programming memoizes the previous operation in the storage system while the operations are in progress thanks to memoization. In this way, each operation is repeated once. This saves time and labor. Thanks to this caching process, computer programs run faster and more efficiently. In direct proportion, this makes the computer run more optimized. This is also known as **tabulation** in dynamic program languages.

A cached function remembers the results based on the specific input data coming into the system and presents **the remembered result** as data instead of having to re-analyze and calculate it. This memoization saves a lot of time and effort. Although memory usage increases, computers work much faster thanks to the optimization gained. This speed to be obtained is very beneficial in terms of financial gain in the areas where it is used. All functions in computers are in computational complexity in terms of time and space. In addition to these operations, it facilitates the solution with the addition operation it uses instead of the complex multiplication operations used in problem solving.

Another resolution method is **the chain matrix multiplication** operation. Within these operations, the priority of the order of operations is to solve the problem. It is possible to arrive at a solution of this problem with the **dynamic programming method**. These matrices can be solved by multiplying them in binary by means of mathematical operations. However, this is a very long and difficult way. This slows down computer systems considerably. Dynamic programming systems solve this problem by using a tabulation system to optimize the operation of the computer. This system quickly solves the problem of bracketing and operation priority.

**Types and Properties of Dynamic Programming Methods**

**Dynamic programming** can be used to solve any problem that can be divided into smaller sub-problems. It is possible to examine dynamic programming methods under two different headings. The underlying difference is the type of transformation functions. In the problem solving process, if the operations can be solved in a stochastic manner, there is a problem of deterministic character.

In another case, problems with a dynamic structure may be encountered. The solutions that emerge at the end of this process are also finite. In other words, these problems have certain solutions that are already known. Some processes are **infinite**. The solutions of these processes continue infinitely in the same way. The types mentioned above are as follows:

- A
**deterministic**dynamic programming system is one in which the outcome and what happens in the next stage are always determined by external factors. In other words, all stages of the problem solving process are completely dominated by external factors.

**Stochastic**(probabilistic) dynamic programming system; in such systems, the next step and immediate situations cannot be known or determined by external factors. The only thing known about what will happen in the next steps is probabilities. This probability distribution is distributed in a way that can be known or determined by external factors.

All these methods are processes that enable problems to be solved with **a single method**. All problems are analyzed one by one and the solution and equation of all of them are established. In terms of multi-stage decision-making problems, creating a single problem and finding the appropriate solution to it is done through these systems. The periods that occur in this solution process are continuously interconnected and as a result can be used to solve another problem. Some features of these systems are as follows:

- Problems are broken down into as many pieces as the person managing the system wants. In this way, the desired optimization is achieved.

- Each solution phase should have its own specific solution. This is important for solutions that are not connected to other sub-problems.