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Whether you’re a computer programming student or a data analyst, understanding the power of dynamic programming can help you apply basic mathematical concepts to optimize the solving speed of more intricate computations.

This algorithmic technique makes solving complicated mathematical problems much faster and easier by combining recursive operations with memorization. In doing so, programmers can use previously-solved problems to evaluate increasingly complex calculations more efficiently.

So, what is dynamic programming?

In this article, we will use examples to make this rather abstract concept more tangible by exploring two classic mathematical problems that require recursive solutions and how this clever programming strategy simplifies their calculation.

Let’s get started!

• What is dynamic programming?
• Is dynamic programming the same as recursion?
• Is there a connection of dynamic programming to Python?
• Basic concepts of dynamic programming
• Real-world applications
• Main takeaways

## What is Dynamic Programming?

Dynamic programming means using an algorithmic programming strategy that breaks down problems into sub-problems, then solving each individually and saving the results in a data structure such as an array.  This allows future instances of the same problem to be solved more quickly, as the data structure already contains solutions to prior problems.

The programming technique is often used to solve problems where the same solutions can be reused in different contexts. It also works well for problems with an overlapping substructure – meaning, a solution to one part of the problem could help you find the solution to another part. Dynamic programming means using an algorithmic programming strategy that breaks down problems into sub-problems, then solving each individually and saving the results in a data structure such as an array.

## Is Dynamic Programming the Same as Recursion?

Recursion is a programming technique that involves breaking down complex functions into simpler ones, which are repeated until a result is obtained. On the other hand, dynamic programming combines recursion with memorization; it solves sub-problems once and stores them in memory for later use.

The difference between recursion and dynamic programming lies in how they store solutions to sub-problems. While a recursive algorithm relies on multiple executions of the same function, dynamic programming utilizes only one execution and stores the result for future use.

## Is There a Connection of Dynamic Programming to Python?

Yes, dynamic programming can be used in Python. In fact, the popular language is a great choice for exploring this concept. It offers easy-to-understand syntax and tools to make the implementation of algorithms like dynamic programming simpler than ever before.

Python also offers libraries such as NumPy which allow for more efficient computations to facilitate solving problems using dynamic programming methods. ## Basic Concepts of Dynamic Programming

If this is your first time exploring this concept, it might feel a bit overwhelming, but we’ll show you two examples that will help you apply the two fundamental principles more concretely. These two basic principles include overlapping subproblems and optimal substructures.

Here is a comprehensive analysis:

### Overlapping Subproblems

When a particular mathematical solution involves solving the same problem multiple times, it has overlapping subproblems.

Dynamic programming optimizes data analysis by eliminating the process of calculating the same problem repeatedly by computing the solution once, storing the result, then reusing it as needed.

### Optimal Substructure

For dynamic programming to work, the problem must be such that it can be broken down into subproblems, and the solution to the whole problem is a combination of the solutions for each subproblem.

In that case, the initial problem has an optimal substructure.

### Example #1: The Fibonacci Sequence

The Fibonacci sequence is a frequently found pattern in nature and a mathematical concept that consists of a series of numbers in which each number is the sum of the two numbers before it. The first ten digits are 1, 1, 2, 3, 5, 8, 13, 21, 34, and 55, in which 1 + 1 = 2, 1 + 2 = 3, 2 + 3 = 5, and so forth.

While solving the Fibonacci sequence with small numbers is simple, it gets more complicated as you move into larger and larger digits.

Dynamic programming, however, makes the process more straightforward by saving all previous computations and using them to calculate each proceeding one ad infinitum.

### Example #2: Change-Making Problem

Like the Fibonacci sequence, the Change-Making Problem is a mathematical and programming concept that requires figuring out the minimum number of coins of a particular denomination needed to make a given change.

So, for example, it might ask, “What is the minimum number of nickels, dimes, and pennies a cashier could use to give a customer exactly \$0.36 in change?”

Typically, calculating the solution would require multi-step computations in which the solver multiplies various coins based on each denomination’s maximum total value without exceeding the maximum amount of change.

Dynamic programming eliminates that recursive process by storing the minimum number of coins needed for each smaller amount, then using it to solve more complex problems–  such as \$115.15 in change using quarters, nickels, dimes, and pennies– quickly and efficiently based on each previous problems’ solution.

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Because dynamic programming allows for more efficient problem-solving, it has many real-world applications in any industry that deals with complex, recursive data analysis.

We’ll look at computer science, economics, and business examples.

### Computer Science

Whenever you use a word processing program, you watch dynamic programming in action as the platform organizes the content into neat lines.

Behind the scenes, word processors must constantly decide where to break lines to fill the most space without splitting a word into two parts.

Without dynamic programming, that process would require evaluating every possible place to break the line, then picking the one that looks best. Instead, they use a shortcut that remembers the best breakpoint after each word typed previously, then apply that same rule to every subsequent line.

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### Economics

Economists frequently rely on dynamic programming to optimize financial portfolios by evaluating the best possible assets to invest in to maximize their returns while minimizing risk.

Based on the results of the analysis, they can then adjust the return-to-risk ratio efficiently because the algorithm uses the subproblem calculations to solve the full array of possible asset combinations.

Businesses can use dynamic programming to determine the best way to allocate their resources when deciding specific strategies for maximizing profits while minimizing costs.

For example, a factory that produces “widgets” and “gadgets” can analyze production costs and available inventory to decide the maximum number of each product they can manufacture without requiring additional raw materials.

## Main Takeaways

Now that you know the answer to “What is dynamic programming?” you can simplify even the most cumbersome computations, making it far more straightforward to find optimal solutions without waiting for the code to evaluate every possible outcome individually.

Give it a try with one of the above problems, and once you’ve got it figured out, use it to optimize your programming projects.

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