Also referred to as DP, dynamic programming is a powerful technique for solving complex problems in computer science, engineering, and mathematics.
While it has a reputation for being intimidating, learning dynamic programming is much easier once you understand the basics.
In the simplest terms, dynamic programming is a method of solving problems by breaking them down into smaller sub-problems and reusing the solutions to these sub-problems to build up to the solution for the original problem.
This approach allows us to solve problems much more efficiently.
Dynamic programming techniques can be used to solve a wide range of problems, from finding the shortest path in a graph to solving mathematical optimization problems.
In this introduction to dynamic programming, we’ll explore dynamic programming basics like what it’s used for, steps in the process, and the different dynamic programming techniques, types, and algorithms that make it happen.
Plus, we’ll look at some real-world dynamic programming examples, compare it to other programming methods (like dynamic programming vs recursion and greedy algorithms), and share tips on how to learn dynamic programming yourself!
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What Is Dynamic Programming?
🎮 Think about it like playing a video game.
In a video game, you often have to solve small problems to progress to the next level. Sometimes you encounter a problem that you’ve already solved before—so instead of going through the whole process again, you use what you learned from the last time to solve it faster.
Dynamic programming is similar. It helps you find the solution to a big problem by breaking it down into smaller, easier-to-solve problems.
You’ll solve those problems first, then take what you’ve learned from designing those solutions and apply that knowledge to solve the bigger one.
This way, you don’t have to start from scratch, and the original problem feels less insurmountable.
🌟 As this tweet explains: “It’s like having a time machine for your code; it helps you travel back to the simpler version of a problem to solve it more efficiently.”
🌟 This Redditor also has a good way of explaining what dynamic programming is: “Say I put a bunch of matches down on the table, and ask you to count them. After a few moments you say ‘12,’ then I add another match and say ‘How many are there now?’ You wouldn’t have to count them again to tell me there are 13. You already knew there were 12, and I’ve added one more. DP is this. Each individual subtask is ‘counting a match,’ and you are memorizing the previous result to get the next result, in this case, 12 matches, to count to 13 matches.”
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Dynamic programming vs dynamic programming languages
You might assume that a dynamic programming language is just a coding language used for DP. However, while dynamic programming and dynamic programming languages share the term “dynamic,” they are actually not directly related to each other.
Dynamic programming, as you’ve just learned, is a CS/mathematical optimization technique.
Don’t get confused wondering what the connection is: as far as tech concepts go, they’re not in the same category.
What Is Dynamic Programming Used For?
Dynamic programming is commonly used for solving optimization problems, where the goal is to find the best possible solution given certain constraints.
This obviously becomes useful in a wide range of applications, such as:
- 🧮 Combinatorial mathematics: DP is used for optimization problems such as the traveling salesman problem or the knapsack problem.
- 🧬 Bioinformatics: DP can help align sequences of DNA or protein, where the goal is to find the optimal match between two sequences.
- 📅 Management & operations: Dynamic programming can be leveraged to help ops teams and managers allocate resources efficiently (eg in scheduling).
- 👁️ Computer vision and image processing: It can be used in applications to solve problems such as image segmentation and pattern recognition.
- 🤖 Natural language processing applications: DP can assist with processes like speech recognition and machine translation, to find the optimal solution for problems including sequence alignment and language modeling.
Steps in the Dynamic Programming Process
When you boil down dynamic programming basics to their simplest forms, it doesn’t seem so difficult! There are essentially just five steps:
- Identify sub-problems: The first step is to identify the smaller sub-problems that make up the larger problem.
- Store solutions to sub-problems: Create an array or matrix to store the solutions to each sub-problem.
- Solve sub-problems: Solve each sub-problem, one at a time, and store that solution in the array or matrix.
- Use stored solutions: Use the stored solutions to build up to the solution for the original problem.
- Optimize the solution: Finally, optimize the solution by removing any redundant computations and ensuring that it is as efficient as possible.
Top-Down vs Bottom-Up Dynamic Programming
Bottom-up and top-down dynamic programming are two different ways to solve problems using programmers DP.
Think of them as two different ways to put together a puzzle.
🖼️ In top-down dynamic programming (also known as “memoization”), the algorithm starts by solving the original problem and then recursively breaks it down into smaller sub-problems. You then solve each small piece, one at a time, until the entire puzzle is solved. If you compare it to a jigsaw puzzle, it’s like looking at the box as you put it together.
🧩 Bottom-up dynamic programming (also called “tabulation”) is like starting with the small puzzle pieces and putting them together, piece by piece, until you have the big picture. You start by solving the smallest, easiest sub-problems first, and then use that information to solve bigger and bigger problems, until you’ve solved the whole thing.
Both top-down and bottom-up dynamic programming can be useful, depending on the problem you’re trying to solve. The choice between the two often depends on the size of the problem and how well it can be broken down into smaller parts.
Dynamic Programming Examples: 4 Problems You Can Solve with DP
Let’s look at four specific dynamic programming examples to see how DP can be put into practice to solve CS and mathematical problems!
1. Longest Common Subsequence
This algorithm is used to find the longest sequence of characters that are common to two or more strings.
For example, if you have two strings “ABCD” and “ACDF”, the longest common subsequence is “ACD”.
Breaking it down into sub-problems with dynamic programming helps you solve the LCS problem much more efficiently.
2. Fibonacci Sequence
Fibonacci is a famous sequence of numbers, where each number is the sum of the two previous numbers. The sequence starts with 0, 1, 1, 2, 3, 5, 8, 13, 21, and so on.
To find the nth Fibonacci number, you can use dynamic programming to store the solutions to the sub-problems (e.g. 0+1, 1+2) and build up to the solution for the nth number.
3. Floyd-Warshall algorithm
Used to find the shortest paths between all pairs of vertices in a weighted graph. The algorithm works by breaking the problem down into smaller sub-problems and storing the solutions to these sub-problems so they can be reused later.
When applied to the Floyd-Warshall algorithm, dynamic programming techniques allow you to solve the problem more efficiently and avoid redundant computations by reusing solutions to sub-problems.
4. Bellman Ford algorithm
Used to find the shortest path between a source vertex and all other vertices in a weighted graph. The algorithm works by relaxing the edges of the graph in a specific order and updating the distances between the vertices until the optimal solution is found.
The Bellman-Ford algorithm uses dynamic programming by breaking the problem down into smaller sub-problems and updating the solution for each sub-problem in each iteration of the algorithm.
The algorithm stores the solution for each sub-problem in an array and uses this information to ultimately generate the solution for the original problem.
Dynamic Programming vs Recursion, Greedy Algorithms, & Static Languages
To understand dynamic programming better, it can be helpful to compare and contrast it with other techniques.
Let’s look at recursion, the greedy algorithm vs dynamic programming, and whether there’s such a thing as “static vs dynamic programming.”
Dynamic programming vs recursion
What are some of the similarities and differences in dynamic programming vs recursion?
In computer programming, recursion is when a function calls itself.
Instead of writing a long list of steps to solve a problem, you break the problem down into smaller parts, and have the function solve each smaller part one at a time, until the problem is fully solved.
Sound similar to dynamic programming? That’s because it is similar in a few key ways: 👇
- Both dynamic programming and recursion involve breaking down a problem into smaller sub-problems.
- Both techniques can involve solving the sub-problems recursively (applying the same algorithm to each sub-problem until it’s able to return a value).
- Both techniques can be used to solve problems that would otherwise be difficult or impossible to solve with other methods.
So, let’s highlight a few of the important differences: 👇
- Dynamic programming involves storing the solutions to sub-problems in memory and reusing them later, while recursion does not. This can make DP more efficient.
- They are typically used for different purposes. DP is often used for optimization problems (where the goal is to find the best solution among a set of possible solutions). Recursion is more commonly used for searching or traversal problems (where the goal is to explore a problem in a systematic way and find multiple paths).
- Recursion can use up a lot of memory if it creates too many function calls, potentially causing a stack overflow error. Dynamic programming, on the other hand, is designed to minimize the number of computations needed to reach a solution.
Whether dynamic programming or recursion is “better” depends on the specific problem being solved. In general, it’s a good idea to consider both techniques and choose the one that is best suited to the problem at hand.
Greedy algorithm vs dynamic programming
Greedy algorithms are yet another problem-solving technique that involves breaking problems down into sub-problems. However, the key difference in greedy algorithm vs dynamic programming concepts lies in how it finds solutions.
A greedy algorithm doesn’t consider the past or future—it simply wants instant gratification. At each step, the algorithm selects the best option available to it at that point in time, without considering previous sub-problems or the overall solution.
Dynamic programming is much more well-rounded. It reuses the knowledge from prior solutions as it builds towards the larger solution.
A greedy algorithm:
- Makes locally optimal choices at each step in the hope of finding a global optimum
- Can’t always guarantee the optimal solution
- Doesn’t revisit previously solved sub-problems
- May be faster than dynamic programming algorithms for some problems
- Solves problems by breaking them down into smaller sub-problems, solving each sub-problem once, and storing the results to avoid redundant calculations
- Can guarantee the optimal solution for a wide range of problems
- May be slower than greedy algorithms for some problems
Ultimately, if speed is your #1 concern, you might decide to use a greedy algorithm. If you want the confidence that you’re getting the best solution, DP is a better choice.
Dynamic vs static programming
Earlier, we explained that dynamic programming and dynamic programming languages are distinct concepts with different meanings and applications. Similarly, “static programming” isn’t really a thing, but “static programming languages” are.
Since this isn’t actually related to DP the computer science technique, let’s just quickly cover static vs dynamic programming languages on a high level!
When you’re working with a static programming language (like C, C++, Java, and Pascal), you’re essentially writing a set of instructions that don’t change, no matter what inputs you have.
It’s like following a recipe to bake a cake. You have all the ingredients and steps laid out in front of you, and you follow them exactly as written to get the same result every time.
When you write a program with a dynamic programming language, it can adjust and change the way it solves a problem based on the input it receives.
Continuing with our kitchen example, it’s more like cooking by experimenting and adjusting as you go. Instead of following a set recipe, you make changes and adapt based on what you learn as you cook.
When choosing whether to use a static or dynamic programming language for a project, programmers have to weigh considerations like performance, reliability, maintainability, ease of development, and scalability.
Why Learn Dynamic Programming?
If all of this information is hard to wrap your head around at first, it might feel hard to work up the motivation necessary for learning dynamic programming.
To help with the motivation aspect, what are some of the benefits for you if you learn it?
Learning dynamic programming can help you:
- Improve your problem-solving skills: Dynamic programming is a powerful technique for solving complex problems. Learning it can help you become a better problem-solver.
- Work more efficiently: By breaking down a problem into smaller sub-problems and reusing the solutions to these sub-problems, dynamic programming can reduce the amount of work required to solve the original problem.
- Advance your career: Dynamic programming is widely used in computer science and engineering. Having a strong understanding of it can be beneficial in many careers, such as software development, data analysis, and artificial intelligence.
- Challenge your brain: Learning dynamic programming concepts and skills can be a mentally challenging and rewarding experience. The process of breaking down a problem into smaller sub-problems, solving each sub-problem, and building up to the solution for the original problem can be both intellectually stimulating and satisfying.
Even just familiarizing yourself with dynamic programming basics can be helpful—you don’t have to be an expert right away.
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How to Learn Dynamic Programming
If you’re interested in getting a more detailed introduction to dynamic programming, here’s how to pursue further learning!
1. Start with the basics
Begin by studying the basics of algorithms and data structures, including arrays, matrices, and graphs. A strong understanding of these concepts will be essential for understanding dynamic programming.
2. Study examples of dynamic programming problems and solutions
Go through dynamic programming examples like the ones mentioned above—finding the longest common subsequence, computing Fibonacci numbers, and solving the knapsack problem.
This will help you understand how dynamic programming is used to solve real problems and give you hands-on practice.
3. Watch tutorials and take online courses
One of the best things you can do to teach yourself any new skill, of course, is to learn from the experts!
Watch online tutorials and take courses on dynamic programming. This will give you a deeper understanding of dynamic programming patterns and help you learn how to solve dynamic programming problems (from the simple to the complex).
Some of our favorite resources to learn dynamic programming include:
- Master the Art of Dynamic Programming on Udemy: Teaches you how to solve any dynamic programming problem, step by step.
- Greedy Algorithms, Minimum Spanning Trees, and Dynamic Programming on Coursera: Covers greedy algorithms (scheduling, minimum spanning trees, clustering, Huffman codes) and dynamic programming (knapsack, sequence alignment, optimal search trees).
- Tushar Roy’s playlist on YouTube: It walks you through a ton of dynamic programming problems. Great for visual learners.
4. Practice, practice, practice
Finally, practice as much as you can. The more you practice, the better you will become at using dynamic programming to solve problems.
Take It Slow & Steady to Learn Dynamic Programming Basics
Learning dynamic programming takes time and practice, so be patient and persistent. With dedication and effort, you can become an expert in this powerful problem-solving technique.
Start with free videos and articles, move on to courses, and start looking for ways to use dynamic programming in your day-to-day—you’ll be a pro in no time.