Dynamic Programming 101 | Types, Examples, and Use-Cases
Say you're planning a road trip across the country. You've got a list of cities, but you can't decide on the best route. You want to complete the trip without wasting time driving back and forth. Here's how you could plan your trip in a better way:
Dynamic programming is one of the finest ways to solve a class of problems with sub-problems. Did that sound difficult to understand? Dive in to learn all about it with clear concepts and examples.
Dynamic programming (often abbreviated as DP) is a powerful method for solving complex algorithmic problems by breaking them down into smaller, overlapping subproblems. Instead of re-calculating solutions from scratch, the results of these subproblems are calculated once and saved a core concept known as memoization or tabulation.
If you are learning dynamic programming for beginners, abstract definitions can feel overwhelming. Let's look at a practical, real-world example to see how dynamic programming works in action.
Imagine planning a multi-city cross-country road trip. You want to find the absolute shortest route from your starting location to your final destination, passing through several intermediate stopovers.
Here is how a dynamic programming approach solves this route selection problem:
Identify the overlapping subproblems: The overall journey consists of smaller legs between intermediate cities. Rather than recalculating every possible full-length route across the entire map, you break the trip down into point-to-point sub-routes.
Solve and store intermediate calculations: Using a navigation system, you calculate the shortest distance between intermediate stops (e.g., City A to City B) and store that distance in a table for future lookup.
Combine subproblem solutions into an optimal path: When determining the best overall route, you retrieve the previously saved shortest legs rather than recalculating them. Combining these optimal sub-solutions gives you the most efficient full journey without redundant work.
This step-by-step approach breaking down complex problems, solving each sub-part once, storing the intermediate outputs, and reassembling them forms the foundation of dynamic programming in technical interviews and software engineering.
Table of Contents:
When to use dynamic programming
What is Dynamic Programming
Dynamic Programming (DP) is an algorithmic optimization technique used to solve complex algorithmic problems by breaking them down into simpler, overlapping subproblems. First introduced by mathematician [suspicious link removed] in the 1950s, DP operates on a fundamental principle: never solve the same subproblem twice.
By calculating each subproblem once and storing its output in memory using memoization (top-down) or tabulation (bottom-up), dynamic programming for beginners eliminates redundant recursive computations reducing execution time from exponential $O(2^n)$ down to polynomial $O(n)$ time.
Think of dynamic programming as recursion with a memory bank. Instead of recalculating identical recursive paths over and over, DP saves time by trading a small amount of memory space to instantly look up previously computed sub-solutions
When to use Dynamic Programming?
So, how do you know when to use dynamic programming? Here is how a dynamic programming approach solves this route selection problem:
- Identify Overlapping Subproblems: The overall journey consists of smaller legs between intermediate cities. Rather than recalculating every possible full-length route across the entire map, you break the trip down into point-to-point sub-routes.
- Solve and Store Intermediate Calculations: Using a navigation app, you calculate the shortest distance between intermediate stops (e.g., City A to City B) and store that distance in a lookup table.
- Combine Solutions into an Optimal Path: When determining the best overall route, you retrieve previously saved shortest legs rather than recalculating them from scratch. Combining these optimal sub-solutions gives you the most efficient full journey.
Practical Application: The Fibonacci Sequence
To really get a grip on dynamic programming, let's explore a classic example: The Fibonacci sequence.
It is a series of numbers in which each number is the sum of the two preceding ones, usually starting with 0 and 1.
Fibonacci Series: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34…and so on.
Mathematically, we could write each term using the formula:
F(n) = F(n-1) + F(n-2),
With the base values F(0) = 0, and F(1) = 1. And we’ll follow the above relationship to calculate the other numbers. For example, F(6) is the sum of F(4) and F(5), which is equal to 8.
Let’s look at the diagram for better understanding.
Suppose we’ve to calculate F(10). Going by the formula, F(10) should be the sum of F(8) and F(9). Similarly, F(9) would also be the sum of the subproblems F(7) and F(8). As you can see, F(8) is an overlapping subproblem here.
In the above example, if we calculate the F(8) in the right subtree, then it would result in a increased usage of resources and reduce the overall performance.
The better solution would be to store the results of the already computed subproblems in an array. First, we’ll solve F(6) and F(7) which will give us the solution to F(8) and we’ll store that solution in an array and so on. Now when we calculate F(9), we already have the solutions to F(7) and F(8) stored in an array and we can just reuse them. F(10) can be solved using the solutions of F(8) and F(9), both of which are already stored in an array.
Similarly, at each iteration we store the solutions so we don’t have to solve them again and again. This is the main attribute of dynamic programming.
If you try to compute this sequence with a straightforward recursive function, you'll end up doing a lot of unnecessary work. (Want to understand recursion from scratch?)
Here's a simple Python implementation using DP to calculate a Fibonacci sequence:
def fibonacci(n):
# Create an array of size (n+1) to store the computed values
dp = [0, 1] + [0] * (n - 1)
for i in range(2, n + 1):
# Compute the ith Fibonacci number
dp[i] = dp[i - 1] + dp[i - 2]
return dp[n]
print(fibonacci(10)) # Outputs: 55
Step-by-Step Approach to DP
Let's explore how to implement dynamic programming step-by-step:
- Grasp the Problem
- Find the Overlapping Subproblems
- Compute and Store Solutions
- Construct the Solution to the Main Problem
Types of Dynamic Programming
Dynamic programming is divided into two main approaches: top-down (memoization) and bottom-up (tabulation). Both of these methods help in solving complex problems more efficiently by storing and reusing solutions of overlapping subproblems, but they differ in the way they go about it.
Let's dive into these two approaches:
Top-Down DP (Memoization)
In the top-down approach, also known as memoization, we start with the original problem and break it down into subproblems. Think of it like starting at the top of a tree and working your way down to the leaves.
Here, problems are broken into smaller ones, and the answers are reused when needed. With every step, larger, more complex problems become tinier, less complicated, and, thus, faster to solve, and the results of each subproblem are stored in a data structure like a dictionary or array to avoid recalculating them. The ‘memoization’ (a key technique in DP where you store and retrieve previously computed values) process is equivalent to adding the recursion (any function that calls itself again and again) and caching steps.
Some parts can be reused for the same problem and solved when requested, making them easier to debug. However, this approach results in more memory in the call stack being occupied, which can result in a reduction in overall performance and stack overflow.
Let's revisit the Fibonacci sequence example:
def fibonacci(n, memo = {}):
if n <= 2:
return 1
elif n in memo:
return memo[n]
else:
memo[n] = fibonacci(n-1, memo) + fibonacci(n-2, memo)
return memo[n]
print(fibonacci(10)) # Outputs: 55
Here, memo is a dictionary that stores the previously computed numbers. Before we compute a new Fibonacci number, we first check if it's already in memo. If it is, we just return the stored value. If it's not, we compute it, store it in memo, and then return it.
Bottom-Up DP (Tabulation)
The bottom-up approach, also known as tabulation, takes the opposite direction. This approach solves problems by breaking them up into smaller ones, solving the problem with the smallest mathematical value, and then working up to the problem with the biggest value. Solutions to its subproblems are compiled in a way that falls and loops back on itself. Users can opt to rewrite the problem by initially solving the smaller subproblems and then carrying those solutions for solving the larger subproblems.
Here, we fill up a table (hence the name "tabulation") in a manner that uses the previously filled values in the table. This way, by the time we come to the problem at hand, we already have the solutions to the subproblems we need.
Let's use the Fibonacci sequence again to illustrate the bottom-up approach:
def fibonacci(n):
fib_table = [0, 1] + [0]*(n-1)
for i in range(2, n+1):
fib_table[i] = fib_table[i-1] + fib_table[i-2]
return fib_table[n]
print(fibonacci(10)) # Outputs: 55
In this case, fib_table is an array that stores the Fibonacci numbers in order. We start by filling in the first two numbers (0 and 1), and then we iteratively compute the rest from these initial numbers.
In contrast to the top-down approach, the bottom-up approach relies on eliminating recursion functions. There is no stack overflow, and memory space is saved with reduced timing complexity, making it more efficient and preferred when the order of solving subproblems is not critical.
Which approach to choose?
Both top-down and bottom-up dynamic programming can be useful, and your choice depends on the problem at hand and the specific requirements of your program.
The top-down approach might be easier to understand because it follows the natural logic of the problem, but it can involve a lot of recursion and may have a larger memory footprint due to the call stack.
On the other hand, the bottom-up approach can be more efficient because it avoids recursion and uses a loop instead, but it might require a better understanding of the problem to build the solution iteratively.
What are the signs of DP suitability?
Identifying whether a problem is suitable for solving with dynamic programming (DP) involves recognizing certain signs or characteristics that suggest DP could be an effective approach. Here are some common signs that indicate a problem might be a good fit for dynamic programming:
- Overlapping Subproblems: A problem that can be broken down into smaller subproblems that are solved independently, and the same subproblems encountered multiple times strongly indicates DP suitability.
- Optimal Substructure: Problems that exhibit optimal substructure can often be solved using DP. This means that the optimal solution for a larger problem can be constructed from the optimal solutions of its smaller subproblems.
- Recursive Nature: Problems that can be naturally expressed using recursion are often well-suited for DP.
- Memoization Opportunities: If you notice that you can improve a recursive algorithm by memoizing (caching) intermediate results, DP might be a good fit.
- Sequential Dependencies: Problems where the solution depends on the results of previous steps or stages are often candidates for DP. DP is particularly useful when solving problems involving sequences, such as strings, arrays, or graphs.
- Optimization or Counting: DP is often applied to optimization problems (maximizing or minimizing a value) or counting problems (finding the number of possible solutions).
- Recursive Backtracking Inefficiency: If you encounter a recursive backtracking algorithm that is slow due to repeated calculations, this is a clear indication that DP might be a better approach.
- Subproblem Independence: Some problems have subproblems that are entirely independent of each other. In such cases, DP can be applied to solve each subproblem in parallel or any order, making it an efficient choice.
- Limited Set of Choices: Problems where the number of choices at each step is limited and doesn't grow exponentially can often be tackled with DP. DP can explore all possible choices without leading to an impractical number of computations.
Top Dynamic Programming Patterns for Technical Interviews
Mastering dynamic programming for beginners becomes manageable when you focus on core patterns rather than memorizing individual solutions:
1. 0/1 Knapsack Pattern
- Concept: Given items with weights and values, decide whether to include or exclude each item to maximize value within a weight limit.
- Practice Problems:
2. Longest Common Subsequence (LCS)
- Concept: Find the length of the longest sequence present in two strings in the same relative order.
- Practice Problems:
3. Longest Increasing Subsequence (LIS)
- Concept: Identify the longest strictly increasing subsequence within a given numeric array.
- Practice Problems:
Common Beginner Interview Questions
- Climbing Stairs (LeetCode 70): Path counting using a Fibonacci pattern.
- Coin Change (LeetCode 322): Unbounded Knapsack variant for coin minimization.
- House Robber (LeetCode 198): Decision-making with non-adjacent constraints.
Final Thoughts
Dynamic programming is a little like magic: It turns a daunting problem into a series of manageable tasks, making the impossible possible. But unlike a magic trick, the method behind dynamic programming is logical and grounded in sound reasoning.
Sure, getting the hang of it might take some time. You'll need to practice spotting overlapping subproblems and constructing optimal solutions. But once you've mastered these skills, you'll be able to tackle a wide range of problems with newfound efficiency.
Dynamic programming is a useful but advanced skill to learn if one is a programmer or DevOps engineer, particularly if you specialize in Python. It makes complex algorithmic problems easy to digest and its versatility makes it a must-have in the repertoire of every DevOps learning kit. Remember, the journey of a thousand miles begins with a single step – or in our case, a single subproblem.
Cheers and Happy Coding!
FAQs on Dynamic Programming
When should I use Dynamic Programming?
Use Dynamic Programming when you encounter problems with overlapping subproblems and optimal substructure. Common applications include algorithms for optimization, like finding the shortest path, maximizing profit, or minimizing cost.
Are there different types of Dynamic Programming?
Yes, Dynamic Programming can be categorized into two main types: Memoization (Top-down) and Tabulation (Bottom-up). The choice between them depends on the specific problem and your coding preferences.
What is the difference between recursion and dynamic programming?
Recursion breaks a problem into subproblems by re-evaluating them every time. Dynamic programming optimizes recursion by caching already computed subproblem solutions in memory, preventing redundant function calls.
Is memoization top-down or bottom-up?
Memoization is a top-down approach. It starts with the main target problem, breaks it down recursively, and caches the results. Tabulation is the bottom-up approach, which solves base subproblems first using iterative loops.
How do I know if a problem requires dynamic programming for beginners?
Look for two main properties: Overlapping Subproblems (the same smaller calculations repeat) and Optimal Substructure (the optimal solution to the main problem relies on optimal solutions to smaller subproblems).
What are the best strategies to learn dynamic programming for beginners?
Start by drawing recursion trees on paper to spot repeated nodes. Practice fundamental patterns such as Fibonacci, Knapsack, and Grid Travel in multiple languages, and track time and space complexity differences.
Learn more:
The Art of Debugging: Mastering the Bug Hunt, One Error at a Time
Introduction to Object-Oriented Programming
How Software is Developed? A Step-By-Step Guide