Data Structures & Algorithms
Table of content:
- What Is Data Structure?
- Key Features Of Data Structures
- Basic Terminologies Related To Data Structures
- Types Of Data Structures
- What Are Linear Data Structures & Its Types?
- What Are Non-Linear Data Structures & Its Types?
- Importance Of Data Structure In Programming
- Basic Operations On Data Structures
- Applications Of Data Structures
- Real-Life Applications Of Data Structures
- Linear Vs. Non-linear Data Structures
- What Are Algorithms? The Difference Between Data Structures & Algorithms
- Conclusion
- Frequently Asked Questions
Table of content:
- What Is Asymptotic Notation?
- How Asymptotic Notation Helps In Analyzing Performance
- Types Of Asymptotic Notation
- Big-O Notation (O)
- Omega Notation (Ω)
- Theta Notation (Θ)
- Little-O Notation (o)
- Little-Omega Notation (ω)
- Summary Of Asymptotic Notations
- Real-World Applications Of Asymptotic Notation
- Conclusion
- Frequently Asked Questions
Table of content:
- Understanding Big O Notation
- Types Of Time Complexity
- Space Complexity In Big O Notation
- How To Determine Big O Complexity
- Best, Worst, And Average Case Complexity
- Applications Of Big O Notation
- Conclusion
- Frequently Asked Questions
Table of content:
- What Is Time Complexity?
- Why Do You Need To Calculate Time Complexity?
- The Time Complexity Algorithm Cases
- Time Complexity: Different Types Of Asymptotic Notations
- How To Calculate Time Complexity?
- Time Complexity Of Sorting Algorithms
- Time Complexity Of Searching Algorithms
- Writing And optimizing An algorithm
- What Is Space Complexity And Its Significance?
- Relation Between Time And Space Complexity
- Conclusion
Table of content:
- What Is Linear Data Structure?
- Key Characteristics Of Linear Data Structures
- What Are The Types Of Linear Data Structures?
- What Is An Array Linear Data Structure?
- What Are Linked Lists Linear Data Structure?
- What Is A Stack Linear Data Structure?
- What Is A Queue Linear Data Structure?
- Most Common Operations Performed in Linear Data Structures
- Advantages Of Linear Data Structures
- What Is Nonlinear Data Structure?
- Difference Between Linear & Non-Linear Data Structures
- Who Uses Linear Data Structures?
- Conclusion
- Frequently Asked Questions
Table of content:
- What is a linear data structure?
- What is a non-linear data structure?
- Difference between linear data structure and non-linear data structure
- FAQs about linear and non-linear data structures
Table of content:
- What Is Sorting In Data Structures?
- What Is Bubble Sort?
- What Is Selection Sort?
- What Is Insertion Sort?
- What Is Merge Sort?
- What Is Quick Sort?
- What Is Heap Sort?
- What Is Radix Sort?
- What Is Bucket Sort?
- What Is Counting Sort?
- What Is Shell Sort?
- Choosing The Right Sorting Algorithm
- Applications Of Sorting
- Conclusion
- Frequently Asked Questions
Table of content:
- Understanding Bubble Sort Algorithm
- Bubble Sort Algorithm
- Implementation Of Bubble Sort In C++
- Time And Space Complexity Analysis Of Bubble Sort Algorithm
- Advantages Of Bubble Sort Algorithm
- Disadvantages Of Bubble Sort Algorithm
- Applications of Bubble Sort Algorithms
- Conclusion
- Frequently Asked Questions
Table of content:
- Understanding The Merge Sort Algorithm
- Algorithm For Merge Sort
- Implementation Of Merge Sort In C++
- Time And Space Complexity Analysis Of Merge Sort
- Advantages And Disadvantages Of Merge Sort
- Applications Of Merge Sort
- Conclusion
- Frequently Asked Questions
Table of content:
- Understanding The Selection Sort Algorithm
- Algorithmic Approach To Selection Sort
- Implementation Of Selection Sort Algorithm
- Complexity Analysis Of Selection Sort
- Comparison Of Selection Sort With Other Sorting Algorithms
- Advantages And Disadvantages Of Selection Sort
- Application Of Selection Sort
- Conclusion
- Frequently Asked Questions
Table of content:
- What Is Insertion Sort Algorithm?
- How Does Insertion Sort Work? (Step-by-Step Explanation)
- Implementation of Insertion Sort in C++
- Time And Space Complexity Of Insertion Sort
- Applications Of Insertion Sort Algorithm
- Comparison With Other Sorting Algorithms
- Conclusion
- Frequently Asked Questions
Table of content:
- Understanding Quick Sort Algorithm
- Step-By-Step Working Of Quick Sort
- Implementation Of Quick Sort Algorithm
- Time Complexity Analysis Of Quick Sort
- Advantages Of Quick Sort Algorithm
- Disadvantages Of Quick Sort Algorithm
- Applications Of Quick Sort Algorithm
- Conclusion
- Frequently Asked Questions
Table of content:
- Understanding The Heap Data Structure
- Working Of Heap Sort Algorithm
- Implementation Of Heap Sort Algorithm
- Advantages Of Heap Sort
- Disadvantages Of Heap Sort
- Real-World Applications Of Heap Sort
- Conclusion
- Frequently Asked Questions
Table of content:
- Understanding The Counting Sort Algorithm
- Conditions For Using Counting Sort Algorithm
- How Counting Sort Algorithm Works?
- Implementation Of Counting Sort Algorithm
- Time And Space Complexity Analysis Of Counting Sort
- Comparison Of Counting Sort With Other Sorting Algorithms
- Advantages Of Counting Sort Algorithm
- Disadvantages Of Counting Sort Algorithm
- Applications Of Counting Sort Algorithm
- Conclusion
- Frequently Asked Questions
Table of content:
- Understanding Shell Sort Algorithm
- Working Of Shell Sort Algorithm
- Implementation Of Shell Sort Algorithm
- Time Complexity Analysis Of Shell Sort Algorithm
- Advantages Of Shell Sort Algorithm
- Disadvantages Of Shell Sort Algorithm
- Applications Of Shell Sort Algorithm
- Conclusion
- Frequently Asked Questions
Table of content:
- What Is The Binary Search Algorithm?
- Conditions For Using Binary Search
- Steps For Implementing Binary Search
- Iterative Method For Binary Search With Implementation Examples
- Recursive Method For Binary Search
- Complexity Analysis Of Binary Search Algorithm
- Iterative Vs. Recursive Implementation Of Binary Search
- Advantages & Disadvantages Of Binary Search
- Practical Applications & Real-World Examples Of Binary Search
- Conclusion
- Frequently Asked Questions
Table of content:
- What Is Linked List In Data Structures?
- Types Of Linked Lists In Data Structures
- Linked List Operations In Data Structures
- Advantages And Disadvantages Of Linked Lists In Data Structures
- Comparison Of Linked Lists And Arrays
- Applications Of Linked Lists
- Conclusion
- Frequently Asked Questions
Table of content:
- What Is A Singly Linked List In Data Structure?
- Insertion Operations On Singly Linked Lists
- Deletion Operation On Singly Linked List
- Searching For Elements In Single Linked List
- Calculating Length Of Single Linked List
- Practical Applications Of Singly Linked Lists In Data Structure
- Common Problems With Singly Linked Lists
- Conclusion
- Frequently Asked Questions (FAQ)
Table of content:
- What Is A Linked List?
- Reverse A Linked List
- How To Reverse A Linked List? (Approaches)
- Recursive Approach To Reverse A Linked List
- Iterative Approach To Reverse A Linked List
- Using Stack To Reverse A Linked List
- Complexity Analysis Of Different Approaches To Reverse A Linked List
- Conclusion
- Frequently Asked Questions
Table of content:
- What Is A Graph Data Structure?
- Importance Of Graph Data Structures
- Types Of Graphs In Data Structure
- Types Of Graph Algorithms
- Application Of Graphs In Data Structures
- Challenges And Complexities In Graphs
- Conclusion
- Frequently Asked Questions
Table of content:
- What Is Tree Data Structure?
- Terminologies Of Tree Data Structure:
- Different Types Of Tree Data Structures
- Basic Operations On Tree Data Structure
- Applications Of Tree Data Structures
- Comparison Of Trees, Graphs, And Linear Data Structures
- Advantages Of Tree Data Structure
- Disadvantages Of Tree Data Structure
- Conclusion
- Frequently Asked Questions
Table of content:
- What Is Dynamic Programming?
- Real-Life Example: The Jigsaw Puzzle Analogy
- How To Solve A Problem Using Dynamic Programming?
- Dynamic Programming Algorithm Techniques
- Advantages Of Dynamic Programming
- Disadvantages Of Dynamic Programming
- Applications Of Dynamic Programming
- Conclusion
- Frequently Asked Questions
Table of content:
- Understanding The Sliding Window Algorithm
- How Does The Sliding Window Algorithm Works?
- How To Identify Sliding Window Problems?
- Fixed-Size Sliding Window Example: Maximum Sum Subarray Of Size k
- Variable-Size Sliding Window Example: Smallest Subarray With A Given Sum
- Advantages Of Sliding Window Technique
- Disadvantages Of Sliding Window Technique
- Conclusion
- Frequently Asked Questions
Table of content:
- Introduction To Data Structures
- Data Structures Interview Questions: Basics
- Data Structures Interview Questions: Intermediate
- Data Structures Interview Questions: Advanced
- Conclusion
Dynamic Programming - From Basics To Advanced (+Code Examples)

Dynamic Programming (DP) is a technique for solving problems by breaking them into smaller, overlapping subproblems and reusing solutions to save time. It is ideal for optimization and recursive problems, ensuring efficiency by avoiding redundant computations.
In this article, we will explore the core concepts of dynamic programming and its key principles and provide step-by-step guides with real-world examples.
What Is Dynamic Programming?
Dynamic programming (DP) is a problem-solving approach used in computer science to solve problems by breaking them into smaller overlapping subproblems. It is particularly effective for optimization problems and those with a recursive structure.
Key Concepts Of Dynamic Programming:
- Optimal Substructure: A problem exhibits optimal substructure if its solution can be constructed efficiently from the solutions of its smaller subproblems.
- Example: Finding the shortest path in a graph involves combining the shortest paths of smaller subgraphs.
- Overlapping Subproblems: Many problems involve solving the same subproblem multiple times. DP solves each subproblem once and stores the results to avoid redundant computations.
- Example: Fibonacci numbers – calculating F(n) involves repeated calculations of F(n-1) and F(n-2).
- Memoization (Top-Down Approach): In this approach, results of subproblems are stored in a data structure (like a dictionary or array) during recursion to avoid recalculating them.
- Example: Recursive Fibonacci with memoization.
- Tabulation (Bottom-Up Approach): In this approach, subproblems are solved iteratively, and the results are stored in a table (usually an array) to build up the solution for the larger problem.
- Example: Iterative Fibonacci using an array.
Real-Life Example: The Jigsaw Puzzle Analogy
Dynamic programming can be understood with the analogy of solving a jigsaw puzzle efficiently. Imagine you are solving a large jigsaw puzzle. Instead of trying random pieces to see what fits (brute force), you use a systematic approach:
- Break It Down:
You group the puzzle pieces by color or pattern (e.g., edges, sky, trees, etc.), making it easier to focus on smaller parts of the puzzle. - Reuse Your Work:
Once you solve a smaller section (e.g., the sky), you don’t revisit it or redo it. Instead, you place it aside, knowing it’s solved. This is like memoization in dynamic programming. - Build Step by Step:
After solving smaller sections, you combine them to form larger sections (e.g., attaching the sky to the trees). This is similar to tabulation, where you solve subproblems iteratively and build up to the final solution. - Avoid Redundancy:
If you’ve already figured out where a piece fits, you don’t test it again in other places. This prevents repetitive work and saves time.
How This Relates To Dynamic Programming:
- Grouping puzzle pieces: Breaking the problem into subproblems.
- Saving smaller sections: Storing solutions of subproblems to avoid redundant work.
- Combining sections: Using smaller solutions to build the final answer.
- Avoiding redundancy: Ensuring each subproblem is solved only once.
Dynamic programming, like solving a puzzle this way, is all about organization, reuse, and efficiency.
How To Solve A Problem Using Dynamic Programming?
Solving a problem using dynamic programming (DP) involves following a structured approach. Here's a step-by-step guide:
Steps to Solve a Problem Using Dynamic Programming:
- Understand the Problem Clearly
- Read the problem statement carefully.
- Identify what needs to be optimized or calculated.
- Check for DP Characteristics
- Optimal Substructure: Can the problem be broken into smaller, simpler subproblems?
- Overlapping Subproblems: Are some subproblems solved multiple times in a naive solution?
- Define the State
- Decide on the variables that represent the subproblem.
- For example, in the Fibonacci problem, the state can be F(n) representing the nth Fibonacci number.
- Formulate the Recurrence Relation
- Write a mathematical relation that connects the solution of the current subproblem to smaller subproblems.
- Example for Fibonacci: F(n)=F(n−1)+F(n−2)
- Choose an Approach
- Memoization (Top-Down): Solve the problem recursively and store results to avoid recomputation.
- Tabulation (Bottom-Up): Solve subproblems iteratively, starting from the smallest and building up to the final solution.
- Define Base Cases
- Establish the solutions for the simplest subproblems, which will act as starting points.
- Example for Fibonacci: F(0)=0, F(1)=1
- Implement the Solution
- Write code using your chosen approach.
- Use appropriate data structures (e.g., arrays, lists, or dictionaries) to store intermediate results.
- Optimize Space (if needed)
- If only a few previous states are required, reduce space usage by storing only those.
- Example: In Fibonacci, reduce space complexity from O(n) to O(1).
- Test the Solution Thoroughly
- Validate the implementation using various test cases, including edge cases.
- Example: For Fibonacci, test with n=0,1,10,50, etc.
Dynamic Programming Algorithm Techniques
Dynamic programming (DP) involves solving complex problems by breaking them into overlapping subproblems. The two main techniques are:
1. Memoization (Top-Down Approach)
Memoization is a top-down technique where recursive calls store results of subproblems in a cache to avoid redundant calculations.
Concept:
- Solve the problem recursively.
- Store the results of already solved subproblems in a cache (e.g., dictionary or array).
- When the same subproblem arises again, retrieve the result from the cache instead of recomputing it.
Key Characteristics:
- Recursive in nature.
- Saves computation time by avoiding redundant calculations.
- Can be slower for problems with deep recursion due to function call overhead.
Code Example: Fibonacci Using Memoization
def fibonacci_memoization(n, memo={}):
# Check if result is already computed
if n in memo:
return memo[n]
# Base cases
if n == 0:
return 0
if n == 1:
return 1
# Recursively calculate and store the result in memo
memo[n] = fibonacci_memoization(n - 1, memo) + fibonacci_memoization(n - 2, memo)
return memo[n]
# Example usage
n = 10
print("Fibonacci number at position", n, ":", fibonacci_memoization(n))
Output:
Fibonacci number at position 10 : 55
Explanation:
In the above code example-
- We start with a function fibonacci_memoization that computes Fibonacci numbers using a technique called memoization, which helps us avoid redundant calculations.
- The function takes two parameters: n (the position in the Fibonacci sequence) and an optional dictionary memo to store previously computed results.
- We first check if the result for the current n is already in memo. If it is, we directly return it, saving computational effort.
- For the base cases: if n is 0, we return 0; if n is 1, we return 1. These are the starting points of the Fibonacci sequence.
- If the result is not already computed, we calculate it recursively by summing the results of fibonacci_memoization(n - 1, memo) and fibonacci_memoization(n - 2, memo).
- After calculating the value, we store it in memo to avoid recomputing it in future calls.
- Finally, we return the computed value.
- When we call fibonacci_memoization(10), the function computes the 10th Fibonacci number. By using memoization, we ensure that intermediate results are reused efficiently, making the function much faster than naive recursion.
2. Tabulation (Bottom-Up Approach)
Tabulation is a bottom-up technique where the solution to a problem is built iteratively from the smallest subproblems up to the main problem, using a table to store intermediate results.
Concept:
- Solve the problem iteratively from the smallest subproblems up to the main problem.
- Use a table (e.g., array) to store results for subproblems.
- Each subproblem builds on the results of previously solved subproblems.
Key Characteristics:
- Iterative in nature.
- Avoids recursion and function call overhead.
- Often uses less space compared to memoization (no recursion stack).
Code Example: Fibonacci Using Tabulation
def fibonacci_tabulation(n):
# Handle base cases
if n == 0:
return 0
if n == 1:
return 1
# Create a table to store Fibonacci values
dp = [0] * (n + 1)
dp[0], dp[1] = 0, 1 # Base cases
# Fill the table iteratively
for i in range(2, n + 1):
dp[i] = dp[i - 1] + dp[i - 2]
return dp[n]
# Example usage
n = 10
print("Fibonacci number at position", n, ":", fibonacci_tabulation(n))
Output:
Fibonacci number at position 10 : 55
Explanation:
In the above code example-
- We define a function fibonacci_tabulation to compute Fibonacci numbers using tabulation, which involves building a table (array) to store intermediate results.
- The function takes one parameter n, the position in the Fibonacci sequence.
- We first handle the base cases: if n is 0, we return 0, and if n is 1, we return 1. These are the starting points of the sequence.
- We then create a list dp of size n + 1 initialized with zeros. This list will store the Fibonacci values at each position.
- We set dp[0] to 0 and dp[1] to 1, as they represent the base cases.
- Next, we use a loop starting from index 2 to n to fill the table. At each step, we calculate dp[i] as the sum of dp[i - 1] and dp[i - 2]. This ensures that each Fibonacci number is calculated based on the two previous numbers.
- After completing the loop, we return dp[n], which holds the Fibonacci number at position n.
- When we call fibonacci_tabulation(10), the function calculates the 10th Fibonacci number by filling up the table iteratively, avoiding recursion and making the process more efficient.
Comparison Of Memoization And Tabulation:
Aspect |
Memoization (Top-Down) |
Tabulation (Bottom-Up) |
Method |
Recursive |
Iterative |
Space Complexity |
O(n) for cache and recursion |
O(n) for table |
Time Complexity |
O(n) |
O(n) |
Function Calls |
Multiple recursive calls |
Single iterative loop |
Ease of Implementation |
Easier for complex problems |
Slightly harder for some problems |
Preferred When |
Problem is naturally recursive |
Iterative solutions are feasible |
Advantages Of Dynamic Programming
Some of the common advantages of dynamic programming algorithm are:
- Optimal Substructure: DP ensures that a problem can be broken down into simpler subproblems, and solutions to those subproblems are combined to solve the overall problem optimally.
- Avoids Redundant Calculations: By storing the results of previously solved subproblems (either in a table or cache), DP significantly reduces the computation time, especially in problems with overlapping subproblems.
- Improves Efficiency: With DP, problems that would otherwise take exponential time (like the Fibonacci sequence or the knapsack problem) can be solved in polynomial time.
- Provides Clear Structure: The recursive structure of the problem and the defined relationship between subproblems make DP easy to model and understand once the recurrence relation is identified.
Disadvantages Of Dynamic Programming
Some of the common disadvantages of dynamic programming are:
- Space Complexity: DP requires additional memory to store the intermediate results, which can sometimes be prohibitive, especially in problems with a large number of subproblems.
- Overhead: The need to store and retrieve intermediate results (in memoization) or iterate over large tables (in tabulation) can introduce overhead, especially in problems where the number of subproblems is very large.
- Complex Implementation: Identifying the optimal substructure, formulating the recurrence relation, and setting up the state and transition can sometimes be difficult, especially for complex problems.
- Not Always Applicable: Not all problems exhibit optimal substructure and overlapping subproblems, making DP unsuitable for many types of problems.
Applications Of Dynamic Programming
Some of the important applications of dynamic programming are:
- Fibonacci Numbers: Classic example where DP reduces the exponential time complexity of the recursive approach to linear time.
- Knapsack Problem: DP is widely used to solve optimization problems, like the knapsack problem, where we need to maximize or minimize a certain quantity (e.g., profit or cost) while adhering to constraints.
- Longest Common Subsequence (LCS): In string matching problems, DP helps in finding the longest common subsequence between two strings, commonly used in bioinformatics and text comparison.
- Matrix Chain Multiplication: DP is used in problems where the goal is to optimize the order of matrix multiplication to minimize computational cost.
- Shortest Path Problems (e.g., Floyd-Warshall, Bellman-Ford): DP is used to find the shortest paths between vertices in a graph, which is useful in routing algorithms, network optimization, and GPS navigation.
- Edit Distance (Levenshtein Distance): DP helps in calculating the minimum number of operations (insertions, deletions, substitutions) required to convert one string into another, commonly used in spell checkers and text comparison.
- Game Theory: DP can be used to optimize decision-making strategies in games, such as minimizing or maximizing a player's advantage.
- Resource Allocation Problems: In resource scheduling, job scheduling, or project management, DP can optimize allocation to minimize time, cost, or other factors.
Conclusion
Dynamic Programming (DP) is an essential technique for solving complex problems that involve optimization and can be broken down into simpler overlapping subproblems. By leveraging memoization (top-down) or tabulation (bottom-up), DP optimizes the computation process, drastically reducing time complexity and avoiding redundant calculations. However, it is important to note that DP is most effective when the problem exhibits optimal substructure and overlapping subproblems.
While DP offers numerous advantages, such as efficiency and clarity, it can come with challenges like increased space complexity and implementation complexity. Despite these drawbacks, DP remains a powerful tool widely applied in areas such as optimization, pathfinding, string matching, and game theory. Understanding when and how to apply DP can help solve problems that would otherwise be intractable with brute-force approaches.
Frequently Asked Questions
Q. What is Dynamic Programming?
Dynamic Programming algorithm(DP) is a problem-solving technique used to solve optimization problems by breaking them down into smaller overlapping subproblems. DP stores the results of these subproblems to avoid redundant calculations, improving time complexity. It is typically used when a problem has optimal substructure and overlapping subproblems, making it suitable for problems like the Fibonacci sequence, knapsack problem, and shortest path problems.
Q. What are the two main techniques used in Dynamic Programming?
The two main techniques in Dynamic Programming are:
- Memoization (Top-Down): This technique uses recursion to solve subproblems and stores the results in a cache (usually a dictionary or array). It avoids recalculating the same subproblem by checking if the result is already stored.
- Tabulation (Bottom-Up): This technique builds the solution iteratively, starting from the smallest subproblem and solving progressively larger problems. It uses a table to store the results of subproblems.
Q. What is the difference between Memoization and Tabulation?
- Memoization is a top-down approach that uses recursion and caches the results of subproblems, reducing redundant calculations.
- Tabulation is a bottom-up approach where we iteratively solve subproblems in increasing order of size, filling a table to store intermediate results.
The primary difference lies in the way the problem is approached: memoization relies on recursion and is typically easier to implement for complex problems, while tabulation is more iterative and may have less memory overhead due to the lack of recursion.
Q. What are the key characteristics of a problem that make Dynamic Programming applicable?
Dynamic Programming is applicable to problems that exhibit the following two key characteristics:
- Optimal Substructure: The problem can be broken down into smaller subproblems that can be solved independently and combined to form the solution.
- Overlapping Subproblems: The problem contains many instances of the same subproblems, which can be solved once and stored for future use, thus avoiding redundant computations.
Q. What are the main advantages of using Dynamic Programming?
The main advantages of Dynamic Programming include:
- Efficiency: DP reduces time complexity by avoiding redundant calculations through caching or tabulating subproblem results.
- Optimal Solutions: DP guarantees finding the optimal solution to problems by systematically evaluating all possibilities.
- Improved Time Complexity: For problems with overlapping subproblems, DP can reduce exponential time complexity to polynomial time, making complex problems solvable in a reasonable amount of time.
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I’m a Computer Science graduate with a knack for creative ventures. Through content at Unstop, I am trying to simplify complex tech concepts and make them fun. When I’m not decoding tech jargon, you’ll find me indulging in great food and then burning it out at the gym.
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