Home Learning Path

Loading...

Learn DSA

Master Data Structures and Algorithms with interactive theory, visualizations, and hands-on simulations. Build your coding skills with comprehensive learning resources.

Quick Links

Data Structures
Algorithms
About
Contact

Data Structures

Arrays
Linked Lists
Stacks
Queues
Trees
Graphs

Algorithms

Sorting
Searching
Graph Algorithms
Dynamic Programming
Greedy
Divide & Conquer

Interactive Theory

Comprehensive explanations with examples

Visual Learning

Dynamic visualizations and animations

Hands-on Practice

Interactive simulations and exercises

© 2025 Learn DSA. All rights reserved.Made with passion for learning

Privacy Policy Terms of Service Cookie Policy DMCA

Learn DSA is an educational platform designed to help students and professionals master data structures and algorithms. All content is provided for educational purposes only. We are not affiliated with any specific company or institution.

Back to Algorithms

Dynamic Programming

Master the art of solving complex problems by breaking them into simpler subproblems. Learn memoization, tabulation, and optimization techniques for efficient algorithms.

Dynamic Programming Fundamentals

Optimal Substructure

Optimal solution can be constructed from optimal solutions of subproblems

Overlapping Subproblems

Same subproblems are solved multiple times in naive recursive approach

Memoization

Store and reuse results of expensive function calls for efficiency

Common DP Patterns

Linear DP

Problems with 1D state space

Fibonacci • House Robber • Climbing Stairs

2D DP

Problems with 2D state space

Knapsack • LCS • Grid Path

Interval DP

Problems on ranges or intervals

Matrix Chain • Burst Balloons • Optimal BST

Tree DP

DP on tree structures

Tree Diameter • House Robber III • Binary Tree Cameras

Memoization vs Tabulation

Memoization (Top-Down)

Recursive approach with caching of results

Advantages:

• Natural recursive thinking
• Computes only needed subproblems
• Easy to implement

Disadvantages:

• Function call overhead
• Stack space usage
• May hit recursion limits

Best For:

Complex state transitions, when not all subproblems needed

Tabulation (Bottom-Up)

Iterative approach building solutions from base cases

Advantages:

• No recursion overhead
• Better space optimization
• Easier to optimize

Disadvantages:

• Less intuitive
• May compute unnecessary subproblems
• Complex initialization

Best For:

When all subproblems are needed, space optimization important

Classic DP Problems

Fibonacci Sequence

Classic introduction to DP concepts with overlapping subproblems

Memoization • Tabulation • Space Optimized

F(n) = F(n-1) + F(n-2)

Learning DP fundamentals, Mathematical sequences

0/1 Knapsack Problem

Select items with maximum value within weight constraint

O(n × W) / O(W)

2D Table • Space Optimized • Backtracking

dp[i][w] = max(dp[i-1][w], dp[i-1][w-weight[i]] + value[i])

Resource allocation, Budget optimization

Longest Common Subsequence

Find longest subsequence common to two sequences

O(m × n) / O(min(m,n))

2D Table • Space Optimized • Reconstruction

If match: dp[i][j] = 1 + dp[i-1][j-1], else: max(dp[i-1][j], dp[i][j-1])

DNA sequence analysis, Diff utilities

Coin Change Problem

Unbounded Knapsack

Find minimum coins needed or number of ways to make amount

Bottom-up DP • Memoization • Space Optimized

dp[amount] = min(dp[amount], dp[amount - coin] + 1)

Currency systems, Change making

Longest Increasing Subsequence

Find length of longest strictly increasing subsequence

DP O(n²) • Binary Search O(n log n) • Patience Sorting

dp[i] = max(dp[j] + 1) for j < i where arr[j] < arr[i]

Sequence analysis, Stock prices

Edit Distance (Levenshtein)

Minimum operations to transform one string to another

O(m × n) / O(min(m,n))

2D DP • Space Optimized • Operations Tracking

dp[i][j] = min(insert, delete, replace) + cost

Spell checkers, DNA analysis

Matrix Chain Multiplication

Find optimal way to parenthesize matrix multiplications

Interval DP • Memoization • Optimal Parenthesization

dp[i][j] = min(dp[i][k] + dp[k+1][j] + cost(i,k,j))

Compiler optimization, GPU computations

Palindrome Partitioning

Find minimum cuts to partition string into palindromes

O(n³) / O(n²)

Preprocessing • Manacher Algorithm • Optimized DP

dp[i] = min(dp[j] + 1) for j < i where s[j+1:i] is palindrome

Text processing, String analysis

Maximum Subarray (Kadane)

Find contiguous subarray with maximum sum

Kadane Algorithm • DP Approach • Divide & Conquer

maxSum = max(arr[i], maxSum + arr[i])

Stock trading, Signal processing

Space Optimization Techniques

Rolling Array

Use only current and previous rows instead of entire 2D table

Example: 2D DP → O(2 × n) space

State Compression

Use bit manipulation to represent states compactly

Example: Subset problems with bitmask

Coordinate Compression

Map large coordinate space to smaller space

Example: Large range → array indices

DP Problem-Solving Steps

1

Identify the Problem

Look for optimal substructure and overlapping subproblems

2

Define State

What parameters uniquely identify a subproblem?

3

Write Recurrence

Express solution in terms of smaller subproblems

4

Implement & Optimize

Choose memoization or tabulation, optimize space

Master Dynamic Programming

Practice with interactive visualizations and step-by-step solutions

Start with Fibonacci

Back to Graph Algorithms Next: Greedy Algorithms