¶ Dynamic Programming
¶ Competitive programming
¶ Introduction to Dynamic Programming in Competitive Programming
There’s a moment, somewhere in every competitive programmer’s journey, when a problem appears on the screen—deceptively simple at first glance, almost welcoming—and then suddenly becomes an unsolvable maze. You try brute force. You try greedy. You tweak this, adjust that, and still, the solution feels just out of reach. You stare at it for so long that the numbers start dancing. And then someone mentions two words that have intimidated countless programmers and saved just as many: Dynamic Programming.
Dynamic Programming, or DP as most people lovingly (or fearfully) call it, sits at the heart of competitive programming. It is that one domain where logic meets patience, where understanding outweighs memorization, and where creativity takes the shape of patterns, states, and transitions. DP can seem cryptic at first, almost like a language spoken by seasoned coders who have unlocked a secret. But once you start learning it step by step, you realize that it’s not mysterious at all. It’s simply a way of reorganizing your thoughts to avoid doing the same work twice.
This course is built to help you enter that world smoothly, without fear and without frustration. Across the next hundred articles, you’ll discover the beauty of DP, its logic, its intuition, and the techniques that make it an essential pillar of competitive programming. You’ll see how problems that once looked impossible transform into something almost enjoyable once you understand the underlying patterns. You’ll learn how thinking in terms of states opens up solutions you couldn’t have imagined with brute force. And most importantly, you’ll learn how to recognize when a problem is waiting for DP—because half the challenge lies in knowing where to use it.
Before diving into the intricacies of optimization, transitions, or recurrence relations, it’s important to understand what makes Dynamic Programming so special. At its core, DP is about efficiency born from wisdom. It does not try to reduce the steps by taking shortcuts. Instead, it carefully observes that many problems overlap in their computations. DP simply asks, “Why solve the same subproblem again and again when you can solve it once and reuse the answer?” And just like that, an exponential brute-force solution shrinks into something manageable—O(n²), O(n log n), or sometimes even O(n).
But Dynamic Programming is more than just memoization or storing values in a table. It’s a mindset. A way of seeing problems not as linear sequences but as branching possibilities with shared paths. Once that mindset becomes part of your thinking, you start recognizing subproblems everywhere. You begin to see where repetition occurs, where the future depends on past decisions, where choices require remembering something. And DP gives you the tools to handle all of that gracefully.
Many people think DP is difficult because it involves math or complicated formulas. But the real challenge isn’t mathematics—it’s perspective. When you first try to think in “states,” it feels abstract. When you attempt writing recurrence relations, they look big and messy. But then, with practice, something clicks. You start to appreciate how DP breaks large problems into smaller ones, how transitions map beautifully to logical flows, and how each piece fits together like a puzzle. Suddenly, DP doesn’t feel like a burden. It feels like a superpower.
This course is designed to take you toward that moment. It doesn’t rush. It doesn’t assume you know anything beforehand. Whether you have tried DP before or avoided it up until now, these articles will guide you through both the comfort and chaos that come with learning it properly. You’ll see classic problems, modern variations, interesting twists, and unusual challenges—each one helping you strengthen your understanding in a real, applicable way. You won’t be memorizing recipes; you’ll be building intuition.
One truth about DP that becomes clearer as you grow in competitive programming is that it appears in every corner of the field. Whether you’re working with sequences, strings, graphs, trees, grids, probability problems, or even game theory, DP has a role to play. At times it hides beneath the surface, and you won’t immediately recognize it. But once you have practiced enough, you’ll start spotting patterns that others miss. That’s what makes DP such an essential skill—it extends your problem-solving ability into dimensions you didn’t even know existed.
A lot of beginners approach DP with anxiety because they think they need to understand complex formulas right from the start. But DP isn’t about complication. It’s actually about simplification. It transforms problems into smaller, manageable parts. The challenge isn’t the technique—it’s training the mind to break problems down in the right way. That’s exactly what this course helps you master: not memorizing but understanding, not forcing but discovering.
Another part of DP’s reputation comes from the variety of directions it takes: bottom-up, top-down, recursion with memoization, iterative tabulation, optimization tricks, space reductions, transitions involving multiple states, bitmasking, graph DP, digit DP, DP on trees, DP on subsets, and others. It might sound like a lot, but the magic is that once you understand the fundamentals, each new technique becomes easier to grasp. You begin to realize that all DP techniques follow the same DNA—identify states, understand transitions, and use memory wisely. Once you know this core pattern, learning new types feels natural.
You’ll also explore the emotional side of DP, something most tutorials never discuss but every coder experiences. The hesitation when faced with a DP problem. The frustration when transitions don’t make sense. The sudden clarity when the recurrence finally clicks. DP problems are not solved by typing faster—they’re solved by thinking clearly. This course helps you train that clarity by guiding you through examples, giving you the space to understand mistakes, and encouraging you to build the patience that DP demands.
Dynamic Programming rewards those who truly take time to think. It rewards those who revisit problems. It rewards those who value elegance in logic. And during competitive contests, it rewards you with correct solutions that run within tight time limits while others struggle. That feeling—the moment your DP solution gets accepted on a problem that stumped many participants—is immensely satisfying. It makes the journey worth it.
You’ll also see how DP enhances your understanding of algorithms as a whole. Once you start thinking in terms of states and transitions, you become better at analyzing constraints, better at predicting performance bottlenecks, better at identifying patterns even outside of DP. The skill you build here will spill into other areas of competitive programming—divide & conquer, greedy, graph theory, and beyond. DP doesn’t just teach you a technique; it teaches you a refined way of reasoning.
Online judges today host thousands of problems where DP is either the intended solution or a viable optimized approach. Knowing DP not only increases your chances of solving more problems but also gives you the confidence to try problems you once avoided. You stop fearing constraints. You stop fearing exponential behaviors. You stop fearing complexity. Because you know how to break things down and rebuild them efficiently.
Through the hundred articles in this course, you’ll gradually move from simple foundational ideas to highly advanced concepts. You’ll see how to identify overlapping subproblems, how to design recurrence relations, how to use memoization effectively, how to convert recursive ideas into iterative tables, and how to optimize space when constraints demand it. You’ll also learn the art of thinking abstractly—how to represent states smartly, how to compress them when necessary, how to handle tricky transitions, and how to understand the intuition behind the approaches instead of memorizing patterns blindly.
This course won’t rush you through the topics. Instead, it will nurture your understanding slowly and naturally. Every article is crafted so you feel the progression—your ability evolving, your intuition sharpening, your confidence growing. By the time you reach the later stages, you’ll be ready for the most challenging types of DP problems, the kinds that appear in major competitions and often intimidate even experienced programmers.
One of the most empowering things about DP is that it makes you feel in control of complexity. Problems that start with exponential computations shrink gracefully under the influence of thoughtful DP design. You no longer worry about millions of combinations or billions of paths. With the right DP formulation, you reduce them into something completely manageable. That sense of control—of turning chaos into structure—is deeply motivating.
As you move forward, you’ll also realize that DP is as much an art as it is a technique. There’s creativity in defining states, in building transitions, in optimizing memory usage. There’s elegance in the way you can shape a brute-force idea into something linear or polynomial. There’s beauty in the way patterns emerge, in the way repetition becomes opportunity, in the way subproblems weave together into the final result.
By the end of this course, DP will no longer look like a giant wall standing between you and advanced problem-solving. It will look like a reliable tool—one that you understand, respect, and know how to use with precision. You’ll begin to enjoy tackling DP problems. You’ll feel comfortable with the reasoning process. You’ll approach contests with a stronger sense of preparedness. And most importantly, you’ll have expanded your capacity to think like a true competitive programmer.
Dynamic Programming is not something you master instantly. It’s something you grow into, step by step, one idea at a time. But once it clicks, it transforms the way you think forever. Through these hundred articles, you’ll walk that journey in a deliberate, enjoyable, and deeply enriching way. So take a breath, let go of any fear you may have about DP, and step in with curiosity. You're about to gain a skill that will stay with you long after this course is over.
Let’s begin the journey into this universe of states, transitions, patterns, and elegant logic—one that has shaped countless successful competitive programmers and one that will soon become a natural part of how you solve problems. This is where your transformation into a confident, DP-savvy coder truly begins.
¶ Dynamic Programming
I. Foundations (20 Chapters)
1. Introduction to Dynamic Programming: The Power of Memoization
2. Overlapping Subproblems: The Core Idea
3. Optimal Substructure: Building Solutions Recursively
4. Memoization: Top-Down Dynamic Programming
5. Tabulation: Bottom-Up Dynamic Programming
6. Comparing Memoization and Tabulation
7. Basic DP Problems: Fibonacci, Factorial, Climbing Stairs
8. 1D DP: Linear Structures and Sequences
9. Time and Space Complexity Analysis of DP Algorithms
10. Identifying DP Problems: Recognizing Overlapping Subproblems
11. State Definition: Defining the Subproblem Space
12. Transition Function: Defining the Recursive Relationship
13. Base Cases: Handling the Smallest Subproblems
14. DP on Arrays: Subsequence Sum, Longest Increasing Subsequence
15. DP on Strings: Longest Common Subsequence, Edit Distance
16. DP on Grids: Path Finding, Minimum Cost Path
17. DP on Trees: Tree Traversals and DP
18. Bitmasking DP: Representing Subsets
19. Introduction to State Compression
20. Practice Problems: Applying Basic DP Techniques
II. Intermediate Techniques (25 Chapters)
21. DP with Multiple Dimensions: 2D Grids, Matrices
22. DP on Trees: Diameter, Path Sum
23. DP with State Compression: Traveling Salesperson Problem (TSP)
24. DP with Bitmasking: Set Cover Problem
25. DP on Strings: Palindromic Subsequences, Wildcard Matching
26. DP on Arrays: Knapsack Problem (0/1 and Fractional)
27. DP on Arrays: Coin Change Problem
28. DP on Arrays: Maximum Subarray Sum
29. DP on Arrays: Longest Common Prefix
30. DP on Graphs: Shortest Path Algorithms (Bellman-Ford)
31. DP on Graphs: All-Pairs Shortest Paths (Floyd-Warshall)
32. DP with Probability: Expected Value Problems
33. DP with Modular Arithmetic: Handling Large Numbers
34. Optimizing DP Solutions: Reducing Time and Space Complexity
35. State Reduction Techniques: Eliminating Redundant States
36. Convex Hull Trick: Optimizing DP Transitions
37. Divide and Conquer Optimization for DP
38. Meet in the Middle: Combining DP with Other Techniques
39. Dynamic Programming on Trees: Advanced Techniques
40. DP with Bitsets: Efficiently Handling Sets
41. DP with SOS (Sum over Subsets)
42. DP for Game Theory: Minimax Algorithm
43. DP for Combinatorial Problems: Counting Combinations
44. Practice Problems: Applying Intermediate DP Techniques
45. Debugging DP Solutions: Common Errors and Pitfalls
III. Advanced Strategies (30 Chapters)
46. DP with Linear Programming: Connecting DP and LP
47. DP for Geometric Problems: Convex Hull, Line Arrangements
48. DP for String Algorithms: Suffix Trees and Suffix Arrays
49. DP for Network Flow Problems: Min Cost Flow
50. DP with Matrix Exponentiation: Linear Recurrences
51. DP with Fast Fourier Transform (FFT): Polynomial Multiplication
52. DP with Number Theory: Digit DP
53. DP with Game Theory: Sprague-Grundy Theorem
54. DP with Approximation Algorithms: Dealing with NP-Hard Problems
55. DP for Online Problems: Processing Data Streams
56. DP for Parallel Algorithms: Parallelization Techniques
57. DP for Distributed Algorithms: MapReduce and DP
58. DP with Advanced Data Structures: Segment Trees, Fenwick Trees
59. DP with Heavy-Light Decomposition: Tree Queries
60. DP with Link-Cut Trees: Dynamic Tree Operations
61. DP with Persistent Data Structures: Maintaining History
62. DP with Randomized Algorithms: Probabilistic DP
63. DP with Metaheuristics: Combining DP with Search Techniques
64. DP for Machine Learning: Reinforcement Learning
65. DP for Bioinformatics: Sequence Alignment
66. DP for Image Processing: Dynamic Time Warping
67. DP for Natural Language Processing: Parsing
68. DP for Robotics: Motion Planning
69. DP for Finance: Option Pricing
70. DP for Operations Research: Inventory Management
71. Advanced State Compression Techniques
72. Advanced DP Optimization Techniques
73. Advanced DP Problem Solving Strategies
74. Practice Problems: Tackling Challenging DP Problems
75. Analyzing DP Solutions: Optimality Proofs
IV. Expert Level & Applications (25 Chapters)
76. DP for Competitive Programming Contests: Problem Solving Strategies
77. Identifying DP Patterns in Contest Problems
78. Implementing DP Solutions Efficiently
79. Debugging Complex DP Algorithms
80. Advanced DP Techniques: Beyond the Basics
81. DP with Advanced Mathematical Concepts
82. DP for Quantum Computing: Quantum DP
83. DP in Real-World Systems: Case Studies
84. DP in Software Engineering: Compiler Optimization
85. DP in Hardware Design: Circuit Optimization
86. DP in Artificial Intelligence: Planning
87. DP in Economics: Resource Allocation
88. DP in Control Theory: Optimal Control
89. DP in Signal Processing: Filtering
90. DP in Communication Networks: Routing
91. Open Problems in Dynamic Programming: Research Directions
92. The Future of Dynamic Programming: Emerging Trends
93. DP in the Era of Big Data: Scalable DP Algorithms
94. DP for Cloud Computing: Distributed DP Frameworks
95. DP for Edge Computing: Resource-Constrained DP
96. DP for Internet of Things (IoT): Energy-Efficient DP
97. DP for Cybersecurity: Intrusion Detection
98. DP for Social Networks: Influence Maximization
99. DP for Smart Cities: Traffic Management
100. The Impact of Dynamic Programming: A Retrospective