¶ Longest Increasing Subsequence
¶ Competitive programming
The idea of the Longest Increasing Subsequence looks deceptively simple the first time you hear it: given a sequence of numbers, find a subsequence that is strictly increasing and as long as possible. It sounds like something you could solve with a bit of common sense, maybe by scanning the list and picking rising values, the way you’d choose steps while climbing a hill. But once you try, you realize there is something deeper going on—a kind of subtlety that makes this humble problem one of the most iconic topics in competitive programming. LIS is not just an algorithmic challenge; it’s a gateway into a world where patterns, optimizations, and mathematical symmetry blend into one another.
This course of a hundred articles exists because LIS is much more than the single textbook problem most people associate with it. It has variations, extensions, transformations, optimizations, tricks, and interpretations that are rich enough to fill an entire universe of competitive programming insights. Once you understand LIS deeply, you begin to see it not as a task, but as a powerful tool—one that solves scheduling conflicts, ordering constraints, envelope nesting puzzles, multidimensional optimization, string alignment problems, and an astonishing number of disguised challenges that competitors encounter in real contests.
LIS has a special place in algorithmic thinking because it sits at the intersection of dynamic programming, greedy philosophy, binary search optimization, and mathematical structure. It’s the type of problem that grows with your skill level. When you’re a beginner, you learn it as a dynamic programming problem with quadratic complexity. When you’re intermediate, you refine it using binary search to achieve logarithmic improvements. When you’re advanced, you learn how it appears inside segment trees, Fenwick trees, patience sorting, transformations like coordinate compression, and multidimensional LIS. And when you’re truly fluent, you start spotting it naturally in places where nobody else even thinks of it.
Part of the beauty of LIS lies in how it makes you think about sequences—not just as a list of numbers, but as a landscape of relationships. You begin to interpret values in terms of who can follow whom, and which element is compatible with which. You learn that a subsequence isn’t just about picking numbers; it’s about respecting order and balance. LIS teaches you to see structure within chaos. Even in a completely random sequence, there is an order waiting to be discovered. And the elegance of extracting that hidden order is what makes LIS such a timeless algorithmic treasure.
What makes LIS especially meaningful in competitive programming is that it encapsulates the growth many coders go through. You start with brute force intuition. Then you discover DP and feel empowered. Then you discover binary search inside DP and feel enlightened. Then you encounter problems where LIS is just a step, not the answer, and you feel humbled. Then you master those too, and LIS transforms from a problem into a lens. This course is designed to guide you through that entire evolution—not by throwing formulas at you, but by helping you build intuition, insight, and the ability to recognize LIS patterns where they naturally emerge.
Many programmers remember the first time they saw an O(N log N) solution for LIS. It feels like a magic trick. The algorithm builds a helper structure that doesn’t store the actual subsequence but stores the “shape” of one. Each number finds its place in a conceptual pile, and by maintaining these piles correctly, you uncover the length of the longest increasing subsequence far more efficiently than checking every combination. This idea, sometimes called patience sorting, is a perfect example of how algorithms uncover simplicity beneath complexity. It shows you that speed often comes not from cutting corners, but from understanding the underlying structure better.
But LIS is not only about increasing. Once you understand its core mechanics, you can adapt it to longest decreasing subsequence, longest bitonic subsequence, or even sequences with constraints that change the definition of compatibility. You can reshape the problem into two dimensions, three dimensions, or more. A sequence of envelopes becomes a sequence of rectangles. A problem about stacking boxes becomes a problem about sorting pairs. Suddenly you’re exploring a space where sorting and LIS collide, and the result solves problems that thousands of competitors struggle with each year.
One of the most transformative insights that LIS teaches is the importance of ordering—how a simple act of sorting can convert a seemingly impossible multidimensional problem into a clean, manageable one. When you sort the first dimension correctly, you allow LIS to handle the second dimension. This trick appears in problems about Russian dolls, geometry, scheduling, and even database indexing. It shows you that sometimes the hardest problems don’t need new algorithms—they just need the right perspective.
Another dimension of LIS that many coders don’t fully appreciate is its emotional one. LIS is a problem of patience. The DP solution teaches you to consider all possibilities carefully, while the binary search solution teaches you to be selective. It rewards restraint. It punishes haste. If you greedily pick every small increase, you get trapped later. If you hesitate too long, you miss opportunities. In some ways, LIS mirrors the act of learning itself: incremental improvements, occasional setbacks, and decisions that shape the future of a path.
As you progress through the articles of this course, you will discover that LIS shows up in disguise far more often than explicitly. Sometimes you are not given a sequence directly—you must extract it. Sometimes the order comes from geometry. Sometimes from events on a timeline. Sometimes from graph-like structures that can be linearized. Sometimes the entire challenge lies in transforming the problem until it becomes an LIS problem. And this skill—the skill of transformation—is one of the great gifts LIS can give a competitive programmer.
You will also see how LIS interacts with binary search trees, segment trees, coordinate compression, and advanced data structures. These tools enable you to solve LIS even under extreme constraints—millions of elements, updates, queries, modifications. LIS can be computed online, offline, partially, or under dynamic conditions. Once you know how to wield it, it stops being a topic and becomes a technique, a foundational building block that fits into larger algorithmic architecture.
Something else worth appreciating is that LIS teaches you precision. Not just in coding, but in thinking. You start noticing which states matter, which don’t, which transitions are valid, and which are misleading. You learn that brute force transitions aren’t always necessary, and that sometimes one elegant rule can replace hundreds of lines of trial-and-error logic. LIS helps sharpen your instinct for efficiency. It makes you think in terms of structure rather than raw enumeration.
And of course, LIS helps you internalize the soul of dynamic programming: the idea that the solution is crafted one step at a time, using past information intelligently. LIS shows you how DP evolves into something leaner, faster, and more elegant when refined. It becomes a playground for understanding why some DP problems can be optimized with convex hull tricks, divide-and-conquer DP, or monotonic queues. Once you grasp how a simple DP like LIS can be optimized to logarithmic complexity, you open the door to understanding much more advanced optimizations later.
LIS also teaches you how to balance theory with intuition. If you focus only on formulas, you’ll miss the heart of the problem. If you rely only on intuition, you’ll miss the rigor that makes solutions reliable. A good LIS solution is a marriage of both—an understanding of why the algorithm works and a sense of how the numbers flow. This balance is important not only to solve LIS problems, but to grow as a programmer in general.
By the time you complete this course, you won’t just know how to compute LIS efficiently. You’ll understand it at a deeper level—the kind of understanding that lets you spot LIS even when a problem never mentions sequences or increasing relationships. You’ll be able to look at complex multidimensional data and immediately think about sorting strategies and compatibility conditions. You’ll recognize that many problems are simply asking you to find a consistent chain of decisions—a longest increasing subsequence in disguise.
What awaits you across these hundred articles is a journey through perspectives. You’ll explore LIS as a dynamic programming story, as a binary search trick, as a sorting-based technique, as a tree-based structure, and as a conceptual approach to countless unrelated problems. Along the way, you’ll solve puzzles that strengthen intuition, tackle variations that stretch your flexibility, and handle constraints that sharpen your precision.
Most of all, you’ll develop the habit of thinking in terms of order—how to preserve it, how to manipulate it, how to find it where it seems to be absent. LIS is about finding rising patterns in a chaotic world, and there’s something profoundly satisfying about that. It’s more than a technique; it’s a way of approaching problems that rewards patience, clarity, and creativity.
Welcome to this deep dive into the world of LIS. If you're ready for a journey that moves from the basics of subsequences to the heights of multidimensional optimization, then you’re in for something special. Together, we’ll explore not just how to compute the longest increasing subsequence, but how to understand it, apply it, extend it, and eventually, see it everywhere.
¶ Longest Increasing Subsequence
I. Foundations (20 Chapters)
1. Introduction to Longest Increasing Subsequence (LIS)
2. Understanding Subsequences and Substrings
3. Brute-Force Approach to LIS: Time Complexity Analysis
4. Recursive Approach to LIS: Overlapping Subproblems
5. Dynamic Programming Approach to LIS: Memoization
6. Dynamic Programming Approach to LIS: Tabulation
7. Comparing Memoization and Tabulation for LIS
8. 1D Dynamic Programming: LIS as a 1D Problem
9. State Definition for LIS DP: Understanding the States
10. Transition Function for LIS DP: Building the Solution
11. Base Cases for LIS DP: Handling Small Inputs
12. Time and Space Complexity of DP LIS Solutions
13. Implementing LIS DP: Code Examples (C++, Java, Python)
14. Visualizing LIS DP: Understanding the DP Table
15. Constructing the LIS: Backtracking the DP Solution
16. Printing the LIS: Retrieving the Actual Subsequence
17. Variations of LIS: Longest Non-Decreasing Subsequence
18. Variations of LIS: Longest Decreasing Subsequence
19. Applications of LIS: Basic Examples
20. Practice Problems: Basic LIS Implementation
II. Intermediate Techniques (25 Chapters)
21. Optimized LIS using Binary Search: O(n log n) Solution
22. Understanding the Binary Search Approach to LIS
23. Implementing the O(n log n) LIS Algorithm
24. Lower Bound for LIS: Relation to Longest Decreasing Subsequence
25. LIS and Longest Common Subsequence (LCS): Connections
26. LIS and Longest Common Substring: Differences
27. LIS on Permutations: Special Cases
28. LIS with Duplicates: Handling Equal Elements
29. LIS with Negative Numbers: Adapting the Algorithms
30. LIS with Constraints: Problem Variations
31. LIS in 2D Arrays: Extending the Concept
32. LIS on Trees: DP on Trees for LIS
33. LIS on Graphs: Finding LIS in Paths
34. LIS and Interval Problems: Combining Techniques
35. LIS and Segment Trees: Efficient Queries
36. LIS and Binary Indexed Trees (Fenwick Trees)
37. LIS and Sparse Tables: Preprocessing for Queries
38. LIS and Meet in the Middle: Combining Search Strategies
39. LIS and Bitmasking: Representing Subsets
40. LIS and SOS (Sum over Subsets): Related Techniques
41. LIS and Game Theory: Connections to Game Problems
42. LIS and Combinatorial Problems: Counting Subsequences
43. Practice Problems: Intermediate LIS Applications
44. Debugging LIS Code: Common Errors and Pitfalls
45. Optimizing LIS Code: Performance Improvements
III. Advanced Strategies (30 Chapters)
46. LIS and Patience Sorting: A Connection
47. Patience Sorting Algorithm: Detailed Explanation
48. LIS and Longest Increasing Subsequence in a Matrix
49. LIS and Longest Path in a DAG: Graph-based Approach
55. LIS and Network Flow: Max Flow Formulation
51. LIS and Convex Hull: Geometric Approach
52. LIS and Line Arrangements: Geometric Interpretation
53. LIS and Suffix Trees: String-based Approach
54. LIS and Suffix Arrays: String Processing
55. LIS and Dynamic Programming Optimization: Advanced Techniques
56. LIS and Divide and Conquer: Recursive Solutions
57. LIS and Parallel Algorithms: Parallelizing LIS Computation
58. LIS and Distributed Algorithms: Distributed LIS Calculation
59. LIS and Approximation Algorithms: Finding Approximate LIS
60. LIS and Randomized Algorithms: Probabilistic Approaches
61. LIS and Online Algorithms: Processing Data Streams
62. LIS and Competitive Programming Contests: Problem Solving
63. Identifying LIS Problems in Contests
64. Implementing LIS Solutions Efficiently for Contests
65. Advanced LIS Problem Variations: Challenging Problems
66. LIS and Advanced Data Structures: Combining Data Structures
67. LIS and Advanced Algorithm Design Techniques
68. LIS and Number Theory: Connections to Number Sequences
69. LIS and Geometry: Advanced Geometric Applications
70. LIS and Stringology: Advanced String Applications
71. LIS in Machine Learning: Feature Engineering
72. LIS in Data Mining: Pattern Discovery
73. LIS in Bioinformatics: Sequence Analysis
74. LIS in Computer Vision: Object Tracking
75. LIS in Robotics: Motion Planning
IV. Expert Level & Applications (25 Chapters)
76. LIS and Advanced Mathematical Concepts
77. LIS and Quantum Computing: Quantum LIS Algorithms
78. LIS in Real-World Systems: Case Studies
79. LIS in Software Engineering: Code Optimization
80. LIS in Hardware Design: Circuit Design
81. LIS in Cloud Computing: Resource Allocation
82. LIS in IoT: Data Analysis
83. LIS in Cybersecurity: Intrusion Detection
84. LIS in Financial Modeling: Stock Price Analysis
85. LIS in Simulation and Modeling: Event Scheduling
86. LIS in AI and Machine Learning: Advanced Applications
87. LIS and Open Problems: Research Directions
88. The Future of LIS: Emerging Trends
89. LIS and Hardware Acceleration: GPU Implementations
90. LIS and Embedded Systems: Resource-Efficient LIS
91. LIS and Functional Programming: Immutable LIS Data Structures
92. LIS and Object-Oriented Programming: LIS Design Patterns
93. LIS and Design by Contract: Formal Verification of LIS
94. LIS and Testing: Unit Testing LIS Implementations
95. LIS and Performance Tuning: Optimizing LIS Code
96. LIS and Code Optimization: Advanced Techniques
97. LIS and Parallel Computing: Advanced Parallel LIS Algorithms
98. LIS and Distributed Computing: Advanced Distributed LIS
99. LIS and Quantum Information Processing
100. The Impact of LIS: A Retrospective and Future Outlook