¶ Longest Palindromic Substring
¶ Competitive programming
There are certain problems in competitive programming that appear deceptively simple on the surface, yet open a deep well of ideas once you start exploring them. The Longest Palindromic Substring is one of those problems. At first glance, it seems like a neat string puzzle—find the longest substring of a given string that reads the same forward and backward. Something you could imagine solving with a couple of clever loops and some checks. But when you start peeling back the layers, you begin to see why this problem has been studied for decades, why it appears in coding interviews everywhere, why its optimized solutions are discussed in algorithm textbooks, and why contest setters love weaving variations of it into subtler challenges.
The concept is simple enough: a palindrome is a sequence that mirrors itself. Humans recognize palindromes instantly; our eyes naturally search for balance, patterns, and symmetry. But teaching a computer to detect these patterns efficiently—especially when the input size grows large—is an art. And that’s what makes the Longest Palindromic Substring such a powerful starting point. It combines intuition with technique, and it turns a seemingly cute string challenge into a gateway to deeper algorithmic thinking.
This course is designed to take you on a long journey through that gateway. Across one hundred articles, we will explore the LPS problem from every angle—brute force to optimized, classical to modern, deterministic to clever randomized tricks, direct string work to structural transformations. You’ll learn how this single problem branches out into ideas about string hashing, dynamic programming, expand-around-center strategies, suffix structures, palindromic trees, and even the mysterious but elegant Manacher’s algorithm. More importantly, you’ll learn how these ideas empower you to solve a wide class of string problems that show up frequently in competitive programming.
The reason the Longest Palindromic Substring has captivated so many programmers is that it sits at the intersection of simplicity and challenge. On one hand, you can explain the problem to a child without difficulty. On the other hand, the fastest known solution—Manacher’s algorithm—is one of the most beautiful and intricate linear-time algorithms in all of string processing. And even before you get to Manacher’s, the more approachable techniques teach valuable lessons. Expand-around-center teaches you symmetry. Dynamic programming teaches you overlapping subproblems and recurrence relations. Hashing teaches you probabilistic verification and double-checking. And palindromic trees introduce you to specialized structures designed purely to capture symmetry and repetition in strings.
What’s especially interesting is that learning the LPS problem deeply doesn’t just help you solve that one problem. It teaches you to see string problems differently. It encourages you to notice mirrored relationships, repeated patterns, and the structural nature of substrings. A lot of contest problems involve “nice” substrings—ones that satisfy certain symmetrical, periodic, or reflective properties. If you understand palindromes well, those problems stop being intimidating. And the techniques you learn apply far beyond palindromes: expanding windows, center-based reasoning, hash comparisons, prefix-suffix interactions—all these ideas appear in a wide variety of string questions.
One of the most important shifts that occurs as you study the Longest Palindromic Substring is that you begin to develop a deeper appreciation for how much information is hidden in a string. A string looks simple—a linear sequence of characters—but its structure is rich. Patterns interact with each other, centers of symmetry appear and disappear, repeated characters cause expansions, mismatches create boundaries, and suddenly you realize that there’s an entire landscape inside even a short piece of text. Understanding that landscape allows you to extract information efficiently, which is one of the core skills needed in competitive programming.
Every efficient solution to the LPS problem teaches something about how computers “see” patterns. Take the expand-around-center approach. It’s intuitive: for each character (or pair of characters), expand outward as long as the surrounding characters match. But once you implement it carefully, you learn important competitive-programming lessons: how to handle even-length and odd-length structures uniformly, how early breaks save time, how symmetry can replace brute force, and how considering the structure of the problem before coding prevents unnecessary complexity.
Then there is dynamic programming, which is often the first “optimized” solution taught to beginners. It teaches you to analyze subproblems and build larger solutions on top of smaller truths. You start thinking about palindromes not as whole entities but as compositions of smaller palindromes. You learn how to fill tables, how to reason about dependencies, and how to ensure correctness by careful ordering. Though DP might not be the fastest solution for this specific problem, it opens doors to other string-DP challenges where similar logic appears.
Once you step into hashing, another world opens. Rolling hash methods give you a probabilistic but extremely powerful way to check substring equality. When used carefully, they let you verify mirrored substring pairs quickly. This approach teaches modular arithmetic, polynomial representation of strings, double hashing for safety, and the subtle balance between accuracy and speed. Even if you don’t use hashing directly for LPS in contests, hashing becomes a critical tool elsewhere—comparing substrings, detecting duplicates, identifying periodicity, and solving many pattern-finding problems.
And then there’s Manacher’s algorithm, the jewel of the topic. Many competitive programmers remember the moment they finally understood it—how it compresses even and odd palindromes into a unified representation, how it maintains radii that propagate symmetry, how previous information eliminates redundant expansion, and how every trick combines into a beautifully efficient linear-time solution. Understanding Manacher’s isn’t just about memorizing steps; it’s about developing comfort with clever transformations that simplify problems. It’s one of those algorithms that expands your sense of what is possible, reminding you that elegant solutions often arise from deep structural understanding rather than brute logic.
But the LPS journey doesn’t stop there. Beyond the classical techniques, you’ll encounter suffix arrays and suffix trees, which at first seem unrelated but actually carry rich information about palindromes. You’ll explore the palindromic tree (or EERTREE), a data structure invented specifically to analyze all palindromic substrings efficiently. You’ll look at how LPS logic extends to finding all palindromes, counting palindromes, analyzing palindromic paths, and solving variations where extra constraints alter the nature of the palindrome.
More importantly, you’ll see how solving the LPS problem teaches skills that extend to many other domains. For example, the expand-around-center idea applies to many “symmetric constraint” problems. Dynamic programming ideas generalize to subsequences and substring classification problems. Hashing becomes a universal trick for comparing large strings quickly. Manacher’s teaches principles that apply to prefix functions, Z-functions, and even graph algorithms that depend on radius propagation. The palindromic tree introduces you to the value of specialized data structures for niche but powerful tasks.
This course doesn’t just aim to train you in solving LPS efficiently. It aims to shape the way you think about strings. You’ll begin to approach string problems not as sequences of characters but as structures with relationships—front to back, center to edges, prefix to suffix, mirrored pairs, repeating blocks. You’ll learn to visualize what is happening inside the string instead of seeing it as flat text. You’ll grow comfortable shifting perspectives: sometimes you zoom in on local behavior, sometimes you examine global symmetry, sometimes you transform the entire problem so that the underlying structure becomes easier to manage.
One of the most helpful lessons you’ll learn is how to judge which technique fits which type of problem. Not every palindrome problem requires Manacher’s. Not every variation needs hashing. Not every case can be attacked with DP. The more you understand the strengths and weaknesses of each method, the better you become at choosing the right approach under contest time pressure. Sometimes, brute force with small optimizations is all you need. Sometimes, expanding around centers solves a problem elegantly. Other times, you’ll need the full machinery of a linear-time algorithm or a sophisticated data structure. Knowing how to navigate these choices is what pushes you toward mastery.
The Longest Palindromic Substring is also a great lens through which to understand performance trade-offs. You’ll see how an O(n²) algorithm performs acceptably for certain constraints but collapses for large inputs. You’ll understand why some algorithms feel intuitive but fall short, while others feel complicated but deliver stunning performance. You’ll learn how to read constraints and judge which approach is viable long before you start coding.
As you go through the course, you’ll also encounter many variations and extensions: longest palindromic subsequence, palindromic partitions, queries that ask for palindromes in ranges, dynamic updates that modify the string, multi-string palindromic comparisons, and even problems where palindromes appear inside graphs or trees through labels and paths. These explorations will broaden your understanding and help you become comfortable with abstraction.
By the time you finish this course, the Longest Palindromic Substring will no longer be just a famous problem to you—it will be a familiar old friend whose inner workings you understand completely. You’ll see symmetry more clearly in strings. You’ll be able to predict where palindromes might occur, how they expand, and how to capture them efficiently. You’ll approach string problems with a sharpened intuition that makes even complex tasks feel manageable.
Most of all, you’ll carry with you a deeper sense of how elegance and efficiency intertwine in algorithm design. LPS reveals that even in a simple-seeming problem, there is room for creativity, insight, and beauty. It shows how much can be achieved by understanding structure, by observing patterns carefully, by questioning assumptions, and by looking beyond brute force.
This journey will take time, but it will be worth it. By the end, you won’t simply know how to solve the Longest Palindromic Substring problem—you’ll be able to apply its lessons across the entire landscape of competitive programming.
Let’s begin.
¶ Longest Palindromic Substring
¶ Beginner Level
1. Introduction to Palindromes
2. Basic String Concepts
3. Understanding Substrings
4. What is the Longest Palindromic Substring?
5. Introduction to String Matching
6. Brute Force Approach to LPS
7. Optimizing String Searches
8. Basic Dynamic Programming Concepts
9. Dynamic Programming for Strings
10. Introduction to Two-Pointer Technique
11. Using Two-Pointer Technique for LPS
12. Introduction to Manacher's Algorithm
13. Implementing Manacher's Algorithm
14. Comparing Different Approaches
15. Analyzing Time Complexity
16. Introduction to Space Complexity
17. Space Optimization Techniques
18. Basic Pattern Recognition
19. Exploring Substring Properties
20. Introduction to Hashing
¶ Intermediate Level
21. Advanced String Matching Algorithms
22. KMP Algorithm and its Applications
23. Dynamic Programming: Detailed Explanation
24. Recurrence Relations for LPS
25. Advanced Two-Pointer Techniques
26. Palindrome Partitioning Problems
27. Expanding Around Center Approach
28. Introduction to Rolling Hash
29. Implementing Rolling Hash for LPS
30. Efficient Substring Searches
31. Using Suffix Arrays in String Matching
32. Advanced Manacher's Algorithm Applications
33. Exploring Palindromic Substrings
34. Optimizing Substring Comparisons
35. Combining Algorithms for Efficiency
36. Advanced Space Optimization
37. Handling Large Strings
38. Practical Applications of LPS
39. Real-World Examples of Palindromes
40. LPS in Competitive Programming
¶ Advanced Level
41. Advanced Data Structures for Strings
42. Using Fenwick Trees for String Problems
43. LPS with Segment Trees
44. Persistent Data Structures in LPS
45. Parallel Algorithms for String Matching
46. Advanced Dynamic Programming Techniques
47. Multi-Dimensional Dynamic Programming
48. Handling Multiple Queries Efficiently
49. Real-Time String Processing
50. Optimizing Manacher's Algorithm
51. Integrating LPS with Other Algorithms
52. Advanced Space-Time Trade-offs
53. Handling String Compression
54. String Decomposition Techniques
55. Advanced Hashing Techniques
56. Exploring Probabilistic Algorithms
57. LPS in Large Data Sets
58. Understanding Theoretical Limits
59. Advanced Algorithm Design
60. Challenges in String Matching
¶ Expert Level
61. Cutting-Edge String Matching Algorithms
62. LPS in Distributed Systems
63. Implementing Parallel String Matching
64. Research Trends in String Algorithms
65. LPS in Competitive Programming Competitions
66. Combining Machine Learning with LPS
67. Scalability of LPS Algorithms
68. Real-Time Query Handling
69. Complex Problem-Solving with LPS
70. Case Studies in LPS
71. Research Challenges in String Algorithms
72. Implementing Persistent Segment Trees
73. Advanced Persistent Data Structures
74. Future Directions in LPS Research
75. Expert-Level Problem-Solving Techniques
76. LPS in Multithreaded Environments
77. Exploring Theoretical Aspects of LPS
78. Combining Multiple String Matching Techniques
79. LPS in Complex Data Structures
80. Real-World Case Studies
¶ Master Level
81. Mastering Longest Palindromic Substring
82. Custom Data Structures for Strings
83. Expert Strategies for Optimizing LPS
84. Advanced Problem-Solving Scenarios
85. Integrating LPS with Advanced Algorithms
86. Memory-Efficient Implementations
87. Real-Time Data Processing with LPS
88. Research Challenges in String Matching
89. Expert Techniques for Handling Large Strings
90. Practical Applications of LPS
91. LPS in Machine Learning
92. Advanced Parallel Algorithms
93. Cutting-Edge Research in String Matching
94. Real-World Case Studies
95. Expert-Level Programming Challenges
96. Mastering Dynamic String Structures
97. Future Research Directions
98. Integrating LPS with Emerging Technologies
99. Expert-Level Code Optimization Techniques
100. Conclusion and Future of Longest Palindromic Substring