¶ Rod Cutting
¶ Competitive programming
Rod cutting is one of those rare problems in competitive programming that feels simple at first glance yet quietly contains an entire universe of ideas. At its surface, it sounds almost like a small business puzzle: given a rod of a certain length and a list of prices for different possible cut sizes, how do you cut the rod to earn the maximum revenue? It doesn’t look like dynamic programming at first. It doesn’t feel like an optimization problem that holds deeper meaning. But the moment you start solving it seriously, you discover that beneath its modest definition lies one of the richest and most insightful topics in algorithmic thinking.
This course of one hundred articles is dedicated to exploring everything that rod cutting teaches you—not just the problem itself, but the entire ecosystem of reasoning, strategies, patterns, and variations that have made it a classic in the world of competitive programming. Rod cutting is more than a DP problem. It’s a story about choices, substructure, recursion, optimization, and the delicate balance between intuition and technique.
Most people’s first encounter with rod cutting comes when they are just stepping into dynamic programming. At that stage, the mind still thinks in brute-force terms, imagining every possible combination of cuts and hoping one of them turns out to be the best. Then you’re confronted with the explosion of possibilities: cut here, don’t cut there, cut again, don’t cut again. The number of ways to break the rod grows so quickly that brute force collapses almost immediately. And this is where the true nature of the problem reveals itself—because although the number of possible ways is huge, the structure of the problem is wonderfully simple. Each cut you make divides the rod into smaller rods, and each of those smaller rods is just another version of the same problem you started with.
This recursive nature is what makes rod cutting such a perfect introduction to dynamic programming. It illustrates subproblems beautifully. It teaches you how to break something complicated into pieces that look exactly like the original. You begin to see why memoization matters. You begin to understand how overlapping subproblems appear everywhere, even when the narrative of the problem doesn’t mention repetition. And you see how a clever DP table quietly handles what brute force cannot.
The elegance of this problem lies in the way it forces you to rethink how you evaluate possibilities. In everyday reasoning, you might be tempted to choose big cuts for big profit or avoid too many small cuts because they “feel” inefficient. But rod cutting ignores your instincts. It invites you to let the algorithm show which option is truly optimal, sometimes in ways that surprise even experienced programmers. A small cut repeated several times might outperform a single large cut. A combination of uneven cuts might earn more than the intuitive symmetrical approach. Rod cutting teaches you to trust computation over guesswork and structure over intuition.
But the problem isn’t just about maximizing revenue. It’s about understanding how optimization works. It’s about learning how to build solutions from the ground up. It’s about recognizing the patterns that appear again and again in DP problems—patterns that reappear in knapsack, coin change, matrix chain multiplication, and countless others. Rod cutting is one of the earliest and clearest expressions of these ideas, and because it is so clear, it becomes a foundation on which more advanced techniques can be built.
As you dig deeper, rod cutting reveals another truth: optimization problems often live in a world where tradeoffs matter. Cutting creates opportunities but also removes them. Choosing one path prevents another. The problem becomes a conversation between what you can do now and what you might need later. And dynamic programming becomes the language that mediates that conversation. It shows you how to compare possibilities fairly, how to build solutions consistently, and how to treat every decision with respect for its consequences.
Rod cutting also gives you a new appreciation for bottom-up thinking. At first, the recursive definition feels natural—solve the big problem by solving smaller versions of it. But over time, you start to feel the efficiency of building a solution from the smallest rods upward. You fill the table, length by length, understanding each size thoroughly before climbing to the next. By the time you reach the full rod, you’re no longer searching blindly; you’re assembling an answer using a map you built yourself. That shift—from recursion to tabulation—is more than a technical adjustment. It’s a change in how you think about problems, how you value steps, how you interpret progress.
Throughout the course, you’ll also explore the many variations that spring from the humble core idea. What if you limit the number of cuts? What if cuts have costs? What if the rod has to meet certain constraints, like minimum segment size or maximum number of pieces? What if the rod cutting problem is disguised as something else entirely—a scheduling conflict, a partitioning challenge, a resource allocation task? Rod cutting becomes a framework for reasoning about how to divide something valuable into smaller units without losing sight of the whole.
One of the most fascinating aspects of rod cutting is how frequently it appears in disguised forms in actual contests. Many problems that seem unrelated turn out to be rod-cutting variants wrapped in story-like narratives. A problem about splitting a line of tasks, a problem about breaking a string, a problem about cutting boards or cables, a problem about dividing workloads—they often reduce to the same DP essence. Once you recognize this, a wide range of problems becomes instantly more accessible. Rod cutting teaches you how to see through the surface of a challenge and identify the structure underneath, which is one of the most valuable skills in competitive programming.
Something else rod cutting teaches—often quietly—is discipline. When working on DP, it is tempting to jump into coding, but rod cutting encourages a different approach. It makes you slow down and think. What exactly is the recurrence? What does each state represent? What choices lead to that state, and what outcomes follow? Conceptually, this problem trains you not to fear complexity but to organize it. It shows you that even the most tangled decision process becomes manageable once you describe it with clarity.
While exploring the problem in depth, you also gain a deeper sense of the relationship between memory and computation. Memoization shows how forgetting results leads to redundant calculations. Tabulation shows how structured computation can solve huge problems efficiently. As the course progresses, you’ll build an instinct for when to choose which approach and how to mix them when necessary. You’ll learn to identify the shape of the solution space, to understand where the algorithm starts and where it ends, and to appreciate how each subproblem is a small, essential tile in the mosaic of the final result.
Rod cutting also offers a powerful lesson about generalization. Once you understand how to cut a rod optimally, you start seeing how to break down other objects optimally. The insights from this problem echo in areas like memory segmentation, load balancing, batch processing, string splitting, array partitioning, and even polynomial multiplication. The idea of dividing something in a way that maximizes overall value is a recurring theme across disciplines, and rod cutting gives you a clean, understandable starting point.
As you journey through the articles in this course, you’ll move from the simplest forms of the problem to highly complex scenarios involving multiple variables, constraints, and objectives. You’ll see how the same basic DP structure adapts gracefully to new demands. You’ll experiment with greedy heuristics that sometimes work and sometimes fail, gaining insight into why dynamic programming is necessary in the first place. You’ll explore optimization techniques that reduce time complexity, memory usage, and overhead. You’ll examine variations that involve binary search, prefix computations, state compressions, and clever transformations.
By the time you complete the full course, rod cutting will no longer feel like a small puzzle from an introductory textbook. It will feel like a versatile, powerful paradigm—one that sharpens your intuition for DP, improves your ability to analyze decisions, strengthens your understanding of optimization, and expands your toolkit for solving diverse competitive programming challenges. You’ll be able to recognize rod-cutting structure instantly, even when hidden. You’ll develop the confidence to approach new problems with clarity and grace. And you’ll see how this seemingly tiny problem becomes a gateway to deeper algorithmic mastery.
What began as a simple question about cutting a rod becomes much more—a journey into the heart of dynamic programming, a study in decision making, a lesson in balancing parts and wholes, and ultimately, a transformative way of thinking about how problems are built and how they can be solved.
Welcome to this course. The rod is long, but so is the list of insights waiting along its length. By the time you finish cutting through all one hundred articles, you won’t just know the rod cutting problem—you’ll understand the beauty and power of dynamic programming at a level few ever reach.
¶ Rod Cutting
¶ Beginner Level
1. Introduction to Regular Expressions in Competitive Programming
2. What Are Regular Expressions? Basic Definitions
3. Basic Syntax of Regular Expressions
4. Literals and Metacharacters: Understanding the Foundation
5. Character Classes: Matching Sets of Characters
6. The Dot (.) Character: Matching Any Character
7. Using the Caret (^) and Dollar Sign ($) for Anchors
8. Matching Digits and Non-Digits: \d, \D
9. Matching Word Characters and Non-Word Characters: \w, \W
10. Whitespace Characters: \s and \S
11. Quantifiers: Using * (Zero or More), + (One or More), ? (Zero or One)
12. Character Ranges: Using Brackets [ ]
13. Negating Character Classes with ^ inside Brackets
14. Using Parentheses for Grouping
15. Basic String Matching with Regex in Programming Languages
16. Simple Examples: Matching Email Addresses and Phone Numbers
17. Escaping Special Characters in Regex
18. Understanding and Using \b for Word Boundaries
19. Matching Numbers and Floating Point Values with Regex
20. Regex in Python, C++, and Java: Basic Syntax Comparison
¶ Intermediate Level
21. Introduction to Regex Engine Types: NFA vs. DFA
22. *Greedy vs Lazy Quantifiers: Using ?, +?, ??
23. Using Alternation: The Pipe (|) Operator
24. Capturing Groups and Backreferences in Regex
25. Named Capturing Groups and Their Applications
26. Lookahead and Lookbehind: Positive and Negative Assertions
27. Understanding Non-Capturing Groups (?
28. Using Regex for Validating Strings: Email, URLs, Dates
29. Using Regex to Split Strings Efficiently
30. Pattern Matching for Alphanumeric Strings
31. Regex for Matching Dates: YYYY-MM-DD, DD/MM/YYYY
32. Solving String Matching Problems with Regex
33. Regex for Matching Specific String Lengths
34. Regex for Case Insensitive Matching
35. Complex Pattern Matching: Combining Multiple Patterns
36. Using Regex for Pattern Substitution: Replacing Substrings
37. Advanced Grouping Techniques: Nested and Complex Groups
38. Handling Optional Patterns with Regex
39. Using Regex in Validation: Password and Username Constraints
40. Regex for Parsing and Extracting Data from Text Files
¶ Advanced Level
41. Backtracking in Regex Engines: Theory and Practice
42. Optimization Techniques for Regex in Competitive Programming
43. Performance Considerations: Time Complexity of Regex Operations
44. Matching Large Texts Efficiently Using Regex
45. Regex and Finite Automata: Connecting Theory to Practice
46. Handling Unicode and Multilingual Text with Regex
47. Advanced Lookahead and Lookbehind Assertions
48. Zero Width Assertions: ^, $, \b, \B
49. Regex for Complex String Searching Algorithms
50. Using Regex for Pattern Matching in Arrays and Strings
51. Regex for Palindrome Detection in Competitive Programming
52. Regex for Extracting Phone Numbers from Text
53. Regex for Matching HTML and XML Tags
54. Matching Parentheses and Brackets: Balancing Expressions
55. Solving Counting Problems with Regex
56. Regular Expressions for Parsing Log Files
57. Regex for Detecting Invalid Patterns in Strings
58. Recursive Patterns in Regex: Concepts and Applications
59. Performance Profiling of Regex in Large Inputs
60. Regex for Searching and Replacing Multiple Patterns Simultaneously
61. Using Regex to Solve Anagram Problems
62. Regex for Matching Complex Mathematical Expressions
63. Advanced Search and Replace Techniques with Regex
64. Regex for Extracting Key-Value Pairs from Text
65. Regex for Parsing and Analyzing CSV Files
66. Regex for Matching Specific Characters in Large Texts
67. Finding Common Substrings Using Regex
68. Optimizing Regex Patterns for Speed
69. Regex for Parsing and Validating JSON Format
70. Regex for Complex Text Data Extraction
71. Regex for Matching and Validating IP Addresses
72. Advanced Grouping and Backreferences for Pattern Extraction
73. Regex for Solving Cryptographic Challenges
74. Regex for Matching Complex Graph Structures
75. Handling Multiple Lookaheads in Regex Efficiently
76. Regex for Matching Non-ASCII Characters in Strings
77. Regular Expressions in Algorithms for Sorting and Searching
78. Using Regex for Efficient Data Compression
79. Regex for Solving the Knuth-Morris-Pratt String Matching Problem
80. Regex for Performing Syntax Highlighting in Code Editors
81. Parsing Nested Structures Using Regex: Parentheses, Brackets, etc.
82. Combining Regex with Other Algorithms for Faster String Matching
83. Regex for Validating Complex Formats in Input Validation
84. Regex for Extracting Keywords from Text
85. Regex in Real-Time Text Processing
86. Handling Non-Deterministic Finite Automata (NFA) in Regex
87. Advanced String Matching Algorithms with Regex
88. Using Regex in Dynamic Programming Problems
89. Regex in Genetic Algorithms for Pattern Matching
90. Regex for Parsing Mathematical Expressions in Coding Contests
91. Regex in File Systems: Searching for Specific File Types
92. Regex for Solving Search and Filter Problems in Databases
93. Regex for Parsing and Analyzing Code Syntax Trees
94. Efficient Regex Search Algorithms in Large Databases
95. Regex for Extracting Structured Data from Unstructured Text
96. Regex Optimization in String Manipulation Problems
97. Regex for Solving Complex Validation and Parsing Tasks
98. Regex for Matching Patterns in Data Streams
99. Using Regex in Advanced Algorithmic Problems: Dynamic Programming and Greedy
100. Final Thoughts: Mastering Regex in Competitive Programming