¶ Custom Hash Functions
¶ Competitive programming
¶ Introduction to Custom Hash Functions in Competitive Programming
If you’ve spent enough time solving problems on any competitive programming platform, you’ve probably run into that frustrating moment when your beautifully written solution keeps failing on hidden test cases — not because of logic errors, but because your hash table decides to betray you at the worst possible moment. Collisions pile up, operations slow down, and suddenly the solution you expected to run in linear time takes exponentially longer.
This is often the silent villain behind TLE verdicts in problems involving hashed data structures. And once you realize this, you begin to understand why experienced coders don’t just rely on the default hash functions provided by the language. They build their own. They protect their code with custom hashing the same way a locksmith trusts their own handmade lock over a mass-manufactured one.
This course begins with something that may look like a niche trick but is actually a major competitive programming survival skill: writing custom hash functions.
They’re not some mysterious magic that only high-rated coders whisper about. They’re simply small, precise, clever adaptations that let you control how your data behaves in hash-based containers — unordered maps, unordered sets, hash tables, maps of pairs, vectors, tuples, custom structs, and even large composite keys. And once you have this skill, a whole new section of the competitive programming world becomes easier, more stable, and far more efficient to navigate.
¶ Why Hash Functions Matter More Than You Realize
When beginners learn about hashing, they’re usually told that hash tables give average O(1) insertion, lookup, and deletion. But what they’re not told is something more uncomfortable: the default hash functions built into many languages are not designed for hostile environments.
Competitive programming is a hostile environment.
A bad test case is enough to break the illusion of constant time. Many problem-setters and testers know how easy it is to bring default hashing to its knees. If a platform uses adversarial test cases — or even just unlucky distributions of input — a default hash can collapse into a nightmare of collisions.
This course on custom hash functions exists because we want you to avoid this category of hidden pitfalls entirely.
You will see why default hashing can behave poorly, how input patterns can sabotage you, and why high-level abstractions like unordered_map<int, int> or unordered_set<long long> sometimes need reinforcement. When you learn to design strong hash functions, you stop playing defense. Instead, your code becomes robust in every situation — from normal inputs to malicious stress tests.
In top-tier competitions, this robustness often decides who solves the problem and who times out.
¶ The Real Story Behind “Custom Hash” in CP
When competitive programmers talk about custom hashing, they’re usually referring to techniques like:
- using randomized seeds to avoid predictable collisions
- designing consistent but unpredictable transformations
- building hash functions for pairs, structs, and complex objects
- preventing worst-case attack patterns on STL containers
- hashing large numbers, vectors, and sequences efficiently
- improving the performance of
unordered_mapandunordered_set - designing hash mixers that distribute bits more effectively
One of the best examples is the infamous “ordered vector hash” attack, which forces C++’s default hash to produce collisions by choosing values that map to the same bucket. This isn’t theoretical — it has happened countless times during contests.
So advanced coders began using custom hash utilities, often inspired by algorithms from fast cryptographic hash mixers, random bit manipulation strategies, and splitmix hashing techniques.
In this course, your journey will begin with understanding why these strategies work and how you can adapt them for competitive programming where speed is everything.
¶ Two Big Reasons Custom Hashing Gives You an Edge
1. You eliminate the unpredictability of collisions
Default hash functions can be predictable. If you’re unlucky, your input triggers a chain of collisions that slows everything down dramatically. Custom hashes introduce randomized behavior, making it nearly impossible for collisions to systematically occur.
2. You can hash anything — literally anything
Many problems involve combinations of data — pairs, triplets, custom states, DP transitions, geometric coordinates, graph nodes with metadata, or any structure that doesn’t fit neatly into built-in hashable types.
With custom hashing, you can create hashable versions of all of them. That means using unordered_set<pair<int,int>> becomes just as easy as using unordered_map<int,int>. More importantly, it becomes efficient.
¶ Custom Hashing Isn’t Just About Avoiding TLE — It’s About Designing Faster Solutions
Once freed from the limitations of default hashing, you begin to notice a shift in how you think about problem-solving.
A large portion of competitive programming involves lookup-heavy tasks:
- Coordinate compression
- Frequency counting
- Detecting duplicates
- Storing visited states
- Graph algorithms with hash maps (common in problems with large or dynamic node states)
- Memoization of dynamic programming states
- Handling millions of small queries
- Tracking patterns or values in arrays, grids, or trees
When each of these operations becomes stable, fast, and collision-resistant, your solutions gain a measurable edge.
This course is designed to refine your understanding of hashing to the point where even the most complex problems feel manageable. You'll be surprised at how often hash functions become the unsung heroes in your code.
¶ A Skill That Separates Good Coders From Great Ones
Mastering custom hashing is very much like learning mental shortcuts used by grandmasters in chess. It isn’t obvious at first glance. Most beginner and intermediate coders use STL containers without ever thinking about their internal mechanics. But as the problems get tougher, those who understand these internals begin to write smarter, faster, and more reliable code.
This course will guide you into that deeper level.
You’ll not only learn how hashing works, but also how seasoned competitive programmers think about it. You’ll see why some people use 64-bit hashing, others prefer certain mixing functions, and why clever bit manipulations can dramatically improve distribution.
And most importantly, you’ll learn to choose the right hashing strategy for different kinds of inputs — integers, pairs, vectors, strings, long long values, and even complex objects that you'd never imagine could be hashed efficiently.
¶ Hashing Is More Than a Trick — It’s a Philosophy of Writing Safer Code
One of the beautiful parts of competitive programming is how it forces you to understand the hidden layers beneath abstractions. Hashing is a perfect example. On the surface, it looks like a simple black-box mechanism. But the moment you push boundaries — large constraints, tricky input distributions, or millions of operations — the cracks begin to show, unless you reinforce them with your own design.
Custom hashing is about taking control.
Instead of waiting for STL’s built-in hash to fail you, you build an approach that is stable and efficient, regardless of the input pattern. You don’t let worst-case scenarios ruin your solution. You don’t leave performance to luck.
In a field where milliseconds decide rankings, this mindset gives you a significant advantage.
¶ What Makes a Hash Function Good?
You’ll explore this in detail throughout the course, but it’s worth previewing the core principles here:
- Uniform distribution: The function should map similar keys to very different hash values.
- Low collision probability: Even with millions of operations, collisions should remain minimal.
- Speed: The hash must be extremely fast. Every nanosecond counts in CP.
- Randomization: Adding a bit of unpredictability makes your program resistant to adversarial cases.
- Stability: Same input should always yield the same hash, regardless of external factors.
- Flexibility: It should support composite structures seamlessly.
You’ll study real-world examples of good and bad custom hash functions to understand how small design choices make a big difference.
¶ Where You’ll Use These Skills in Competitive Programming
Almost everywhere.
Hashing appears in:
- hashing coordinates for geometric problems
- storing compressed graph states
- DP with large or complex state transitions
- memoization in recursive search
- string hashing
- rolling hash variations
- building hash maps for subarray problems
- dealing with multiset-like structures efficiently
- comparing complex objects through hashed signatures
- grid problems that require storing (x, y) states
- game theory problems with memoized configurations
Every time you need fast lookups on non-trivial keys, custom hashing becomes an important tool.
¶ The Hidden Confidence That Comes With Mastery
There’s a very real psychological benefit that coders rarely talk about: once you learn custom hashing, you stop feeling afraid of using unordered maps or sets in large constraints. You no longer avoid them out of fear that the hidden test cases will punish your choice of data structure.
Instead, you write solutions with confidence.
You use the right tool at the right moment, knowing that you can control its behavior. This removes hesitation and frees up your creativity for solving the actual problem. More importantly, it sharpens your competitive instincts — you think faster, code faster, and trust your solutions more.
This confidence comes from mastery, and that’s what this course aims to instill.
¶ A Journey From Basics to Deep Expertise
By the end of this course, you won’t just know how to write custom hash functions — you’ll understand hashing as a deep concept. You’ll feel comfortable crafting hashes for any data type, designing mixing strategies tailored to specific constraints, and optimizing them for speed while maintaining safety.
You’ll also learn when hashing is not the answer, and alternative approaches outperform even the best custom hash — a nuance that separates surface-level knowledge from true mastery.
The goal is to make you not just capable but exceptional in using hashing as part of your competitive programming toolkit.
¶ Closing Thoughts
Custom hashing may seem like a technical detail, but it has shaped the strategies of competitive programmers for years. It’s a quiet cornerstone of efficient solutions, the difference between theoretical constant time and actual constant time, and a skill that turns potentially fragile code into rock-solid implementations.
This introduction marks the beginning of a deeper journey. As you move through the course, you’ll uncover layers of understanding that will reshape how you build solutions, analyze complexity, and interact with data structures.
Hashing isn’t just a technique — it’s one of the most elegant forms of control you can exert as a competitive programmer.
And by the time you complete all 100 articles, you won’t just write custom hash functions — you’ll wield them like a craftsperson who knows exactly how to shape performance, precision, and reliability with a few lines of code.
¶ Custom Hash Functions
¶ Beginner Level (1-30)
1. Introduction to Hash Functions in Competitive Programming
2. Understanding the Basics of Hashing
3. What is a Hash Function? Definition and Properties
4. Common Use Cases of Hash Functions in Programming
5. Built-in Hash Functions in Programming Languages
6. Introduction to Collisions in Hashing
7. Handling Collisions: Separate Chaining
8. Handling Collisions: Open Addressing
9. Introduction to Custom Hash Functions
10. Why Use Custom Hash Functions?
11. Basic Properties of a Good Hash Function
12. Simple Hash Functions: Sum of Characters
13. Simple Hash Functions: Product of Characters
14. Simple Hash Functions: XOR-Based Hashing
15. Simple Hash Functions: Polynomial Rolling Hash
16. Introduction to Rolling Hash Technique
17. Rolling Hash for Strings: Basic Implementation
18. Rolling Hash for Substrings: Sliding Window Technique
19. Introduction to Rabin-Karp Algorithm
20. Implementing Rabin-Karp Algorithm for Pattern Matching
21. Hashing Integers: Basic Techniques
22. Hashing Pairs of Integers: Basic Techniques
23. Hashing Tuples: Basic Techniques
24. Hashing Custom Objects: Basic Techniques
25. Introduction to Hash Tables
26. Implementing a Custom Hash Table
27. Using Hash Tables for Frequency Counting
28. Using Hash Tables for Duplicate Detection
29. Using Hash Tables for Two-Sum Problem
30. Basic Problems on Hashing in Competitive Programming
¶ Intermediate Level (31-60)
31. Advanced Properties of Hash Functions
32. Uniform Distribution in Hash Functions
33. Minimizing Collisions in Custom Hash Functions
34. Choosing a Good Modulus for Hash Functions
35. Choosing a Good Base for Polynomial Rolling Hash
36. Double Hashing: Reducing Collisions Further
37. Universal Hashing: Introduction and Applications
38. Perfect Hashing: Introduction and Applications
39. Rolling Hash with Multiple Bases: Reducing Collisions
40. Rolling Hash with Multiple Moduli: Reducing Collisions
41. Rabin-Karp Algorithm with Multiple Hashes
42. Hashing Large Integers: Techniques and Challenges
43. Hashing Large Strings: Techniques and Challenges
44. Hashing Graphs: Basic Techniques
45. Hashing Trees: Basic Techniques
46. Hashing Matrices: Basic Techniques
47. Hashing Sets: Basic Techniques
48. Hashing Multisets: Basic Techniques
49. Hashing Sequences: Basic Techniques
50. Hashing Custom Data Structures: Intermediate Techniques
51. Using Hash Functions for Memoization
52. Using Hash Functions for Caching
53. Using Hash Functions for Load Balancing
54. Using Hash Functions for Sharding
55. Using Hash Functions for Distributed Systems
56. Using Hash Functions for Consistent Hashing
57. Using Hash Functions for Bloom Filters
58. Using Hash Functions for Count-Min Sketch
59. Using Hash Functions for HyperLogLog
60. Intermediate Problems on Hashing in Competitive Programming
¶ Advanced Level (61-90)
61. Cryptographic Hash Functions: Introduction
62. Comparing Cryptographic and Non-Cryptographic Hash Functions
63. Designing Custom Cryptographic Hash Functions
64. Designing Custom Non-Cryptographic Hash Functions
65. Advanced Techniques for Minimizing Collisions
66. Advanced Techniques for Uniform Distribution
67. Advanced Techniques for Choosing Modulus and Base
68. Advanced Techniques for Double Hashing
69. Advanced Techniques for Universal Hashing
70. Advanced Techniques for Perfect Hashing
71. Advanced Techniques for Rolling Hash
72. Advanced Techniques for Rabin-Karp Algorithm
73. Advanced Techniques for Hashing Large Integers
74. Advanced Techniques for Hashing Large Strings
75. Advanced Techniques for Hashing Graphs
76. Advanced Techniques for Hashing Trees
77. Advanced Techniques for Hashing Matrices
78. Advanced Techniques for Hashing Sets
79. Advanced Techniques for Hashing Multisets
80. Advanced Techniques for Hashing Sequences
81. Advanced Techniques for Hashing Custom Data Structures
82. Using Hash Functions for Advanced Memoization
83. Using Hash Functions for Advanced Caching
84. Using Hash Functions for Advanced Load Balancing
85. Using Hash Functions for Advanced Sharding
86. Using Hash Functions for Advanced Distributed Systems
87. Using Hash Functions for Advanced Consistent Hashing
88. Using Hash Functions for Advanced Bloom Filters
89. Using Hash Functions for Advanced Count-Min Sketch
90. Advanced Problems on Hashing in Competitive Programming
¶ Expert Level (91-100)
91. Designing Hash Functions for Specific Applications
92. Designing Hash Functions for Real-Time Systems
93. Designing Hash Functions for Large-Scale Systems
94. Designing Hash Functions for Parallel Systems
95. Designing Hash Functions for Distributed Systems
96. Designing Hash Functions for Secure Systems
97. Designing Hash Functions for High-Performance Systems
98. Open Problems in Hashing and Competitive Programming
99. Research Trends in Custom Hash Functions
100. Future Directions in Custom Hash Functions