Hash Tables Demystified — Hash Functions, Chaining, and Open Addressing Explained | Chapter 11 of Intro to Algorithms

Efficient data storage and retrieval are fundamental challenges in computer science. Chapter 11 of introduction to algorithms explores hash tables—an essential data structure enabling near-constant-time INSERT, SEARCH, and DELETE operations for dynamic sets. This chapter dives into how hash functions work, why collisions are inevitable, and how strategies like chaining and open addressing maintain performance under real-world constraints.

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From Direct Addressing to Hash Tables

Direct addressing uses the key itself as an array index, making access extremely fast. However, this is impractical for large key universes (e.g., billions of possible keys but only thousands in use). Hash tables compress the universe of keys into a manageable array size using a hash function h(k), which maps each key k to a slot in a table of size m.

Hash Functions and Handling Collisions

A well-designed hash function distributes keys uniformly across the table. Yet, with more possible keys than table slots, collisions (two keys mapping to the same slot) are inevitable. Two primary strategies address collisions:

• Chaining: Each slot contains a linked list of entries. New elements are inserted at the front. SEARCH and DELETE only need to scan one chain. The load factor α = n/m (n = elements, m = slots) determines average chain length and search time—ideally kept low.
• Open Addressing: All entries are stored within the table, and collisions are resolved by probing for alternate slots. Probing sequences (like linear or double hashing) dictate how alternative locations are chosen. This method is highly memory efficient but requires careful deletion logic and good hash function design to avoid clustering.

Chaining in Depth

In chaining, each table slot holds a pointer to a linked list of all entries mapping to that slot. Operations:

• INSERT: Place new item at the head of its chain.
• SEARCH: Traverse the list at the appropriate slot.
• DELETE: Remove item from the chain using its pointer.

With a low load factor and independent uniform hashing, average-case search and insert times are Θ(1 + α).

Open Addressing Explained

Open addressing stores all elements inside the array itself, using probing to find alternative slots on collision. Key probing strategies:

• Linear Probing: Tries the next slot in sequence (h(k, i) = (h₁(k) + i) mod m), but can suffer from clustering.
• Double Hashing: Uses a secondary hash (h(k, i) = (h₁(k) + i × h₂(k)) mod m) to create better-distributed probe sequences and avoid clustering.

Deleting elements is more complex; "DELETED" markers or rehashing are often needed to preserve table structure.

Hash Function Design & Universal Hashing

A good hash function distributes keys evenly and efficiently. Popular methods:

• Division Method: h(k) = k mod m (best when m is prime).
• Multiplication Method: h(k) = floor(m × (kA mod 1)), for 0 < A < 1.
• Multiply-Shift: For tables where m is a power of two, use bitwise operations for speed.

Universal hashing selects the hash function randomly from a family of functions, guaranteeing low collision probability regardless of input distribution—a powerful theoretical and practical safeguard. Modern hardware also enables random hashing for variable-length or cryptographic keys.

Performance, Practical Considerations & Modern Trends

Hash table performance is deeply tied to the load factor (α = n/m). For open addressing, expected probes for unsuccessful search are ≤ 1/(1–α); for successful search, ≤ (1/α) × ln(1/(1–α)). Linear probing benefits from memory cache-friendliness, while double hashing offers superior clustering resistance. When dealing with strings or variable-length keys, cryptographic hash functions may be needed for uniform distribution and security.

On modern CPUs, hash functions are designed for fast computation (like the Wee Hash Function) to minimize overhead and maximize data throughput. Understanding these trade-offs is essential for building scalable, high-performance software systems.

Key Terms and Glossary

• Hash Table: Array-based structure for dynamic sets
• Hash Function: Maps keys to slot indices
• Load Factor (α): n/m, average elements per slot
• Collision: Two keys mapping to the same slot
• Chaining: Linked lists to resolve collisions
• Open Addressing: Probing within the array to resolve collisions
• Linear Probing: Sequential probing strategy
• Double Hashing: Uses two hash functions for probe sequence
• Universal Hashing: Randomized function selection for low collision
• Cryptographic Hash Function: Pseudorandom, fixed-output mapping
• Multiply-Shift Hashing: Fast hashing using bit shifts

Conclusion: The Power of Hash Tables

Chapter 11 of introduction to algorithms makes it clear: hash tables are indispensable for dynamic data management and are used everywhere from database indexing to caching, symbol tables, and beyond. Mastering hash function design and collision resolution strategies unlocks high-performance, scalable algorithms for real-world systems.

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