Hash Table

A hash table (also known as a hash map) is a fundamental data structure that enables efficient storage and retrieval of key-value pairs. It works by transforming keys into array indices using a hash function, achieving constant-time O(1) average case performance for insertions, deletions, and lookups.

Core Concepts

Hash Function

The heart of a hash table is its hash function, which:

• Converts keys of any size into fixed-size integers
• Distributes values uniformly across the available space
• Generates the same output consistently for identical inputs
• Minimizes collision between different inputs

Collision Resolution

When multiple keys hash to the same index, collision resolution becomes necessary. Common strategies include:

1. Chaining: Each array slot contains a linked list of elements
2. Open Addressing: Probing for next available slot using methods like:
   • Linear probing
   • Quadratic probing
   • Double hashing

Performance Characteristics

Time Complexity

• Average case: O(1) for insert, delete, search
• Worst case: O(n) when many collisions occur
• Space Complexity: O(n) where n is the number of stored items

Load Factor

The load factor (α) = n/m, where:

• n = number of stored elements
• m = size of the hash table
• Typically maintained below 0.7 for optimal performance

Applications

Hash tables are extensively used in:

• Database for indexing
• Cache implementations
• Symbol Table in compilers
• Dictionary implementations in programming languages
• Set data structure implementations

Implementation Considerations

Resizing

• Dynamic Array is needed when load factor exceeds threshold
• Usually doubles table size when capacity is reached
• Requires rehashing all existing elements

Key Requirements

Keys must be:

• Immutable while in the hash table
• Implement consistent equals and hashCode methods
• Distribute well across the hash space

Common Operations

INSERT(key, value):
    index = hash(key) % tableSize
    store (key,value) at index (handling collisions)

SEARCH(key):
    index = hash(key) % tableSize
    return value at index (following collision chain if necessary)

DELETE(key):
    index = hash(key) % tableSize
    remove (key,value) at index (handling collision chain)

Trade-offs

Advantages

• Constant-time average case operations
• Flexible key types
• Space-efficient for sparse data

Disadvantages

• No ordering of keys
• Performance degrades with poor hash functions
• Extra space needed for collision handling

Hash tables represent a crucial Algorithm tool in computer science, balancing theoretical elegance with practical utility in countless applications.