Chaining

Chaining is a fundamental collision resolution strategy used in hash table implementations to handle situations where multiple keys hash to the same array index. This method maintains the integrity and efficiency of hash table operations while gracefully managing collisions.

Core Mechanism

When implementing chaining, each slot in the hash table's underlying array contains a linked list (or similar data structure) rather than a single key-value pair. When a collision occurs:

1. The new element is simply appended to the list at the hashed index
2. Each node in the chain stores both the key and value
3. Lookups traverse the chain to find the matching key

class HashNode:
    key
    value
    next

class HashTable:
    array[size] of HashNode

Performance Characteristics

Time Complexity

• Average case: O(1 + α) where α is the load factor
• Worst case: O(n) when all elements hash to the same index
• Space Complexity: O(n + m) where:
  • n = number of elements
  • m = size of hash table array

Load Factor Impact

The performance of chaining is directly related to the load factor:

• Lower load factors (< 0.7) maintain short chains
• Higher load factors increase average chain length
• Unlike open addressing, can exceed load factor of 1.0

Advantages and Disadvantages

Advantages

• Simple implementation
• Cache for small chain lengths
• Handles clustering better than open addressing
• Never requires table resizing due to collisions

Disadvantages

• Extra memory overhead for link pointers
• Poor locality of reference for long chains
• Memory Management overhead for nodes

Implementation Considerations

Chain Data Structure Choice

Several options exist for implementing the chains:

1. Singly Linked List (most common)
2. Self-Balancing BST for improved worst-case
3. Dynamic Array for small chains

Chain Operations

INSERT(key, value):
    index = hash(key) % tableSize
    node = new HashNode(key, value)
    node.next = array[index]
    array[index] = node

SEARCH(key):
    index = hash(key) % tableSize
    current = array[index]
    while current != null:
        if current.key == key:
            return current.value
        current = current.next
    return null

Optimizations

Chain Length Management

• Monitor maximum chain length
• Consider rehashing when chains exceed threshold
• Implement dynamic resizing based on load factor

Advanced Techniques

1. Ordered Chains: Maintain sorted chains for better search
2. Head-and-Tail Chains: Track both ends for faster insertion
3. Hybrid Chaining: Switch data structures based on chain length

Applications

Chaining is particularly effective in:

• Symbol Table in compilers
• Cache implementations with high collision rates
• Database handling non-unique keys
• Set implementations in programming languages

The simplicity and reliability of chaining make it a popular choice for hash table implementations, especially when collision patterns are unpredictable or when memory overhead is not a primary concern.