Principles of Hash Tables and Collision Resolution

Description
A Hash Table is a data structure that allows direct access to memory storage locations based on a key. It maps keys to specific indices of an array via a hash function, achieving average time complexity of O(1) for insertion, deletion, and lookup operations. The core challenge is hash collision (different keys mapping to the same index), which must be resolved through strategies.

1. Role of the Hash Function
The hash function converts a key of arbitrary size (e.g., a string, an object) into an integer within a fixed range (i.e., an array index). An ideal hash function should satisfy:

• Fast computation
• Uniform distribution (to reduce collisions)
  For example, for an integer key using modulo: index = key % table_size.

2. Causes of Hash Collisions
A collision occurs when two different keys compute to the same index via the hash function. For example:

• Table size is 10, keys 15 and 25 both map to index 5 via key % 10.
  Collisions are inevitable (due to the pigeonhole principle), hence resolution strategies are required.

3. Collision Resolution: Separate Chaining

• Principle: Each array index corresponds to a linked list (or other data structure). All key-value pairs mapping to the same index are stored in that list.
• Operation Example:
  • Insert key 15 (index 5): Add a node to the end of the linked list at index 5.
  • Insert key 25 (index 5): Continue adding to the same linked list.
  • Lookup key 25: Compute index 5, traverse the linked list comparing keys.
• Advantages: Simple and effective, performance approaches O(1) when lists are short.
• Disadvantages: Degrades to O(n) for long lists, requires dynamic resizing for optimization.

4. Collision Resolution: Open Addressing

• Principle: All elements are stored directly in the array. On collision, a probing sequence is followed to find the next empty slot.
• Linear Probing: If index i is occupied, try i+1, i+2... until an empty slot is found.
  • Example: Insert key 25 (index 5 occupied) → try index 6.
  • Disadvantage: Prone to clustering (consecutive occupied slots), reducing efficiency.
• Quadratic Probing: Avoids clustering by probing with a quadratic sequence (e.g., i+1², i+2²).
• Double Hashing: Uses a second hash function to calculate the step size, further dispersing elements.

5. Performance Optimization and Resizing

• Load Factor: Number of elements / Table size. When the load factor exceeds a threshold (e.g., 0.75), resizing is required:
  1. Create a new array (typically double the size).
  2. Rehash all keys into the new array.
• Dynamic Resizing: Avoids frequent collisions, ensuring average time complexity remains close to O(1).

Summary
The core of a hash table lies in hash function design (for uniform distribution) and collision resolution strategies (Separate Chaining or Open Addressing). Combined with dynamic resizing, it can efficiently support fast data operations. In practice, Separate Chaining is more common due to ease of implementation, while Open Addressing has advantages in memory-constrained scenarios.