🔁Data Structures Unit 9 – Hash Tables and Hash Functions

What's the Big Idea?

• Hash tables provide an efficient way to store and retrieve data using key-value pairs
• Utilize a hash function to compute an index into an array of buckets or slots from which the desired value can be found
• Enable fast insertion, deletion, and lookup operations with an average time complexity of O(1)
• Consist of two main components: an array (the table where data is stored) and a hash function (used to map keys to specific indexes in the array)
• Ideal for scenarios that prioritize search and retrieval efficiency over the order of elements
• Commonly used in database indexing, caches, unique data representation, and more
• Facilitate the implementation of associative arrays or dictionaries, where keys are mapped to specific values

Key Concepts

• Hash function maps a given key to a specific index in the hash table's array
  • Determines the location where the key-value pair will be stored
• Collision occurs when two different keys hash to the same index in the table
  • Collision resolution techniques, such as chaining and open addressing, handle collisions
• Load factor represents the ratio of the number of entries to the size of the array
  • Influences the performance and likelihood of collisions in the hash table
• Rehashing is the process of resizing the hash table's array when the load factor exceeds a certain threshold
  • Involves creating a new array with a larger size and transferring all elements to the new array
• Uniform distribution is a desirable property of a hash function, ensuring that keys are evenly distributed across the array
• Determinism means that a hash function should always produce the same hash code for the same input key

How It Works

• Hash function takes a key as input and applies a mathematical formula to generate a hash code
  • Hash code is an integer value that corresponds to an index in the hash table's array
• Key-value pair is stored in the array at the index determined by the hash code
• Collisions are resolved using techniques like chaining (linked lists) or open addressing (probing)
  • Chaining maintains a linked list at each index to handle multiple key-value pairs with the same hash code
  • Open addressing probes the array for the next empty slot until a suitable position is found
• Retrieval of a value involves computing the hash code of the given key and accessing the corresponding index in the array
• If collisions were resolved using chaining, the linked list at the index is traversed to find the desired key-value pair
• If open addressing was used, the probing sequence is followed until the key is found or determined to be absent
• Deletion of a key-value pair involves locating the entry and removing it from the hash table
  • In chaining, the entry is removed from the linked list
  • In open addressing, the entry is marked as deleted using a special flag or tombstone value

Types and Variations

• Separate chaining uses linked lists to handle collisions, allowing multiple key-value pairs to be stored at the same index
• Open addressing, including linear probing, quadratic probing, and double hashing, probes the array for the next empty slot in a specific sequence
• Robin Hood hashing is a variation of open addressing that aims to distribute the load more evenly by swapping elements during insertion
• Cuckoo hashing uses multiple hash functions and allows key-value pairs to be moved between different tables to resolve collisions
• Hopscotch hashing combines the ideas of chaining and open addressing, using a neighborhood of buckets for each hash code
• Coalesced chaining combines separate chaining and open addressing by linking together the key-value pairs within the same table
• Perfect hashing uses a two-level hash table structure to ensure O(1) time complexity for lookups without any collisions

Pros and Cons

Pros:

• Efficient insertion, deletion, and lookup operations with an average time complexity of O(1)
• Ideal for scenarios that prioritize fast search and retrieval over the order of elements
• Suitable for implementing associative arrays, dictionaries, and key-value stores
• Can be used for various applications, such as database indexing, caches, and symbol tables

Cons:

• Hash collisions can occur when multiple keys map to the same hash code, impacting performance
• The choice of hash function is crucial for achieving good distribution and minimizing collisions
• Rehashing can be expensive when the hash table needs to be resized due to a high load factor
• The order of elements is not preserved, as the hash function determines the storage location
• Inefficient for scenarios that require sorted data or range queries
• Memory overhead due to the use of an array and additional data structures for collision resolution

Real-World Applications

• Database indexing uses hash tables to enable fast searching and retrieval of records based on specific fields
• Caching systems employ hash tables to store frequently accessed data in memory for quick access
• Compilers utilize hash tables for symbol tables to keep track of identifiers and their attributes
• Password verification systems securely store hashed passwords for efficient authentication
• Spell checkers use hash tables to quickly look up words in a dictionary
• Network routers rely on hash tables for efficient IP address lookup and packet forwarding
• Cryptographic hash functions, such as SHA-256, generate unique fixed-size hash codes for data integrity and security purposes

Coding It Up

• Hash table implementations typically use an array to store key-value pairs and a hash function to compute array indices
• Collision resolution techniques, such as separate chaining or open addressing, are implemented to handle hash collisions
• The choice of hash function depends on the data type of the keys and the desired distribution properties
• Common operations include insertion, deletion, and lookup, which involve computing the hash code and accessing the corresponding array index
• Resizing the hash table is triggered when the load factor exceeds a certain threshold, requiring rehashing of all elements
• Language-specific libraries, such as

  std::unordered_map

  in C++ or

  dict

  in Python, provide built-in hash table implementations
• Custom hash table implementations can be developed to suit specific requirements or to learn the underlying concepts

Common Pitfalls and Tips

• Choosing a weak hash function can lead to poor distribution and frequent collisions, impacting performance
  • Ensure the hash function distributes keys evenly across the array and minimizes clustering
• Not handling collisions properly can result in lost data or incorrect search results
  • Implement appropriate collision resolution techniques, such as separate chaining or open addressing
• Failing to resize the hash table when the load factor becomes too high can degrade performance
  • Monitor the load factor and trigger rehashing when necessary to maintain efficiency
• Overlooking the potential for hash table attacks, such as denial-of-service (DoS) attacks that exploit collisions
  • Employ techniques like salting or using cryptographic hash functions to mitigate vulnerabilities
• Not considering the memory overhead associated with hash tables, especially with large datasets
  • Assess the trade-off between performance and memory usage based on the specific requirements of the application
• Ignoring the possibility of key mutations, which can lead to incorrect behavior if keys are modified after insertion
  • Design the hash table to handle immutable keys or provide safe methods for key updates
• Neglecting to handle null or empty keys appropriately, which may cause unexpected behavior
  • Define a consistent strategy for handling special cases and document it clearly in the implementation.