A hash function is a function that takes an input of arbitrary size and produces a fixed-size output, known as a hash value or hash code. In hash tables, this hash value is used to determine the index where the data element will be stored. The importance lies in enabling efficient data retrieval – ideally, accessing an element by its key takes constant time (O(1)).

Collisions occur when two different keys produce the same hash value. This is unavoidable due to the finite size of the hash table and the potentially infinite number of possible keys. Two common collision resolution techniques are:

• Separate Chaining: Each index in the hash table points to a linked list (or other data structure like a tree). When a collision occurs, the new key-value pair is added to the linked list at that index.

  • Advantages: Simple to implement, can handle a large number of collisions, doesn't require resizing the hash table as frequently.
  • Disadvantages: Performance degrades if linked lists become very long (O(n) lookup time in the worst case). Requires extra memory for the linked lists.

• Open Addressing: When a collision occurs, the algorithm probes for an empty slot within the hash table itself. Common probing methods include linear probing, quadratic probing, and double hashing.

  • Advantages: More space-efficient than separate chaining (no extra memory for linked lists).
  • Disadvantages: Can suffer from clustering (especially linear probing), which degrades performance. Requires careful consideration of load factor to avoid performance issues. Deletion can be complex.

Modular arithmetic is a fundamental component of many hash functions. It involves taking the remainder of a division operation. In a hash function, the hash value is often calculated by taking the input key modulo the size of the hash table. This ensures that the hash value falls within the valid range of indices for the table (0 to Table Size - 1).

The choice of the modulus value significantly affects the distribution of hash values. If the modulus is a prime number, it generally leads to a more uniform distribution of hash values, reducing the likelihood of clustering. If the modulus is a power of 2, it can lead to poor distribution, especially if the input keys have patterns that are multiples of the power of 2.

A universal hash function is a family of hash functions, each chosen randomly from a specified distribution. The key benefit of using a universal hash function is that it provides a provable guarantee that the number of collisions will be limited, regardless of the input keys. This avoids the worst-case collision scenarios that can occur with a single, fixed hash function.

Example of a universal hash function: Consider a universal hash function of the form: h(k) = (a * k + b) mod p, where:

• k is the input key.
• a is a randomly chosen integer between 1 and p-1.
• b is a randomly chosen integer between 0 and p-1.
• p is a prime number larger than the largest possible key value.

By choosing 'a' and 'b' randomly, we ensure that the hash function will distribute keys uniformly across the hash table, even if the keys have patterns that would cause collisions with a fixed hash function.

A good hash function should possess the following properties:

• Uniformity: It should distribute keys evenly across the hash table, minimizing collisions.
• Deterministic: For a given key, it should always produce the same hash value.
• Efficiency: It should be computationally fast to calculate.

The load factor of a hash table is the ratio of the number of elements stored in the table to the size of the table (number of slots). It is calculated as: Load Factor = (Number of Elements) / (Table Size). A high load factor indicates that the table is becoming full, leading to more collisions and slower performance. As the load factor increases, the average lookup time increases towards O(n) in the worst case.

Strategies for managing the load factor include:

• Resizing: When the load factor exceeds a certain threshold (e.g., 0.75), the hash table is resized to a larger table. All elements are rehashed and reinserted into the new table.
• Choosing a suitable table size: Selecting a prime number for the table size can help to reduce collisions, especially when using certain hash functions.

Example of a simple hash function: A very basic hash function could be summing the ASCII values of the characters in a string and taking the result modulo the table size. For example, for the string "abc" with a table size of 10, the hash value would be (97 + 98 + 99) % 10 = 294 % 10 = 4. Weakness: This function is prone to collisions because different strings can easily have the same sum of ASCII values. Strings with similar characters will have similar hash values, leading to clustering.