In hashing, how can hashing functions be classified based on the method used to compute the hash value from a given key?

Introduction / Context:
Hashing functions are at the core of hash table implementations and many other algorithms requiring fast lookup. A hashing function transforms a key into an index in a hash table. Different strategies exist for computing hash values, and these strategies are often grouped into categories. Understanding these categories helps in selecting or designing appropriate hash functions for different use cases. This question asks you to classify hashing functions based on how they compute hash values from keys.

Given Data / Assumptions:

• The goal of a hashing function is to distribute keys uniformly over the address space of the table.
• Keys may be numeric, character based, or composite.
• Textbooks describe several standard hashing methods with specific names.
• The question is about classification by method, not by implementation language or table size.

Concept / Approach:
Hash functions are often grouped into several main methods. The division or modular method computes the hash value as key mod table_size. The multiplication method multiplies the key by a constant fraction and uses the fractional part to determine the index. The mid square method squares the key and extracts some middle bits or digits as the hash value. The folding method splits the key into parts and combines them, often by addition, to form a hash. Digit analysis examines specific digits of the key, discarding or combining them based on patterns. These classifications are independent of programming language and table size and focus on the mathematical approach used to derive the hash from the key.

Step-by-Step Solution:
Step 1: Recall standard hashing methods taught in data structures courses: division, multiplication, mid square, folding, and digit analysis. Step 2: Understand that these methods describe different ways of transforming the key into a hash value. Step 3: Evaluate option a, which lists division or modular method, multiplication method, mid square method, folding method, and digit analysis method. Step 4: Compare this with option b, which oversimplifies classification to only good or bad, which is informal and not the typical textbook classification. Step 5: Reject options c and d, which classify hash functions by programming language or table size, neither of which reflects the method used for computation.

Verification / Alternative check:
Standard data structures textbooks provide sections with headings such as division method, multiplication method, mid square method, and folding method, often accompanied by examples and discussions of advantages and disadvantages. They sometimes mention digit analysis as another method, particularly for numeric keys. These texts treat these methods as distinct categories, confirming that option a accurately reflects recognised classifications.

Why Other Options Are Wrong:
Option b offers a subjective classification into good and bad, which may be used informally but does not help in understanding specific computational techniques. Option c focuses on programming languages, which are irrelevant to the theoretical method; a division based hash function can be implemented in any language. Option d ties classification only to table size, but table size is a parameter of the hash table rather than a property of the hash function method itself.

Common Pitfalls:
A frequent mistake is to think that the specific arithmetic used in code is arbitrary and not part of a broader category. Another pitfall is to rely solely on library provided hash functions without understanding underlying methods, which can be important when designing custom keys or distributed hash tables. For exam questions, remember the standard method names: division or modular, multiplication, mid square, folding, and digit analysis, as summarised in option a.

Final Answer:
Hashing functions can be classified into methods such as division or modular method, multiplication method, mid square method, folding method, and digit analysis method, according to how they compute the hash value from the key.