Question

Load the records of Exercise 13.27 into an expandable hash file, using linear hash. ing. Start with a single disk block, using the hash function \(h_{0}=K \bmod 2^{0},\) and show how the file grows and how the hash functions change as the records are inserted. Assume that blocks are split whenever an overflow occurs, and show the value of \(n\) at each stage.

Short answer

The hash file starts with a single disk block and the hash function \(h_{0}=K \mod 2^{0}\). With each record's insertion, the hash function is applied, the file grows, the value of \(n\) increases by one, and if overflows occur, the block is split and hash function is adjusted. Repeat these steps until all records are inserted.

Step-by-step solution

Initialization

Begin with an empty hash table and a single disk block, and define the hash function \(h_{0} = K \mod 2^{0}\). The parameter \(n\) has the initial value 0, which is equivalent to the number of disk blocks.

Insert the First Record

Insert the first record in the table. Apply the hash function \(h_{0}\) on the key of the record. Since \(h_{0}\) function is \(K \mod 2^{0}\), any number modulo 1 is always 0. Therefore, the first record will go into the initial block. After insertion, \(n\) becomes 1, indicating that one block is in use.

Process of Inserting Subsequent Records and Handling Overflows

For every subsequent record insertion, apply the current hash function. If no overflow happens, simply increase the value of \(n\). If an overflow occurs, proceed to Step 4.

Handle Overflow by Splitting Blocks

Whenever an overflow happens, split the block where the overflow occurred. Create a new disk block, and for the split block and the new block, adjust the hash function. The number \(n\) increases by one each time a block is split.

Repeat the Process

Continue the process of inserting records, adjusting the hash function, managing overflows, and increasing the value of \(n\) each time a record is inserted or a block is split until all records have been inserted.

Key concepts

Expandable Hash File

Expandable hash files are a dynamic way to store data that grows with the dataset. Instead of setting a fixed number of buckets or blocks like traditional hash tables, expandable hash files start small and grow as needed. This flexibility prevents wasted space and optimizes performance.

• Initially, you begin with a minimal setup, often just one disk block.
• As more records are inserted, the system adapts by adding blocks, ensuring each insertion is efficiently managed.

The beauty of this method is its scalability. The hash file expands seamlessly by adding new blocks, reducing the need for constant rehashing of the entire dataset. With each block addition, new bucket pointers may be introduced, elegantly accommodating more records.

Overflow Handling

Overflow handling is crucial in hashing, especially when multiple records hash to the same location. As records are added, the allocated block can exceed its capacity, leading to an overflow problem. In linear hashing, this is managed by splitting blocks:

• When a block reaches overflow, it is split, creating a new block to distribute the records.
• The records from the overflowed block are rehashed to either remain in the original block or move to the newly created block.
• This process helps maintain efficient access times and avoids long search times, which would otherwise occur if many records shared a single block.

Efficient overflow handling ensures the hash table remains well-balanced, optimizing both storage space and retrieval times.

Hash Functions

Hash functions are the backbone of a hashing system. They take an input (or 'key') and return a number, which determines where the key-value pair is stored in the hash file. This system is key to achieving quick data access. In linear hashing:

• An initial hash function, such as \(h_0 = K \mod 2^0\), is defined.
• As the dataset grows, the hash function is adjusted to accommodate more records and reduce collisions. This often involves increasing the modulus, effectively increasing the number of addressable blocks.

These alterations are crucial for maintaining efficient file management, ensuring that no single block becomes too congested as more data is added.

Disk Blocks

Disk blocks are the storage units used in hashing systems. Think of them as containers for records. In the context of linear hashing and expandable hash files, disk blocks play a vital role:

• Initially, the setup begins with a limited number of disk blocks, often just one.
• As records are inserted and overflow is managed by splitting blocks, new disk blocks are added to accommodate additional data.
• These blocks ensure data is stored efficiently, balancing load and mitigating the risk of overflowing any single block.

Each block aims to minimize disk read operations, ensuring the hash file remains efficient in storage and retrieval, leveraging block splits and rehashing to uniformly distribute records across available blocks.