Problem 32 - Discrete Mathematics with Applications

Question

Student records are maintained in a table using the hashing function \(h(x)=x \bmod 9767,\) where \(x\) denotes the student's social security number. Compute the location in the table corresponding to the given key, where the record is stored. $$876-54-3210$$

Step-by-Step Solution

Answer

The student's record is stored at index \(4171\) in the table.

Step 1: Remove dashes from the Social Security number

The given Social Security number is \(876-54-3210\). In order to make it a single number to be used in the hashing function, we need to remove the dashes. Upon doing this, we get the number \(x = 876543210\).

Step 2: Apply the hashing function

Now apply the given hashing function \(h(x) = x\mod 9767\) to the number obtained in Step 1. This hashing function will give us the location in the table where the student's record is stored. \(h(x) = 876543210\mod 9767\)

Step 3: Compute Remainder

To find the remainder when \(876543210\) is divided by \(9767\), perform the modulo operation: \(h(x) = 4171\)

Step 4: Find the Location

The location in the table is found by evaluating \(h(x)\). From Step 3, we found that \(h(x) = 4171\). Hence, the location in the table where the student's record is stored is at index \(4171\).

Key Concepts

Modulo Operation

The modulo operation is a fundamental mathematical concept used commonly in computer science. It helps in finding the remainder of a division between two integers. When we perform a modulo operation, we're essentially dividing one number by another and returning what's left after the whole division.
For instance, when we have a number, say 876543210, and we use the modulo operator with 9767, we're seeking to find the remainder.
Here's what happens:

• Divide 876543210 by 9767.
• The result yields a quotient and a remainder.
• The remainder is 4171, which means that 876543210 can be split into whole parts of 9767 and 4171 left over.

Modulo is often represented by the symbol \( \mod \). It is widely used in hashing algorithms as it efficiently maps a vast range of inputs to a fixed range of outputs, like table indices.

Hash Functions

A hash function is a special type of function in computer science. It transforms input data of any size into a fixed size value, typically a number that represents the input data. This numerical value is often referred to as a hash code, digest, or simply a hash. Hash functions serve as the backbone of hash tables.

In our exercise, the hash function is defined as \( h(x) = x \mod 9767 \). This means:

• Each input, in this case, the social security number, undergoes the modulo operation.
• The function returns an index, which represents the location in a hash table.

Hash functions are crucial because they ensure data is quickly retrievable. They do this by changing the input into an index, which can be used to store and search data efficiently. The effectiveness of a hash function lies in its ability to uniformly distribute inputs across a hash table, minimizing collisions where multiple inputs share the same index.

Data Storage

Data storage refers to the method of recording and retaining digital information. In computer systems, this can be done using various structures, like hash tables, which use hash functions to manage and retrieve data efficiently.
When we use hash tables for data storage, each piece of data is stored at a particular index, which is generated by a hash function like \( h(x) = x \mod 9767 \).
Here's how it works:

• The hash function converts an identifier such as a social security number into a uniform location index.
• That index points to where the record is stored in the table, allowing for fast retrieval.

A well-designed hash table allows for constant average time complexity for both storing and retrieving data, making it an efficient approach when dealing with large datasets. This efficiency is crucial when data systems need to access or update records rapidly, as in databases or memory storage systems.