Hash tables are powerful data structures used in computer science to efficiently store and retrieve key-value pairs. They work by using a hash function to map keys to indices in an array, allowing for constant-time access on average. In this explanation, we'll delve into the workings of a hash table with a specific hash function and a small array size.

Let's imagine we have a small data array with a size of 11, and our keys are 3-character strings composed of uppercase letters from A to Z. To map these strings to indices in our array, we'll define a hash function. Our hash function will convert each character of the string into a number (0 to 25, corresponding to their position in the alphabet) and then combine these numbers using a base-26 conversion scheme.

Here's how it works:

1. Each character in the string is converted to a number from 0 to 25 based on its position in the alphabet. For example, 'A' becomes 0, 'B' becomes 1, and so on up tob 'Z' becoming 25.

2. We treat the string as a base-26 number, where the first character's value is multiplied by 26₂ (262), the second character's value is multiplied by 26₁ (26), and the third character's value is multiplied by 26₀ (1). We sum these products to get a single integer value representing the entire string.

3. To ensure this integer value falls within the range of our array size (0 to 10), we take its modulo with 11 (k % 11).

Let's consider an example to illustrate this process:

Suppose we want to hash the string "ABC".
'A' maps to 0, 'B' maps to 1, and 'C' maps to 2.
Using our base-26 conversion, the integer value for "ABC" is calculated as (0 * 26₂) + (1 * 26₁) + (2 * 26₀) = 0 + 26 + 2 = 28.
Taking modulo 11, we get 28 % 11 = 6.

So, "ABC" would be hashed to index 6 in our array.

Now, let's proceed with inserting some keys into our hash table:

1. Inserting Keys:

We have a list of 3-letter airport codes: PHL, ORY, GCM, HKG, GLA, AKL, FRA, LAX, DCA.

2. Hashing the Keys:

3. Handling Collisions:
We encounter collisions when two keys hash to the same index. To address collisions, we need a strategy for resolving them. One common approach is to use separate chaining, where each index in the array holds a linked list of key-value pairs that hashed to that index.

4. Final Hash Table:
After handling collisions, our hash table might look like this:

In this hash table, keys that hashed to the same index are stored in the same slot, forming a linked list structure. This allows for efficient retrieval of values even in the presence of collisions.

Hash tables are fundamental data structures used in various applications such as database indexing, caching, and symbol tables in programming languages. Understanding how they work and how to handle collisions is crucial for designing efficient algorithms and systems.