Question

Draw the 11 -entry hash table that results from using the hash function, \(h(i)=(3 i+5) \bmod 11,\) to hash the keys 12,44,13,88,23,94,11,39,20 \(16,\) and \(5,\) assuming collisions are handled by chaining.

Short answer

The hash table entries result in chains at the indices calculated based on the given hash function and keys.

Step-by-step solution

- Apply the hash function

Calculate the hash value for each key using the formula: \[ h(i) = (3i + 5) \bmod 11 \]

- Calculate hash for each key

- For key 12: \[ h(12) = (3 \times 12 + 5) \bmod 11 = 41 \bmod 11 = 8 \]- For key 44: \[ h(44) = (3 \times 44 + 5) \bmod 11 = 137 \bmod 11 = 5 \]- For key 13: \[ h(13) = (3 \times 13 + 5) \bmod 11 = 44 \bmod 11 = 0 \]- For key 88: \[ h(88) = (3 \times 88 + 5) \bmod 11 = 269 \bmod 11 = 5 \]- For key 23: \[ h(23) = (3 \times 23 + 5) \bmod 11 = 74 \bmod 11 = 8 \]... Continue for each key: 94, 11, 39, 20, 16, 5.

- Handle collisions

Use chaining to handle collisions. Place keys sharing the same hash value into a linked list (chain) at that hash index.

- Build the hash table

Construct the hash table with 11 entries, inserting each key into the calculated index while forming chains where necessary.

- Finalize the hash table

Verify all keys are placed correctly and chains are maintained at each index. The complete hash table will look as follows:

Key concepts

hash function

A hash function is an important concept in computer science. It takes an input (or 'key') and returns a fixed-size string of bytes. The primary purpose of a hash function is to map data of arbitrary size to data of a fixed size. In hashing tables, this function helps determine the index at which a key should be stored. For example, in the given exercise, the hash function used is: \[ h(i) = (3i + 5) \bmod 11 \] This function will ensure that each key is assigned an index between 0 and 10 (since there are 11 entries), leveraging the modulo operation.

collision handling

Hashing often results in more than one key mapping to the same index. This situation is known as a collision. Handling collisions efficiently is crucial to maintain the performance of a hash table. In our exercise, collisions occur multiple times. For instance, both keys 44 and 88 map to the index 5. To handle these collisions, chaining is used. Chaining involves constructing a linked list (or chain) at each index of the hash table. Thus, when a collision occurs, instead of replacing the existing key, the new key is added to the list. This way, each index of our hash table can store multiple keys, maintaining the integrity of the table.

chaining in hash tables

Chaining is an effective method to handle collisions in hash tables. When two keys hash to the same slot, they are stored in a linked list at that slot. This method is simple and easy to implement. In the provided exercise, keys 23 and 12 both hash to index 8. Using chaining, these keys (along with any future keys that hash to the same index) will be linked together. So, at index 8, we'll have a chain starting with key 23, followed by key 12. Chaining offers several advantages, such as:

• Flexibility in handling multiple collisions.
• Simplified deletion and insertion operations.

However, it's important to design the hash function carefully to minimize collisions and, consequently, the length of chains.

C++ data structures

C++ provides a variety of data structures to efficiently manage and store data. Some common data structures include arrays, linked lists, stacks, queues, and hash tables. Using the correct data structure can greatly improve the performance and efficiency of your program. In our exercise, we utilize the hash table data structure with chaining. To implement this in C++, we'd typically use an array of linked lists. Each entry in the array corresponds to the hash table slot, and each linked list handles collisions at that slot by storing all entries that hash to that specific index.

• Arrays: These provide O(1) time complexity for accessing elements but lack the flexibility required for collision handling.
• Linked Lists: These allow dynamic memory allocation and can efficiently handle variable-sized data.

Combining arrays with linked lists (as we do in chaining) results in an effective structure where each element of the array points to a linked list, offering both the advantages of fast indexing and flexible collision handling.