Suppose a hash table is constructed using linked lists to store the data within each bin, and that collisions are resolved by chaining. Suppose n elements have been inserted into the hash table. What is the worst case time complexity of querying whether an object exists within the hash table? You may assume that the hash table will not need to be resized, and that evaluating the hash function takes Θ(1) time.

In a hash table that uses linked lists to resolve collisions (a method known as chaining), the time complexity for querying whether an object exists depends on the number of elements that hash to the same bin (i.e., have the same hash value).

### Given:
- \( n \) elements are inserted into the hash table.
- The hash function evaluation takes \( \Theta(1) \) time.
- The hash table will not need to be resized.

### Worst-Case Scenario:
- All \( n \) elements hash to the same bin. This is an extreme case of collision where every element is stored in the same linked list.

### Querying an Object:
- To query whether an object exists, you would:
  1. Evaluate the hash function in \( \Theta(1) \) time.
  2. Traverse the linked list associated with that bin to find the element.

In the worst-case scenario, the length of the linked list is \( n \) (since all elements are in the same bin). Thus, the time complexity for traversing the linked list to search for the element is \( \Theta(n) \).

### Conclusion:
The **worst-case time complexity** for querying whether an object exists in the hash table is **\( \Theta(n) \)**.

This happens when all elements hash to the same bin, leading to a single long linked list.

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Would you like further details or have any questions?

Here are 5 related questions:
1. How does the average-case time complexity of querying differ from the worst-case?
2. What strategies can be employed to minimize the likelihood of worst-case time complexity in a hash table?
3. What is the impact of using a poor hash function on the performance of a hash table?
4. How does open addressing differ from chaining in terms of time complexity?
5. What are some advantages of using a balanced binary search tree over a hash table?

**Tip:** When designing hash tables, choosing a good hash function is critical to evenly distribute elements across bins, reducing the likelihood of collisions.

Understand the worst-case time complexity of querying whether an object exists in a hash table using linked lists for collision resolution. Explore how this complexity depends on the number of elements and the distribution within the hash bins.

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