How is the time complexity of the Celebrity Problem affected by the number of people in the group?

The Celebrity Problem is a popular problem in computer science and is often used to illustrate the concept of algorithms and time complexity. The problem is defined as follows: in a group of people, a celebrity is someone who is known by everyone but does not know anyone else. The task is to identify the celebrity, if there is one, given a function that tells you whether person A knows person B.

The time complexity of the Celebrity Problem is affected by the number of people in the group in the following way:

1. The naive approach to solve the Celebrity Problem would be to check for every person if they are a celebrity. This would involve checking, for each person, if they know no one and if everyone else knows them. This would result in a time complexity of O(n^2), where n is the number of people in the group.

2. However, there is a more efficient approach to solve the Celebrity Problem. This approach involves using a stack to eliminate non-celebrities. First, push all the people into the stack. Then, pop off two people from the stack, and check if they know each other. The person who knows the other is not a celebrity, so they can be discarded. The person who does not know the other could potentially be a celebrity, so they are pushed back onto the stack. This process is repeated until there is only one person left in the stack, who is the potential celebrity. This person is then checked against all the other people to confirm if they are a celebrity. This approach results in a time complexity of O(n), where n is the number of people in the group.

So, the time complexity of the Celebrity Problem is directly proportional to the number of people in the group, and the exact time complexity depends on the approach used to solve the problem.