Question

Given an \(n\) -element array \(X\) of integers, Algorithm \(C\) executes an \(O(n)\) time computation for each even number in \(X,\) and an \(O(\log n)\) -time computation for each odd number in \(X\). What are the best-case and worst-case running times of Algorithm C?

Short answer

Best-case: \(O(n \text{ log } n)\), Worst-case: \(O(n^2)\)

Step-by-step solution

Identify the Types of Computations

Determine the type of computation based on whether the number in array X is even or odd. Even numbers result in an \(O(n)\) computation, while odd numbers lead to an \(O(\text{log } n)\) computation.

Best-Case Scenario

In the best-case scenario, all elements in the array X are odd numbers. Since each element leads to an \(O(\text{log } n)\) computation and there are \(n\) elements, the total time complexity would be: \[n \times O(\text{log } n) = O(n \text{ log } n)\]

Worst-Case Scenario

In the worst-case scenario, all elements in the array X are even numbers. Since each element leads to an \(O(n)\) computation and there are \(n\) elements, the total time complexity would be: \[n \times O(n) = O(n^2)\]

Key concepts

Best-Case Scenario

Imagine you have an array filled with numbers, and you need to run an algorithm on it. The best-case scenario refers to the situation where the algorithm performs the fastest. This occurs when your input is as favorable as possible for the algorithm.

In our exercise, Algorithm C runs quickest when all elements in the array are odd numbers. Why? Because each odd number in the array leads to a computation time of only \(O(\text{log } n)\). When every number is odd, the algorithm doesn't have to deal with the more time-consuming \(O(n)\) computations that are required for even numbers. Therefore, the best-case scenario for Algorithm C results in a total time complexity of:

\[n \times O(\text{log } n) = O(n \text{ log } n)\]

This is the fastest the algorithm can run for a given array size of \(n\).

Worst-Case Scenario

The worst-case scenario is the exact opposite of the best-case scenario. Here, we look at the situation where the algorithm performs the slowest. This happens when the input is the least favorable.

For Algorithm C, the worst-case scenario occurs when every element in the array is an even number. Each even number triggers a computation that takes \(O(n)\) time. This is significantly slower compared to the \(O(\text{log } n)\) computations for odd numbers.

In this case, for an array with \(n\) elements, the total time complexity would be:

\[n \times O(n) = O(n^2)\]

This represents the slowest possible running time for Algorithm C, given the specified types of elements in the array.

Big O Notation

Big O Notation is a mathematical concept used to describe the efficiency of an algorithm in terms of its time and space requirements as the input size grows. It's like a way to measure how an algorithm's run time or space requirements grow as the size of the input increases.

Big O focuses on the largest part of the time complexity, ignoring constants and smaller terms. This helps us get a sense of the scalability of the algorithm.

For example:

• \(O(1)\) - The time remains constant, no matter how large the input.
• \(O(n)\) - The time increases linearly with the input size.
• \(O(n^2)\) - The time increases quadratically with the input size.

In our exercise, we used Big O notation to express the time complexities \(O(\text{log } n)\) for odd numbers and \(O(n)\) for even numbers. This notation simplifies understanding and comparing the efficiency of different parts of the algorithm.

Time Complexity

Time complexity is a way to express an algorithm's performance as a function of the input size. It tells us how the computation time grows as the size of the input increases.

In our example, Algorithm C has different time complexities depending on whether the number is even or odd. The key takeaway is:

• For even numbers: The computation takes \(O(n)\) time.
• For odd numbers: The computation takes \(O(\text{log } n)\) time.

So, depending on the input (whether the array consists mainly of even or odd numbers), the overall time complexity of the algorithm changes. This illustrates how essential it is to consider different input scenarios while analyzing an algorithm's performance.