Question

Algorithm \(A\) executes an \(O(\log n)\) -time computation for each entry of an \(n\) -element array. What is the worst-case running time of Algorithm \(A\) ?

Short answer

The worst-case running time of Algorithm A is \(O(n \thinspace \text{log}\thinspace n)\).

Step-by-step solution

Understand the Problem

Algorithm A has a computation time of \(O(\text{log}\thinspace n)\) for each entry in an array of size \(n\). We need to find the total worst-case running time.

Number of Computations

Since the array has \(n\) elements, there will be \(n\) computations each taking \(O(\text{log}\thinspace n)\) time.

Multiply Computations

The total running time is \(n\) computations times the time for each computation: Total running time = \(n \times O(\text{log}\thinspace n)\).

Simplify the Expression

Using Big-O notation properties, the final expression for the worst-case running time can be simplified to: \(O(n \thinspace \text{log}\thinspace n)\).

Key concepts

Big-O notation

Big-O notation is a mathematical concept to describe the upper bound of an algorithm's running time. It helps us understand how the performance of an algorithm scales with the size of the input data.
The notation is used to express the worst-case scenario in terms of time or space complexity.
For example, if an algorithm has a running time of O(n), it means that in the worst case, the time taken grows linearly with the input size. Other common complexities include O(1) (constant time), O(n²) (quadratic time), and O(log n) (logarithmic time).
Big-O notation helps developers compare algorithms and choose the most efficient one for their needs. It abstracts away constant factors, focusing on the growth rate as the input size increases.

Logarithmic complexity

Logarithmic complexity, denoted as O(log n), is a measurement of how the running time of an algorithm increases logarithmically as the input size increases. Algorithms with logarithmic complexity are very efficient, especially for large input sizes.
In our exercise, the computation time for each element in the array is O(log n), where n is the number of elements. This means that the time required increases very slowly compared to the size of the input.
Some common algorithms with logarithmic complexity include binary search and certain operations in balanced binary search trees like AVL or Red-Black Trees. These algorithms divide the problem into smaller parts and solve each part independently, reducing the overall time required.

Algorithm analysis

Algorithm analysis is the process of determining the computational complexity of algorithms, focusing on their time and space requirements. By analyzing algorithms, developers make informed decisions about which algorithms to use based on their performance.
In our exercise, we analyzed an algorithm with O(log n) time complexity for each element in an array of size n. We determined the total worst-case running time by multiplying the number of elements (n) by the time complexity for each element (O(log n)), giving us a final worst-case running time of O(n log n).
This process involves breaking down the algorithm into smaller steps, understanding the time complexity of each step, and combining these complexities to get the overall running time. This helps in optimizing the algorithm and ensures that it performs efficiently even for large inputs.