Question

Suppose you have a computer that requires I minute to solve problem in stances of size \(n=1000 .\) What instance sizes can be run in 1 minute if you buy a new computer that runs 1000 times faster than the old one, assuming the following time complexities \(T(n)\) for our algorithm? (a) \(T(n) \in \Theta(n)\) (b) \(T(n) \in \Theta\left(n^{3}\right)\) (c) \(T(n) \in \Theta\left(10^{n}\right)\)

Short answer

For the new computer, with a speed 1000 times faster: (a) \(n = 1,000,000\) for \(\Theta(n)\) (b) \(n = 10,000\) for \(\Theta(n^{3})\) (c) \(n = 1003\) for \(\Theta(10^{n})\).

Step-by-step solution

Solve for \(T(n) \in \Theta(n)\)

The time complexity of \(\Theta(n)\) represents a linear time complexity. This means that if the new computer is 1000 times faster, it will also be able to process 1000 times more instances in the same amount of time. Therefore, for a time complexity of \(\Theta(n)\), the new computer could process problems of size \(n = 1000 \times 1000 = 1,000,000\) in one minute.

Solve for \(T(n) \in \Theta\left(n^{3}\right)\)

The time complexity \(\Theta\left(n^{3}\right)\) is cubic, which means that increasing the instance size by a certain factor will increase the time by the cube of that factor. Therefore, if the new computer is 1000 times faster, the instance size it can handle will be the cubed root of 1000 times bigger. This can be calculated as \(n = 1000 \times \sqrt[3]{1000} \approx 1000 \times 10 = 10,000\). Therefore, the new computer could process problems of size \(n = 10,000\) in one minute when the time complexity is \(\Theta\left(n^{3}\right)\).

Solve for \(T(n) \in \Theta\left(10^{n}\right)\)

The time complexity \(\Theta\left(10^{n}\right)\) is exponential, meaning that the time doubles with every additional instance. Even though the new computer is 1000 times faster, with exponential growth like this, the increase in instance size the new computer can handle will be small. We can calculate this as \(n = 1000 + \log_{10}{1000} \approx 1000 + 3 = 1003\). Therefore, the new computer could process problems of size \(n = 1003\) in one minute when the time complexity is \(\Theta\left(10^{n}\right)\).

Key concepts

Algorithm Efficiency

Understanding algorithm efficiency is crucial to optimizing programs and ensuring they run effectively. Algorithm efficiency, often referred to as the performance or complexity of an algorithm, determines how well a program uses resources such as time and memory. When we talk about how an algorithm scales with the increase of the input size, we're usually concerned with its time complexity, because it directly impacts the run time. For example, if a new computer runs 1000 times faster, an algorithm with linear time complexity (Theta(n)) will allow for an input size increase of 1000 times the original, indicating high efficiency.

Computational Complexity

Computational complexity is a field of study within computer science that focuses on classifying computational problems based on their inherent difficulty and quantifying the amount of resources necessary to solve them. Different complexities are categorized as constant, linear, polynomial, or exponential, among others. The complexity determines how the computation time increases with the size of input data.
For instance, with a 1000-fold increase in a machine's speed, a problem with cubic complexity (Theta(n^3)) doesn't simply increase the problem size by 1000, but rather by the cubed root of 1000, demonstrating a rapidly scaling resource requirement compared to linear complexity.

Exponential Growth

Exponential growth in the context of computational complexity describes an algorithm whose time complexity grows exponentially with the input size, denoted as Theta(10^n) for our example. The primary characteristic of exponential growth is that the computation time doubles with each additional unit increase in input size, making it unsustainable for large inputs. In our scenario, even a 1000 times faster computer does not significantly increase the manageable input size. This underlines the impracticality of exponential algorithms for large-scale problems due to their explosive growth rate.

Big Theta Notation

Big theta notation (Theta) is a mathematical symbol used to describe an algorithm's time complexity with a tight bound, which means it includes both upper and lower limits. It gives a way to express the limitation on the growth of the run time. Specifically, if an algorithm has a complexity of Theta(f(n)), there exist constants c1, c2, and n0 such that for all n > n0, c1*f(n) <= T(n) <= c2*f(n), where T(n) is the actual time complexity of the algorithm. This notation is crucial for understanding the efficiency and scalability of algorithms, as it provides a more precise measurement than the upper-bound Big O notation alone.

For instance, when we say an algorithm has a time complexity of Theta(n), it suggests that its growth rate is linear and precisely bounded, providing a clear expectation of how performance will change with increasing inputs.