Question

Presently we can solve problem instances of size 100 in 1 minute using algorithm \(A,\) which is a \(\Theta\left(2^{n}\right)\) algorithm. On the other hand, we will soon have to solve problem instances twice this large in 1 minute. Do you think it would help to buy a faster (and more expensive) computer?

Short answer

No, buying a faster computer would not significantly help in this case due to the exponential time complexity of the algorithm. The problem size increase outweighs any reasonable speed increase a new computer might offer.

Step-by-step solution

Understand the problem and algorithm time complexity

Firstly, we need to understand that the time complexity of algorithm \(A\) is \(\Theta\left(2^{n}\right)\). This means that as the size of the problem \(n\) increases, the time taken to solve the problem doubles. Right now, the algorithm can solve a problem of size 100 in 1 minute.

Apply the size increase to the time complexity equation

Next, let's apply the problem size increase to the time complexity. As per the problem, we will soon need to solve problems twice as large in 1 minute. So, the problem size \(n\) will increase to 200. Given the time complexity of the algorithm, this would mean that the time taken to solve the problem doubles with each increase in the problem size, i.e., the algorithm will take \(2^{200-100} = 2^{100}\) times as long as to solve a problem of size 200 as it does to solve a problem of size 100.

Determine the usefulness of a faster computer

For the final step, we consider if a faster (more expensive) computer would help. A faster computer could possibly execute the algorithm more quickly, but considering the time complexity of the algorithm, it wouldn't help enough to offset the exponential growth. The increase is so large (\(2^{100}\) times larger) that merely doubling the speed of processing (which is a reasonable expectation for a new computer) would still leave us with a task that takes much longer than 1 minute.

Key concepts

Exponential Growth

Exponential growth is a concept that describes how quickly a quantity increases by a constant factor in each time step or iteration. In the context of algorithm time complexity, exponential growth occurs when the time taken to solve a problem doubles every time the input size increases by a fixed amount. This is described by the expression \( \Theta(2^n) \) for an algorithm, indicating that the time required grows exponentially as the problem size, \( n \), increases.
For instance, consider algorithm \( A \), which can solve a problem of size 100 in one minute. If the problem size doubles to 200, the time required to solve it doesn't just double; it grows exponentially. In this specific example, the formula \( 2^{n} \), applied to \( n = 200 \), suggests an enormous increase in time, specifically by a factor of \( 2^{100} \).
Understanding exponential growth helps in assessing the viability of using certain algorithms as problem sizes increase. Algorithms with exponential time complexity can quickly become impractical for large datasets due to their inherently rapid increase in computational demand.

Algorithm Efficiency

Algorithm efficiency refers to the effectiveness of an algorithm in terms of two main resources: time and space. In the exercise provided, time efficiency, which is how quickly an algorithm can complete its task, is the primary concern.
An efficient algorithm is one that can solve a problem effectively without using excessive computational resources. When an algorithm's time complexity is \( \Theta(2^n) \), like in algorithm \( A \), it is not considered efficient for large inputs because it requires a substantial increase in time with each increase in problem size.
When evaluating algorithm efficiency, consider both the current performance and the scaling potential. For instance:

• Can the algorithm handle larger datasets without a significant loss in performance?
• Is there an alternative algorithm with lower time complexity that could be used instead?
• Would optimizing code or using better data structures achieve the necessary performance?

Thus, even though faster hardware could improve execution speed, it may not overcome the innate inefficiencies of an algorithm with exponential growth.

Computational Complexity

Computational complexity studies the amount of computational resources needed to solve a problem, usually in terms of time (how long it takes) and space (how much memory is used).
An important part of computational complexity is understanding the type of complexity (e.g., constant, logarithmic, linear, polynomial, exponential) an algorithm presents. This determines how well it scales as input sizes increase.
In the given problem, algorithm \( A \) has exponential time complexity, \( \Theta(2^n) \). This means for larger inputs, the computation time can grow faster than what is feasible, as seen with a problem size of 200 requiring \( 2^{100} \) times more time than a problem size of 100. This massive requirement makes it nearly impossible to solve within a reasonable time frame, even with increased computational power.
Understanding computational complexity helps in choosing the right algorithm and preparing for the challenges posed by different types of complexities. For large-scale problems, selecting algorithms with polynomial time complexity or less can often provide significantly better performance.