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 most likely not help. The algorithm's time complexity is such that when the problem size doubles, the time it takes to solve increases exponentially. This means that to solve a problem twice as large within one minute, the new computer would need to be \(2^{100}\) times faster. This speed increase is currently impossible and if it were, would likely be cost-prohibitive.

Step-by-step solution

Understand the Algorithm

Algorithm A is a \(\Theta\left(2^{n}\right)\) algorithm, meaning that the time it takes to complete grows exponentially with the size of the input. Note that if the size of our input doubled, because of our algorithm's nature, the problem's complexity would not just double, but would increase exponentially.

Determine Time for Larger Problem

First, we would need to determine how much time it currently takes to solve a problem of twice the size. With our existing computational capabilities and using Algorithm A, a problem twice the size would take approximately 2 times as long to solve. Since our current problem size of 100 takes 1 minute to solve, a problem of size 200 would approximately take \(2^{200}/2^{100}\) times longer, that is, \(2^{100}\) times longer. This would translate into \(2^{100}\) minutes.

Evaluate the Cost-Benefit

Recognizing that our new computer system would have to be \(2^{100}\) times faster to solve problems twice as large in 1 minute shows the magnitude of the speed increase required. Given the currently possible technological advancements, this is most likely impossible. Even if it were possible, the cost would likely be prohibitive. The efficiency of the algorithm is a crucial consideration when sizing problems and this algorithm's exorbitant growth demonstrates why an exponential-time algorithm is unsustainable in the long run.

Key concepts

Algorithm Efficiency

Algorithm efficiency is a crucial consideration when it comes to solving computational problems. Efficient algorithms use less time and resources to solve problems, which is important when the problem size increases. In our case, Algorithm A has a time complexity of \( \Theta(2^n) \), which indicates it is an exponential algorithm.
However, the efficiency of an algorithm is intrinsically linked to its time complexity. Exponential time algorithms like Algorithm A become impractical for larger problem sizes because they require significantly more time to process as the input size increases. Therefore, understanding and selecting a more efficient algorithm is often more beneficial than just upgrading hardware.

Time Complexity

Time complexity helps us understand how the computational work of an algorithm grows as the size of the input grows. It is usually expressed using Big O notation, which gives an upper bound of the running time or space used by an algorithm.
For Algorithm A, the time complexity is \( \Theta(2^n) \), meaning that its running time grows exponentially with the increase of the input size, \( n \). An exponential time complexity indicates poor scalability because each additional element increases the running time dramatically. When \( n \) doubles, as in our example, the time taken to complete the task doesn't merely double; it becomes exponentially larger, specifically \( 2^n \times (2^n/2) = 2^{n/2} \).

• Exponential algorithms are often slow and inefficient for large inputs.
• Careful consideration is needed when choosing or designing algorithms with better time complexity.

Problem Size Doubling

Doubling the problem size has a distinct effect on different types of algorithms. When dealing with exponential algorithms like Algorithm A, doubling the input size results in an unsustainable increase in the processing time.
Using Algorithm A, a current problem size of 100 takes about 1 minute. If the problem size is doubled to 200, the completion time increases exponentially, roughly to \( 2^{100} \) minutes. This result is due to the nature of exponential functions, where each increment doubles the effect rather than just increments it.
This phenomena shows the limits of using such algorithms in practical settings where input sizes can increase rapidly. Instead of considering hardware upgrades, exploring algorithms with better time complexity, such as those with polynomial or linear growth rates, can provide more lasting and cost-effective solutions.