Question

There are two algorithms called Algl and Alg2 for a problem of size n. Algl runs in \(n^{2}\) microseconds and Alg2 runs in 100 n log \(n\) microseconds. Algl can be implemented using 4 hours of programmer time and needs 2 minutes of CPU time. On the other hand, Alg2 require 15 hours of programmer time and 6 minutes of CPU time. If programmers are paid 20 dollars per hour and CPU time costs 50 dollars per minute, how many times must a problem instance of size 500 be solved using Alg2 in order to justify its development cost?

Short answer

The short answer will depend on the exact value of \(\log(500)\) and the result of the final calculation in step 4.

Step-by-step solution

Calculate the cost of Algl

First, the cost of Algl needs to be calculated. The cost of Algl consists of the programming cost and the CPU cost. The programming cost is 4 hours times $20 per hour, which equals $80. The CPU cost is 2 minutes times $50 per minute, resulting in $100. Therefore, the total cost of Algl in dollars is $180.

Calculate the cost of Alg2

Next, the cost of Alg2 is calculated. The programming cost is 15 hours times $20 per hour, making it $300. The CPU cost is 6 minutes times $50 per minute, coming out to be $300. Thus, the total cost of Alg2 is $600.

Calculate the execution cost of Alg2 for problem size n=500

The amount of time Alg2 takes to solve a problem of size 500 is \(100 \times 500 \times \log(500)\) microseconds. This needs to be converted into minutes since the CPU time cost is calculated in minutes. Therefore, the execution time is \(\frac{100 \times 500 \times \log(500)}{60 \times 10^6}\) minutes. The CPU cost for execution is this time multiplied by $50 per minute.

Find out how many times Alg2 must be implemented to justify its cost

Finally, the cost of Alg2 including both the development and execution cost needs to be less than or equal to the cost of using Algl for the same number of times. Setting up an equation \(600 + x \times (\text{execution cost of Alg2}) \leq x \times 180\) and solving for x will give the minimum number of times Alg2 needs to be implemented to justify its cost.

Key concepts

Cost Analysis

To understand the financial implications of using different algorithms, we start with a cost analysis. This analysis involves breaking down the various expenses associated with each algorithm's development and runtime. In our exercise, we compare two algorithms, Algl and Alg2, based on the programmer’s time cost and CPU time cost.

The programmer's time cost for Algl is calculated as 4 hours at \(20 per hour, totaling \)80. For Alg2, it’s considerably more with 15 hours at the same rate, totaling \(300. Meanwhile, the CPU cost is computed as \)50 per minute, leading to \(100 for Algl's 2 minutes and \)300 for Alg2's 6 minutes. Thus, Algl's total cost is \(180, and Alg2's is \)600, not including the cost of executing the algorithms on a problem of size n=500, which will add to Alg2's total expense.

In cost analysis, such considerations enable us to assess the trade-offs between initial investments versus long-term operational costs. By calculating these costs meticulously, we ensure that we make the most economically sound decisions in the context of algorithm selection and implementation.

Algorithm Performance

The performance of an algorithm is typically evaluated by how efficiently it solves a problem, especially as the problem size scales. In our example, Algl runs in (n^2) microseconds while Alg2 runs in 100 n log(n) microseconds. The performance of both algorithms differs significantly as the input size n grows.

To appreciate this difference, consider that a quadratic time complexity, such as that of Algl, will become inefficient as n becomes large. On the other hand, the (n log n) time complexity of Alg2, despite a higher initial constant factor (100), scales better for large n. As such, Alg2 may be the preferred choice for efficiency in large problem instances despite its higher initial development costs.

Analyzing algorithm performance is crucial in the field of computer science, as it can greatly affect the viability of a software solution. Efficient algorithms can save time and resources, thereby also reducing operational costs in the long term.

Algorithm Execution Time

Diving deeper into the notion of efficiency, the execution time of an algorithm is a direct measure of its performance. It indicates the duration an algorithm takes to complete its task. For Alg2, when dealing with a problem of size 500, we calculate its execution time in microseconds and convert it to minutes to align with the billing rate for CPU time.

Using the formula provided, \(\frac{100 \times 500 \times \log(500)}{60 \times 10^6}\) minutes, we quantify the algorithm's execution time, which, when multiplied by the CPU cost per minute, yields the cost of running Alg2 for a single instance of size 500. This step is crucial as it factors into the total cost when trying to determine how many times Alg2 has to be run to justify the difference in development costs compared to Algl.

Understanding the execution time is vital, as it directly translates to user experience and operational expenses. A faster algorithm can lead to shorter wait times for users and lower costs for computations, a particularly important consideration when running compute-intensive tasks at scale. Matching the algorithm to the problem context and size can result in significant savings and performance enhancements.