Question

Agorithms A and B porfom the same tardk. Cn inpur of s coe n, sigorithm A cwocutes 0.003 \(m^{2}\) instructions, and algorithm B expcutes 243 n inetructions. Find the approwimate value of \(n\) above ahich algorithm \(B\) is mono efficient. Prou may use a calculator or spreadsheet.

Short answer

Algorithm B is more efficient for \(n > 81000\).

Step-by-step solution

Understand the Problem

We are given two algorithms, A and B. We need to find the value of \(n\) above which algorithm B is more efficient than algorithm A. This means we need to solve the inequality: Algorithm B's execution count should be less than Algorithm A's execution count.

Set up the Inequality

Algorithm A executes \(0.003 \cdot n^2\) instructions. Algorithm B executes \(243 \cdot n\) instructions. We want to find the smallest value of \(n\) such that \(243n < 0.003n^2\).

Simplify the Inequality

Rearrange the inequality \(243n < 0.003n^2\) to \(0.003n^2 - 243n > 0\).

Solve Quadratic Inequality

Factor the quadratic inequality. Start by dividing everything by 0.003: \(n^2 - 81000n > 0\). Simplify to find \(n(n - 81000) > 0\).

Identify Critical Points

The critical points of the inequality \(n(n - 81000) > 0\) are \(n = 0\) and \(n = 81000\). Test intervals around these points to find where the inequality is satisfied.

Determine Solution Interval

For \(n(n - 81000) > 0\), test the intervals \((0, 81000)\) and \((81000, \infty)\). At \(n = 81000\), expressions change sign, showing \(n > 81000\) satisfies the inequality.

Key concepts

Quadratic Inequality

A quadratic inequality involves expressions where a quadratic term, typically denoted as \( ax^2 + bx + c \), is compared to a value, to determine an interval on the number line. In our problem, we worked with an inequality resulting from comparing the computational efficiency of two algorithms. We had to solve \( 0.003n^2 - 243n > 0 \). Simplifying this inequality helps us find the critical points by factoring or using the quadratic formula.
Quadratic inequalities differ from quadratic equations. Inequalities require us to find solution sets or intervals. Critical points (or roots) are often located where the expression changes signs. These situations may demand testing the inequality on different intervals between and beyond these roots.
To solve, first divide and simplify the inequality. For example, our inequality turned into \( n(n - 81000) > 0 \), marking \( n = 0 \) and \( n = 81000 \) as the critical points. Testing the solution in intervals, we identified \( n > 81000 \) as where algorithm B becomes more efficient.

Algorithm Comparison

Algorithm comparison is a process of evaluating different algorithms against one another to determine which performs more efficiently under given conditions. This is essential in computer science for optimizing operations. In our example, we needed to compare algorithm A and B to find out when algorithm B becomes more efficient based on their instruction count.
Algorithm A's computational requirement grows quadratically, represented by \(0.003n^2\), which means its performance will decline more significantly as \(n\) increases. In contrast, algorithm B's linear growth \(243n\) remains steadier on increases of \(n\).
The tipping point where B overtakes A in efficiency lies beyond a particular \(n\). This point, found through inequality solving, demonstrated that algorithm B becomes more efficient when \( n > 81000 \). Through comparison, you understand what makes an algorithm faster or slower depending on problem size \(n\).

Computational Complexity

Computational complexity offers a lens into the efficiency of algorithms, describing how the number of resources required for an algorithm scales with the input size \( n \). In solving our problem, computational complexity helped us understand the efficiency variance between two algorithms.
With algorithm A growing at \( O(n^2) \) and algorithm B at \( O(n) \), we see they have different efficiencies. The potential for decreased efficiency due to inputs increasing illustrates why understanding complexity is pivotal when evaluating algorithms. Where one might expect B's linear complexity to dominate A's quadratic complexity over large inputs, our task precisely verifies that for \( n > 81000 \), algorithm B indeed requires fewer computational resources than algorithm A.
The analysis aids decision-making on which algorithm to implement, depending on acceptable input sizes. This complexity-based forethought enables informed choices in designing or selecting algorithms tailored to specific environments or constraints.