Question

A student was asked to develop an algorithm to find and output the largest of three numerical values \(x, y\), and \(z\) that are provided as input. Here is what was produced: Input: \(x, y, z\) Algorithm: Check if \((x>y)\) and \((x>z)\). If it is, then output the value of \(x\) and stop. Otherwise, continue to the next line. Check if \((y>x)\) and \((y>z)\). If it is, then output the value of \(y\) and stop. Otherwise, continue to the next line. Check if \((z>x)\) and \((z>y)\). If it is, then output the value of \(z\) and stop. Is this a correct solution to the problem? Explain why or why not. If it is incorrect, fix the algorithm so that it is a correct solution.

Short answer

The original algorithm is incorrect for tied values. Include >= conditions.

Step-by-step solution

Understand the Problem

We need to determine if the algorithm correctly outputs the largest of three values: \(x\), \(y\), and \(z\).

Analyze the Provided Algorithm

The algorithm checks each pair of comparisons: \((x>y)\) and \((x>z)\), \((y>x)\) and \((y>z)\), \((z>x)\) and \((z>y)\). If a value satisfies both conditions, it is outputted.

Identify the Flaw

The issue arises when two or more values are equal and are the largest. The current checks will not output the correct largest value if any values are equal, since they do not cover equal comparisons, such as \((x \geq y)\) and \((x \geq z)\).

Propose a Correct Solution

Incorporate equalities into the comparisons to account for cases where two or more values may be equal: \((x \geq y)\) and \((x \geq z)\), \((y \geq x)\) and \((y \geq z)\), and \((z \geq x)\) and \((z \geq y)\). Modify the algorithm to include these checks.

Implement the Correct Algorithm

1. Check if \(x\) is greater than or equal to \(y\) and \(x\) is greater than or equal to \(z\). If true, output \(x\).2. Else, if \(y\) is greater than or equal to \(x\) and \(y\) is greater than or equal to \(z\), output \(y\).3. Else, output \(z\). This checks all possible scenarios including ties, ensuring a correct output of the largest value.

Key concepts

Comparison Operators

In algorithm design, comparison operators are crucial for making decisions based on the values of variables. A comparison operator checks the relationship between two values. Common operators include equality ('=='), inequality ('!='), greater than ('>'), less than ('<'), greater than or equal to ('≥'), and less than or equal to ('≤'). These operators return a Boolean result - either true or false.

When designing an algorithm to find the largest of three numbers, you typically use 'greater than' (>) or 'greater than or equal to' (≥) operators to compare the numbers. For instance, the expression \(x > y\) checks whether \(x\) is greater than \(y\). Similarly, \(y ≥ z\) checks if \(y\) is at least as large as \(z\).

• The benefit of using comparison operators is they directly influence the flow of decision-making in an algorithm.
• They provide clear paths for the algorithm based on predetermined conditions.
• Proper use of these operators ensures that the algorithm checks all necessary conditions.

In the discussed exercise, comparison operators determine which value among \(x, y,\) and \(z\) is the largest.

Equal Values Handling

Handling equal values in algorithms is essential, especially when determining the largest of multiple inputs. In the initial algorithm, only strict greater than (>) comparisons were used. Hence, it failed to account for situations where two or more values might be equal. To address this, using the greater than or equal to (≥) operator allows the algorithm to consider equality.

For example, if \(x\) and \(y\) are equal and larger than \(z\), a strict greater than comparison would miss this, but a greater than or equal to comparison correctly identifies \(x\) or \(y\) as the largest.

• Applying greater than or equal to (≥) checks ensures that tie values aren't overlooked.
• It enhances the robustness of the algorithm by covering all potential outcomes.
• This way, any value among \(x, y, z\) that meets the equality criteria can be recognized as the largest.

By integrating equal values handling, the algorithm becomes more comprehensive and reliable in identifying the largest number in all scenarios.

Algorithm Correction

Correcting an algorithm involves identifying flaws and adjusting the logic to ensure accurate outputs. In this case, the algorithm initially failed to consider scenarios where two or more inputs might be equally largest.

To correct this algorithm, extend the comparisons to include equal values. By modifying the conditions to:
1. Check if \(x ≥ y\) and \(x ≥ z\) to ensure \(x\) is at least as large as both other values.2. If the first condition fails, check if \(y ≥ x\) and \(y ≥ z\) to determine if \(y\) should be chosen.3. Only output \(z\) in the last step, when neither \(x\) nor \(y\) meet their respective conditions.

• Each step prioritizes the largest value, including ties, ensuring that equality is considered alongside traditional greater comparisons.
• This revised approach guarantees that the algorithm can accurately output the largest value between the three without ambiguity.
• Such corrections refine the algorithm by addressing logical gaps that might cause incorrect outputs.

Implementing these corrections makes the algorithm not only correct but also more robust and adaptable to diverse input scenarios.