Question

In a scheduling program, we want to check whether two appointments overlap. For simplicity, appointments start at a full hour, and we use military time (with hours 0-24). The following pseudocode describes an algorithm that determines whether the appointment with start time startl and end time end 1 overlaps with the appointment with start time start2 and end time end2. H start1 s start2 \(s=\) start 1 the \(s=\) start2 If end \(1<\) end2 \(e\) * end 1 The \(e=\cos 2\) If see The appointionents overlap. Flue The appointwents don't overlap. 'Trace this algorithm with an appointment from 10-12 and one from 11-13, then with an appointment from 10-11 and one from 12-13.

Short answer

10-12 and 11-13 overlap; 10-11 and 12-13 do not overlap.

Step-by-step solution

Understand the Problem

We need to determine if two appointments overlap by comparing their start and end times. Appointments are represented in 24-hour time format.

Analyze the Pseudocode Conditions

The pseudocode checks which of the two appointments start earlier and assigns that appointment's start time and end time to variables \( s \) and \( e \). It then compares the end times of the appointments and assigns the earlier end time to \( e \). If \( s < e \), there is an overlap.

Trace with First Pair of Appointments (10-12 and 11-13)

- Start by comparing start times: \(\text{start1} = 10\), \(\text{start2} = 11\).- \(\text{start1} < \text{start2}\), so \( s = 10 \).- Compare the end times: \(\text{end1} = 12\), \(\text{end2} = 13\).- \(\text{end1} < \text{end2}\), so \( e = 12 \).- Since \( s < e \) (10 < 12), the appointments overlap.

Trace with Second Pair of Appointments (10-11 and 12-13)

- Compare the start times: \(\text{start1} = 10\), \(\text{start2} = 12\).- \(\text{start1} < \text{start2}\), so \( s = 10 \).- Compare the end times: \(\text{end1} = 11\), \(\text{end2} = 13\).- \(\text{end1} < \text{end2}\), so \( e = 11 \).- Since \( s ot< e \) (10 ot< 11), the appointments do not overlap.

Key concepts

Pseudocode

Writing algorithms in pseudocode is a helpful way to create a clear and easy-to-understand plan for problem-solving. Pseudocode is a simplified version of a programming language that allows you to outline your logic before implementing it in a real programming language. It serves as a blueprint that both programmers and non-programmers can understand.

• In our scheduling problem, the pseudocode helps determine whether two appointments overlap by focusing on start and end times.
• By using simple terms and operations, pseudocode abstracts complex details away, allowing you to focus on logic and sequence of operations.
• Precise conditions and steps outline the problem-solving methodology, such as comparing start times and determining which appointment ends earlier.

Pseudocode is neither formal code nor is it too verbose; it's the middle ground that simplifies complex problems into manageable tasks. This approach speeds up the planning stage of coding and ensures that everyone involved understands the workflow.

Problem-Solving

Problem-solving is the core of algorithm design. It involves breaking down the problem into smaller and more manageable parts, and then addressing each part systematically. In our example of checking appointment overlaps:

• The first step is to clearly understand what needs to be solved: determining overlap based on start and end times.
• The algorithm then uses comparisons like checking which appointment starts first.
• Understanding the relationship between start and end times is crucial, as it influences the overlap status.
• By logically applying conditions, we arrive at a conclusion – whether the appointments overlap or not.

By taking a logical and systematic approach, problem-solving becomes much more efficient and effective. Debugging and optimizing the solution afterward becomes easier, as each step clearly contributes to the final outcome.

Time Complexity

Time complexity is an important aspect of algorithm design, as it helps in determining how scalable the algorithm is when the size of input data increases. For our appointment scheduling problem:

• Time complexity focuses on evaluating how many steps the algorithm takes relative to the sizes of the input pairs, in this case, comparing two sets of times (start and end times).
• The pseudocode essentially consists of constant time operations, as it always evaluates a fixed number of conditions – comparing start times and end times just once.
• Therefore, in this scenario, the time complexity can be described as O(1), which means the time taken is constant, regardless of the input size.

Understanding and evaluating time complexity enables you to choose the most efficient algorithm, especially crucial for problems where large amounts of data are involved. This knowledge ensures that the solution can handle larger scales without degrading performance.