Question

Bob loves foreign languages and wants to plan his course schedule for the following years. He is interested in the following nine language courses: LA15, LA16, LA22, LA31, LA32, LA126, LA127, LA141, and LA169. The course prerequisites are: LA15: (none) • LA16: LA15 • LA22: (none) • LA31: LA15 • LA32: LA16, LA31 • LA126: LA22, LA32 • LA127: LA16 • LA141: LA22, LA16 • LA169: LA32 Find the sequence of courses that allows Bob to satisfy all the prerequi- sites.

Short answer

LA15, LA22, LA16, LA31, LA32, LA141, LA127, LA126, LA169.

Step-by-step solution

Identify Courses Without Prerequisites

List the courses that can be taken without any prerequisites. These are the starting points.Courses with no prerequisites: LA15, LA22.

List Courses Dependent on Initial Courses

Identify which courses have prerequisites that can be met by completing the initial courses (LA15 and LA22).Dependents on LA15: LA16, LA31.Dependents on LA22: LA126, LA141.

Complete Courses Dependent on LA15

Complete the courses that require LA15 as a prerequisite. Ensure to list these in order.1. LA152. LA163. LA31

Find Courses Dependent on Multiple Prerequisites

Identify courses that have multiple prerequisites, some of which may already have been identified.1. LA32: Requires LA16 and LA31, both prerequisites met.2. LA127: Requires LA16, prerequisite met.3. LA126: Requires LA22 and LA32, LA32 prerequisite met after LA22 and LA32 completion.4. LA141: Requires LA22 and LA16, prerequisites met.

Complete Remaining Courses

Finish the sequence of courses ensuring all prerequisites are met in order.1. LA22 (previous list)2. LA1413. LA32 4. LA1275. LA1266. LA169: Requires LA32, prerequisite already met.

Final Course Sequence

Compile all steps into a final sequence ensuring all prerequisites are satisfied in order.LA15, LA22, LA16, LA31, LA32, LA141, LA127, LA126, LA169.

Key concepts

Topological Sorting

To solve course scheduling with prerequisites, we use a concept called topological sorting. Imagine you have a list of tasks where some tasks must be done before others; topological sorting arranges these tasks in a linear order. This is crucial in computer science and helps in dependency resolution, like figuring out our courses.
To apply topological sorting:

• Identify tasks with no prerequisites. They are our starting points.
• Remove these tasks from the list and note that they are done.
• Identify the next set of tasks whose prerequisites are now met.
• Repeat until all tasks are completed.

In the context of course scheduling, the tasks are courses, and the prerequisites are dependencies. This method ensures you don't violate any prerequisite relationships.

Dependency Resolution

Dependency resolution involves addressing the dependencies (i.e., prerequisites) among a set of tasks. For instance, in our exercise, each language course might depend on completing another course first.
This process is essential for:

• Avoiding circular dependencies, which cause impossible-to-complete sequences.
• Ensuring each task (course) is started only after its prerequisites are fulfilled.

Applying dependency resolution, we:
• Identify courses with no prerequisites, allowing us to start somewhere.
• Progressively move to dependent courses once their prerequisites are completed.
• Avoid conflicts by orderly handling all dependencies.

Effective dependency resolution leads to a smooth sequence of task completion without any conflicts or unmet requirements.

Graph Theory

Graph theory provides the foundation for topological sorting and dependency resolution in scheduling. Think of each course as a node, and each prerequisite relationship as a directed edge, forming a Directed Acyclic Graph (DAG).
In our specific problem:

• Nodes: LA15, LA16, LA22, LA31, LA32, LA126, LA127, LA141, LA169.
• Edges: Represent prerequisites, like an edge from LA15 to LA16.

Using graph theory concepts, we can:

• Visualize the problem as finding a route through the graph where each edge is followed in its direction.
• Apply algorithms like Depth-First Search (DFS) or Kahn's Algorithm to obtain a topological sort of the nodes.

Understanding the graph representation makes it easier to devise solutions for complex scheduling and dependency problems.