Trees For Modeling Real-World Situations In This Discussion

Trees for Modeling Real-World Situations In this discussion, you will continue considering the real-world contexts presented by you and your classmates in the Unit 7 Discussion Board

Post 1: Initial Response. Using your graph from the Unit 7 Discussion, start this discussion by addressing the following to explore the modeling of your context even further: Update your Unit 7 Discussion graph by adding a weight to each of your edges. Present your updated graph with all weighted edges.

Based on the real-world context of your graph, briefly explain what these weights represent. Present a second illustration where you have identified a spanning tree and its total weight within your weighted graph. Describe how you know this subgraph meets the requirements of a spanning tree.

Post 2: Reply to a Classmate. Review a classmate’s graph and address all of the following items completely: Apply either Prim’s algorithm or Kruskal’s algorithm (not presented in the text so you would need to look this up elsewhere) to find a minimum spanning tree for your classmate’s weighted graph. Explain the steps taken and present the minimum spanning tree with a visual. In the context of your classmate’s real-world context: What is the total weight of this spanning tree? What is the difference between your minimum spanning tree and your classmate’s spanning tree (from their initial response)? How can you interpret the total weight for this spanning tree within the real-world context?

Post 3: Reply to Another Classmate. Review a different classmate’s graph and address all of the following items completely. Suppose your objective has been updated from spanning the graph. Now you only need to find an efficient path between any two vertices on your classmate’s weighted graph. Write your own step-by-step process (i.e., in pseudocode or a list of steps) which you propose will find the shortest path between any two vertices. Provide enough detail about the steps so that someone else would be able to apply your idea. Select a starting and ending vertex (which are not adjacent vertices) on your classmate’s weighted graph. Apply your algorithm and present the path with a visual. In the context of your classmate’s real-world context: What is the total weight of your proposed shortest path? How could the shortest path be useful to your classmate, given the real-world context? What steps could you take to test whether or not your algorithm works for determining the shortest paths in all graphs (e.g., for graphs other than this one and for different starting/ending vertices)?

Paper For Above instruction

The assignment involves engaging with the concepts of graph theory to model real-world situations through the creation, analysis, and application of weighted graphs, spanning trees, and shortest path algorithms. The task is structured into three main posts that progressively deepen the understanding of these concepts, encouraging practical application and comparative analysis among classmates' graphs.

In the initial post, students are asked to refine their existing graph from Unit 7 by adding weights to each edge, which numerically represent real-world quantities such as costs, distances, or capacities. The students must then identify a spanning tree within their weighted graph—an acyclic subgraph that connects all vertices—and calculate its total weight, demonstrating understanding of the properties that define a spanning tree and how it models efficient connectivity in real-world applications.

The second post requires students to select a peer’s graph and utilize either Prim’s or Kruskal’s algorithm to determine a minimum spanning tree (MST). Students should explain their procedural steps, illustrate the MST visually, and compare it to the original spanning tree presented by the classmate. They are also expected to interpret the total weight of the MST within the given context, highlighting how the algorithm optimizes overall connection costs or distances, which is critical in network design, transportation, or communication networks.

The third post focuses on shortest path algorithms. Students are to assume a shift in objective—from spanning the entire graph to efficiently connecting two specific vertices—and develop a detailed algorithm (pseudocode or step-by-step instructions) for finding the shortest path. Applying this algorithm to selected vertices on a peer’s graph, students present the optimal path visually and analyze its total weight’s relevance to the real-world scenario. They should also discuss how to validate their algorithm’s effectiveness across different graphs and vertex pairs, emphasizing the importance of robustness and adaptability in practical applications such as routing and navigation systems.

Overall, this activity integrates theoretical concepts with practical modeling, fostering skills in graph construction, algorithm application, and critical analysis within realistic contexts like transportation planning, logistics, and infrastructure optimization.

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