2- The picture below shows a set of points in 2-dimensional space. A grid is provided so that you can compute the distance between points; each grid cell is 1 unit wide and 1 unit tall. You may assume that each data point is placed on a grid intersection. Suppose a weighted distance graph G is constructed from this data set (recall that a distance graph is a complete graph whose nodes represent points in space, and whose edges are weighted by the distance between its endpoints). Then suppose that a minimum spanning tree is computed for G. What will be the weight of the largest edge in this minimum spanning tree? grid details providing data points - (2,6) (5,4) (2,2) (4,2) (12,7) (10,4) (13,5) (12,4) (11,3)

To find the weight of the largest edge in the minimum spanning tree (MST) for the given set of points, we'll follow these steps:

1. **List the points:**
   The points provided in the 2D space are:
   \[
   (2, 6), (5, 4), (2, 2), (4, 2), (12, 7), (10, 4), (13, 5), (12, 4), (11, 3)
   \]

2. **Distance formula:**
   The Euclidean distance \(d\) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by:
   \[
   d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
   \]
   
3. **Constructing the distance graph:**
   For each pair of points, we'll calculate the Euclidean distance and construct a complete graph where each node is a point, and each edge is weighted by the distance between the points.

4. **Minimum spanning tree (MST):**
   The MST will be a subset of the edges that connects all points with the minimum possible total edge weight, ensuring that no cycles are formed.

5. **Largest edge in the MST:**
   Once we have the MST, we will identify the largest edge in this spanning tree.

### Step-by-step process:

Let’s focus on calculating the distances between nearby points and then identify the largest edge in the MST.

#### Key distances between nearby points:

- \( d((2, 6), (5, 4)) = \sqrt{(5 - 2)^2 + (4 - 6)^2} = \sqrt{9 + 4} = \sqrt{13} \approx 3.61 \)
- \( d((5, 4), (2, 2)) = \sqrt{(2 - 5)^2 + (2 - 4)^2} = \sqrt{9 + 4} = \sqrt{13} \approx 3.61 \)
- \( d((2, 2), (4, 2)) = \sqrt{(4 - 2)^2 + (2 - 2)^2} = \sqrt{4} = 2 \)
- \( d((12, 7), (10, 4)) = \sqrt{(10 - 12)^2 + (4 - 7)^2} = \sqrt{4 + 9} = \sqrt{13} \approx 3.61 \)
- \( d((10, 4), (11, 3)) = \sqrt{(11 - 10)^2 + (3 - 4)^2} = \sqrt{1 + 1} = \sqrt{2} \approx 1.41 \)
- \( d((13, 5), (12, 4)) = \sqrt{(12 - 13)^2 + (4 - 5)^2} = \sqrt{1 + 1} = \sqrt{2} \approx 1.41 \)

#### Now, comparing the larger distances:

- \( d((2, 6), (12, 7)) = \sqrt{(12 - 2)^2 + (7 - 6)^2} = \sqrt{100 + 1} = \sqrt{101} \approx 10.05 \)
- \( d((5, 4), (12, 7)) = \sqrt{(12 - 5)^2 + (7 - 4)^2} = \sqrt{49 + 9} = \sqrt{58} \approx 7.62 \)
- \( d((4, 2), (12, 4)) = \sqrt{(12 - 4)^2 + (4 - 2)^2} = \sqrt{64 + 4} = \sqrt{68} \approx 8.25 \)

### Conclusion:

From the distances calculated, the largest distance between any pair of points that are likely to be part of the MST is approximately \(10.05\) units, which is the distance between \((2, 6)\) and \((12, 7)\).

Thus, the weight of the largest edge in the minimum spanning tree is approximately **10.05 units**.

---

Would you like more details on the MST construction process or any other clarifications?

Here are 5 relative questions to deepen your understanding:
1. What is Kruskal's algorithm, and how does it help in finding the MST?
2. How does Prim's algorithm differ from Kruskal's for finding the MST?
3. What would happen if the graph were not fully connected?
4. Can the largest edge in an MST ever form a cycle? Why or why not?
5. How would the result change if the distances were calculated using Manhattan distance instead of Euclidean?

**Tip**: When working with graph problems, understanding the algorithms (Kruskal's and Prim's) for MST construction is crucial, as they handle edge weights differently and are optimized for different scenarios.

The picture below shows a set of points in 2-dimensional space. A grid is provided so that you can compute the distance between points; each grid cell is 1 unit wide and 1 unit tall. You may assume that each data point is placed on a grid intersection. Suppose a weighted distance graph G is constructed from this data set (recall that a distance graph is a complete graph whose nodes represent points in space, and whose edges are weighted by the distance between its endpoints). Then suppose that a minimum spanning tree is computed for G. What will be the weight of the largest edge in this minimum spanning tree?

Explore how to compute the largest edge weight in the minimum spanning tree for a set of 2D points placed on a grid. Learn about Euclidean geometry, graph theory, and algorithms like Kruskal's and Prim's. The problem involves calculating distances between points and constructing a minimum spanning tree. Suitable for high school students in Grades 10-12.

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