Describe the algorithm for constructing the minimum spanning tree of a set of points in Computational Geometry.

The algorithm for constructing the minimum spanning tree (MST) of a set of points in Computational Geometry can be achieved using the Prim's algorithm or Kruskal's algorithm. Both algorithms are commonly used for finding the MST in a graph, and can be adapted for constructing the MST of a set of points.

1. Prim's Algorithm:
- Start by selecting an arbitrary point as the starting point.
- Create a priority queue to store the edges connecting the points.
- Initialize the priority queue with the edges connected to the starting point.
- Create a boolean array to keep track of visited points, initially set to false.
- While the priority queue is not empty:
- Extract the edge with the minimum weight from the priority queue.
- If both the points of the edge are not visited:
- Add the edge to the MST.
- Mark both points as visited.
- Add the edges connected to the newly visited point to the priority queue.
- Repeat the above steps until all points are visited.

2. Kruskal's Algorithm:
- Sort all the edges in non-decreasing order of their weights.
- Create a disjoint set data structure to keep track of the connected components.
- Initialize the MST as an empty set.
- For each edge in the sorted order:
- If adding the edge to the MST does not create a cycle:
- Add the edge to the MST.
- Merge the connected components of the two points of the edge using the disjoint set data structure.
- Repeat the above step until all edges are processed or the MST has n-1 edges, where n is the number of points.

Both algorithms guarantee the construction of the minimum spanning tree, but they differ in their approach. Prim's algorithm starts with a single point and gradually expands the MST by adding the minimum weight edges, while Kruskal's algorithm starts with all points disconnected and progressively connects them by adding the edges in non-decreasing order of their weights.

It is important to note that the choice of algorithm may depend on the specific requirements and characteristics of the given set of points, such as the density of the points or the sparsity of the graph.