Computational Geometry Questions Long
To compute the shortest path between two points in a planar graph using Computational Geometry techniques, we can utilize the Dijkstra's algorithm. This algorithm is widely used for finding the shortest path in a graph with non-negative edge weights. Here is the step-by-step description of the algorithm:
1. Initialize the graph: Start by representing the planar graph as a set of vertices and edges. Each vertex represents a point in the plane, and each edge represents a connection between two points. Assign a weight to each edge, which represents the distance between the connected points.
2. Set the initial conditions: Assign a distance value to every vertex in the graph. Set the distance of the source vertex (starting point) to 0 and the distance of all other vertices to infinity. Create an empty set called the "visited set" to keep track of the vertices that have been visited.
3. Find the shortest path: Iterate through the vertices in the graph. At each iteration, select the vertex with the minimum distance value from the set of vertices not yet visited. Mark this vertex as visited.
4. Update the distances: For the selected vertex, examine all of its neighboring vertices (adjacent vertices) that have not been visited. Calculate the distance from the source vertex to each neighboring vertex through the selected vertex. If this distance is smaller than the current distance assigned to the neighboring vertex, update the distance value.
5. Repeat steps 3 and 4: Continue the iterations until all vertices have been visited or until the destination vertex (target point) has been visited. This ensures that the shortest path to all vertices has been found.
6. Retrieve the shortest path: Once the algorithm terminates, the shortest path from the source vertex to the destination vertex can be obtained by backtracking from the destination vertex to the source vertex using the recorded distances and the graph's structure.
By following these steps, the Dijkstra's algorithm can efficiently compute the shortest path between two points in a planar graph using Computational Geometry techniques.