Python Graph Algorithms: Minimum Spanning Trees Suppose you are an engineer working on designing a road network for a new town. The town has many residential areas and commercial centers that need to be connected efficiently. You decided to represent the town as a connected, undirected graph where each vertex represents a location, and each edge represents a road connecting two locations. The weight of each edge represents the distance between the two locations. To ensure the road network is efficient, you need to find the minimum spanning tree of the graph. However, due to budget constraints, you can only construct roads with a maximum distance limit. You need to determine for how many pairs of locations, the minimum spanning tree of the graph remains the same when the maximum distance limit of a road is increased by 2 units. To solve this, you write a program that takes as input the number of locations, the number of roads, and the weight of each road. Your program will also receive a list of queries where each query contains two locations, and you need to check if the minimum spanning tree of the graph remains the same when the maximum distance limit of the road connecting those two locations is increased by 2. For each query, your program will output "YES" if the minimum spanning tree remains the same, and "NO" otherwise. Input Format: Each test case begins with a line containing two integers n and m, where n is the number of vertices and m is the number of edges in the graph. The vertices are numbered from 1 to n. The next m lines each contain three integers u, v, and w, indicating that there is an undirected edge between vertices u and v with weight w. The next line contains a single integer q, denoting the number of queries. The following q lines each contain two integers u and v, indicating that you need to check if the minimum spanning tree of the graph remains the same when the weight of the edge between vertices u and v is increased by 2. Constraints: 2 ≤ n ≤ 1000 1 ≤ m ≤ min(n*(n-1)/2, 2000) 1 ≤ u, v ≤ n, u ≠ v 1 ≤ w ≤ 1000 1 ≤ q ≤ 1000 Output Format: For each query, output "YES" if the minimum spanning tree remains the same, and "NO" otherwise, separated by a newline. Sample Input 0: 4 5 1 2 1 1 3 2 1 4 3 2 3 6 3 4 5 3 1 2 2 3 1 4 Sample Output 0: YES YES YES Explanation in Sample Input and Output 0: In the given graph, the minimum spanning tree is as follows: Edge (1,2) with weight 1 Edge (1,3) with weight 2 Edge (1,4) with weight 3 For the first query, if we increase the weight of edge (1,2) by 2, it becomes 3. The minimum spanning tree remains the same, and the output is "YES". For the second query, if we increase the weight of edge (2,3) by 2, it becomes 8, which is still greater than the weight of edges (1,2) (1,3), (1,4). Therefore, the minimum spanning tree remains the same, and the output is "YES". For the third query, if we increase the weight of edge (1,4) by 2, it becomes 5, which is equal to the weight of edge (3,4). However, the original minimum spanning tree is still valid. Therefore, the minimum spanning tree remains the same, and the output is "YES".
Python Graph
Suppose you are an engineer working on designing a road network for a new town. The town has many residential areas and commercial centers that need to be connected efficiently. You decided to represent the town as a connected, undirected graph where each vertex represents a location, and each edge represents a road connecting two locations. The weight of each edge represents the distance between the two locations.
To ensure the road network is efficient, you need to find the minimum spanning tree of the graph. However, due to budget constraints, you can only construct roads with a maximum distance limit. You need to determine for how many pairs of locations, the minimum spanning tree of the graph remains the same when the maximum distance limit of a road is increased by 2 units.
To solve this, you write a program that takes as input the number of locations, the number of roads, and the weight of each road. Your program will also receive a list of queries where each query contains two locations, and you need to check if the minimum spanning tree of the graph remains the same when the maximum distance limit of the road connecting those two locations is increased by 2.
For each query, your program will output "YES" if the minimum spanning tree remains the same, and "NO" otherwise.
Input Format:
Each test case begins with a line containing two integers n and m, where n is the number of vertices and m is the number of edges in the graph. The vertices are numbered from 1 to n. The next m lines each contain three integers u, v, and w, indicating that there is an undirected edge between vertices u and v with weight w.
The next line contains a single integer q, denoting the number of queries. The following q lines each contain two integers u and v, indicating that you need to check if the minimum spanning tree of the graph remains the same when the weight of the edge between vertices u and v is increased by 2.
Constraints:
2 ≤ n ≤ 1000
1 ≤ m ≤ min(n*(n-1)/2, 2000)
1 ≤ u, v ≤ n, u ≠ v
1 ≤ w ≤ 1000
1 ≤ q ≤ 1000
Output Format:
For each query, output "YES" if the minimum spanning tree remains the same, and "NO" otherwise, separated by a newline.
Sample Input 0:
4 5
1 2 1
1 3 2
1 4 3
2 3 6
3 4 5
3
1 2
2 3
1 4
Sample Output 0:
YES
YES
YES
Explanation in Sample Input and Output 0:
In the given graph, the minimum spanning tree is as follows:
Edge (1,2) with weight 1
Edge (1,3) with weight 2
Edge (1,4) with weight 3
For the first query, if we increase the weight of edge (1,2) by 2, it becomes 3. The minimum spanning tree remains the same, and the output is "YES".
For the second query, if we increase the weight of edge (2,3) by 2, it becomes 8, which is still greater than the weight of edges (1,2) (1,3), (1,4). Therefore, the minimum spanning tree remains the same, and the output is "YES".
For the third query, if we increase the weight of edge (1,4) by 2, it becomes 5, which is equal to the weight of edge (3,4). However, the original minimum spanning tree is still valid. Therefore, the minimum spanning tree remains the same, and the output is "YES".
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