Query the number of connected components after removing a vertex from the tree

The following query can be used to determine the connected components remaining after removing tree vertices: First consider the tree structure. Then, each connected component is examined by moving through the tree using a breadth-first or depth-first search algorithm. Once the required vertices are evicted, the same traversal method is used to decide the number of connected components. The outcome will be determined based on the change in counts before and after expulsion. This method effectively monitors connection changes and helps in computing updates to connected components in the tree.

usage instructions

• Depth First Search (DFS) Method

• And check the method

Depth First Search (DFS) Method

In the DFS method, we first perform a DFS traversal from any selected node in the original tree to count the connected components after removing the vertices from the tree. During this traversal, we iterate through each connected node, mark each node as visited, and track the number of times DFS was used. We perform a new DFS traversal after deleting a specified vertex, ensuring that deleted vertices are skipped during the exploration phase. We can determine the number of connected components in the updated tree by comparing the number of calls to DFS before and after deletion. This method can efficiently count the number of connected elements while adjusting for changes in the tree structure.

algorithm

• Select any vertex on the original tree as the starting point for DFS traversal.

• Set the variable "count" to start counting connected components. First, set it to 0.

• From the selected starting vertex, use DFS to traverse the tree.

• Mark each vertex visited during the DFS traversal and increment the "count" by 1 for each new DFS call (i.e., for each connected component found).

• After the DFS traversal is completed, the number of connected elements in the original tree will be represented by "count".

• Delete the specified vertex from the tree.

• Repeat the DFS traversal from the same starting vertex, making sure to avoid exploring deleted vertices.

• When running DFS, update the "count" variable similarly to before.

• The number of associated components in the upgraded tree will be determined by subtracting the post-evacuation "count" from the starting "count".

Example

#include <iostream>
#include <vector>

void dfs(const std::vector<std::vector<int>>& tree, int v, 
std::vector<bool>& visited, int& count) {
   visited[v] = true;
   count++;
   for (int u : tree[v]) {
      if (!visited[u]) {
         dfs(tree, u, visited, count);
      }
   }
}

int countConnectedComponents(const std::vector<std::vector<int>>& tree, int startVertex) {
   int n = tree.size();
   std::vector<bool> visited(n, false);
   int count = 0;

   dfs(tree, startVertex, visited, count);
   return count;
}

int main() {
   std::vector<std::vector<int>> tree = {
      {1, 2},
      {0, 3},
      {0, 4},
      {1},
      {2}
   };

   int startVertex = 0;
   std::cout << countConnectedComponents(tree, startVertex) << std::endl;
   return 0;
}

Output

5

And check the method

We first initialize each vertex as a separate component in the union find method so that we can count the connected components after removing the vertex from the tree. To keep track of parts and connections in the original tree, we take and lookup data structures. We update and query the data structure to reflect the change in tree connectivity after deleting the specified vertex. Then count and check the number of different sets in the data structure. The resulting count represents the connectivity of the updated components of the tree. After removing vertices, the search method can efficiently calculate connected components and effectively handle structural changes in the tree.

algorithm

• Create an array or data structure from scratch that represents each vertex as a different part of the tree. Initially, each vertex is its own parent vertex.

• Use the AND lookup operation in the preprocessing step to determine the connected component count of the original tree.

• Use the union data structure to combine the parts of each edge (u, v) in the tree that contain vertices u and v. The initial connectivity of the tree is established in this step.

• Delete the specified vertex from the tree.

• Apply the union lookup operation of the preprocessing step to the modified tree.

• After deletion, calculate and check the number of different parent representatives in the data structure.

• The result count represents the connectivity of the tree update component.

Example

#include <iostream>
#include <vector>

class UnionFind {
public:
   UnionFind(int n) {
      parent.resize(n);
      for (int i = 0; i < n; ++i) {
         parent[i] = i;
      }
   }

   int find(int u) {
      if (parent[u] != u) {
         parent[u] = find(parent[u]);
      }
      return parent[u];
   }

   void unite(int u, int v) {
      int rootU = find(u);
      int rootV = find(v);
      if (rootU != rootV) {
         parent[rootU] = rootV;
      }
   }

   int countDistinctParentRepresentatives() {
      int n = parent.size();
      std::vector<bool> distinct(n, false);
      for (int i = 0; i < n; ++i) {
         distinct[find(i)] = true;
      }
      int count = 0;
      for (bool isDistinct : distinct) {
         if (isDistinct) {
            count++;
         }
      }
      return count;
   }

private:
   std::vector<int> parent;
};

int main() {
   int n = 5;
   UnionFind uf(n);

   uf.unite(0, 1);
   uf.unite(0, 2);
   uf.unite(2, 3);
   uf.unite(2, 4);

   std::cout << uf.countDistinctParentRepresentatives() << 
std::endl;
   return 0;
}

Output

1

in conclusion

In summary, the provided method can efficiently count the number of connected parts in the tree after deleting a specific vertex. Connectivity changes in tree structures can be efficiently handled using depth-first search (DFS) methods and union search methods. The DFS method starts the traversal from the selected vertex, marks each node visited, and counts the connected components. The updated count is obtained by comparing the before and after traversal counts after deleting the vertex, and a new traversal is performed without including the deleted node.

The initial connected component count is determined using a union operation via the Union-Find method, which initializes each vertex as a separate component. Apply the same union operation after deleting the vertices and count the different parent representatives to get an updated count.

Both methods can provide useful insights into the connectivity of the tree after vertices have been removed, and are suitable for a variety of scenarios.

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