Queries to find the Minimum Weight from a Subtree of atmost D-distant Nodes from Node X

When engaged in computer programming, it is sometimes necessary to locate the minimum weight of a subtree that originates from a specific node, with the condition that the subtree must not contain any nodes farther than D units away from the specified node. This problem arises in various fields and applications, including graph theory, tree-based algorithms, and network optimization.

The subtree constitutes a subset of a larger tree structure, with the specified node serving as the root of the subtree. The subtree encompasses all the descendants of the root node and their connecting edges. The weight of a node refers to a particular value assigned to that node, which can denote its significance, importance, or any other relevant metric. In this problem, the objective is to find the minimum weight among all the nodes in the subtree, while limiting the subtree to nodes that are at most D units away from the root node.

In the following article, we shall delve into the intricacies of unearthing the Minimum Weight from a Subtree, which is bounded by no more than D-Distant Nodes, emanating from Node X.

Approaches

• Approach 1 − Depth-First Search (DFS), One of the most common and straightforward approaches to solve this problem is using a depth-first search (DFS) traversal of the subtree.

• Approach 2 − Another approach to solve this problem is using a breadth-first search (BFS) traversal of the subtree.

Syntax

The syntax shown declares a function called findMinimumWeight which takes two arguments. The first argument Node* x is a pointer to a node in a binary tree and the second argument int d is an integer representing a distance. The function returns an integer minWeight. The implementation of the algorithm to find the minimum weight from the subtree of at most d distant nodes from node x is not specified in the given code snippet.

int findMinimumWeight(Node* x, int d) {
   // Implementation of the algorithm
   // ...
   return minWeight;
}

Where −

• Node* x represents the pointer to the root node of the tree.

• int d represents the constraint on the maximum distance of the nodes in the subtree from the root node.

Algorithm

An intricate challenge in computer science involves determining the minimum weight of a subtree that spans no more than D-distant nodes from a given node X. Several algorithms can be employed to tackle this problem. Here, we present a high-level overview of the approach −

Step 1 − Commence with node X as the root of the subtree.

Step 2 − Traverse the subtree in a depth-first manner, meticulously noting the distance of each node from the root node.

Step 3 − Maintain a variable to track the minimum weight encountered thus far.

Step 4 − At each node, assess if the distance from the root node is within the D limit. If so, update the minimum weight variable with the weight of the current node.

Step 5 − Repeat steps 3 and 4 for all nodes within the subtree.

Step 6 − Ultimately, return the minimum weight obtained from the subtree.

Approach 1

One of the simplest and most prevalent solutions to this challenge is to utilize a depth-first search (DFS) exploration of the subtree. In this approach, we traverse the subtree of the given node in a depth-first manner, visiting each node and its offspring before proceeding to the next sibling node. At each node, we assess its distance from the root node and, if it falls within the specified limit, we modify the minimum weight discovered thus far. This approach has an elapsed time intricacy of O(n), where n represents the number of nodes in the subtree, as each node is visited only once.

Example

The presented code constitutes a program aimed at determining the least weight of nodes in a subtree that are at most 'd' distance away from node 'X' in a binary tree. The binary tree is represented by a structure referred to as "Node", which encompasses a weight, a reference to its left child, and a reference to its right child.

The main function 'findMinimumWeightDFS' takes a node 'x' and an integer 'd' as inputs and returns the minimum weight of nodes that are at most 'd' distance away from 'x'. The function employs a helper function 'findMinimumWeightDFS' that executes a depth-first search (DFS) on the binary tree and updates the minimum weight discovered thus far.

The minimum weight is initialized to a large value and modified during the DFS exploration if the current node is at most 'd' distance away from the root node. The function returns the final minimum weight found after the DFS exploration.

#include <iostream>
#include <climits>
using namespace std;

// Definition of Node structure
struct Node {
   int weight;
   Node* left;
   Node* right;
   Node(int w) : weight(w), left(nullptr), right(nullptr) {}
};

// Function to perform DFS traversal and find minimum weight
void findMinimumWeightDFS(Node* x, int d, int distance, int& minWeight) {
   // Base case: if the current node is null or distance exceeds D, return
   if (x == nullptr || distance > d) {
      return;
   }

   // If the current node is at most D-distant from the root node, update minWeight
   if (distance <= d) {
      minWeight = min(minWeight, x->weight);
   }

   // Recursively perform DFS on the left and right children of the current node
   findMinimumWeightDFS(x->left, d, distance + 1, minWeight);
   findMinimumWeightDFS(x->right, d, distance + 1, minWeight);
}

// Function to find minimum weight from subtree of at most D-distant nodes from node X using DFS
int findMinimumWeightDFS(Node* x, int d) {
   int minWeight = INT_MAX; // Initialize minWeight to a large value
   findMinimumWeightDFS(x, d, 0, minWeight); // Perform DFS traversal
   return minWeight; // Return the minimum weight obtained
}

// Driver code
int main() {
    // Create a sample binary tree
   Node* root = new Node(1);
   root->left = new Node(2);
   root->right = new Node(3);
   root->left->left = new Node(4);
   root->left->right = new Node(5);
   root->right->left = new Node(6);
   root->right->right = new Node(7);

   // Test the findMinimumWeightDFS function
   int d = 2;
   int minWeight = findMinimumWeightDFS(root, d);
   cout << "Minimum weight from subtree of at most " << d << "-distant nodes from node X: " << minWeight << endl;

   return 0;
}

Output

Minimum weight from subtree of at most 2-distant nodes from node X: 1

Approach 2

Another strategy to resolve this challenge is to employ a breadth-first search (BFS) exploration of the subtree. In this approach, we traverse the subtree of the given node in a breadth-first manner, visiting all the offspring of a node before proceeding to the next level. At each node, we assess its distance from the root node, and if it falls within the specified limit, we modify the minimum weight discovered thus far. This approach has an elapsed time intricacy of O(n), where n represents the number of nodes in the subtree, as each node is visited only once. However, the spatial intricacy of this approach is greater than the DFS approach, as it necessitates a queue to keep track of the nodes to be explored.

Example

The code constitutes a program aimed at determining the least weight of a node in a binary tree, given a maximum distance "d" from the root node. The strategy involves conducting a Breadth-First-Search (BFS) exploration of the binary tree, and storing each node along with its distance from the root node in a queue. The function initiates with the root node and its distance as 0, and adds it to the queue.

Then, it iteratively retrieves the node from the front of the queue and updates the minimum weight if the node's distance is at most "d". If the node possesses a left or right offspring, it adds the child to the queue with an updated distance. The function continues until the queue is empty. Finally, the function returns the minimum weight obtained from the BFS exploration.

#include <iostream>
#include <queue>
#include <climits>
using namespace std;

// Definition of Node structure
struct Node {
   int weight;
   Node* left;
   Node* right;
   Node(int w) : weight(w), left(nullptr), right(nullptr) {}
};

// Function to perform BFS traversal and find minimum weight
int findMinimumWeightBFS(Node* x, int d) {
   queue<pair<Node*, int>>q; // Create a queue to store nodes and their distances
   q.push({x, 0}); // Push the root node and its distance (0) to the queue
   int minWeight = INT_MAX; // Initialize minWeight to a large value

   while (!q.empty()) {
      Node* curr = q.front().first; // Get the current node from the front of the queue
      int distance = q.front().second; // Get the distance of the current node from the root
      q.pop(); // Pop the current node from the queue

      // If the current node is at most D-distant from the root node, update minWeight
      if (distance <= d) {
         minWeight = min(minWeight, curr->weight);
      }

      // If the current node has left child, push it to the queue with updated distance
      if (curr->left) {
         q.push({curr->left, distance + 1});
      }

      // If the current node has right child, push it to the queue with updated distance
      if (curr->right) {
         q.push({curr->right, distance + 1});
      }
   }

   return minWeight; // Return the minimum weight obtained
}

// Driver code
int main() {
   // Create a sample binary tree
   Node* root = new Node(1);
   root->left = new Node(2);
   root->right = new Node(3);
   root->left->left = new Node(4);
   root->left->right = new Node(5);
   root->right->left = new Node(6);
   root->right->right = new Node(7);

   // Test the findMinimumWeightBFS function
   int d = 2;
   int minWeight = findMinimumWeightBFS(root, d);
   cout << "Minimum weight from subtree of at most " << d << "-distant nodes from node X: " << minWeight << endl;

   return 0;
}

Output

Minimum weight from subtree of at most 2-distant nodes from node X: 1

Conclusion

This article covered how to use C++ to get the least weight from a subtree of nodes that are at most D-distances apart from a particular node X in a binary tree. We investigated the Depth-First Search (DFS) and Breadth-First Search (BFS) methods, and we included implementation codes and example results for each method.