Make a tree with n vertices , d diameter and at most vertex degree k

Given three integers N, D and K. The task is to check whether it is possible to make a tree with exactly N vertices, D diameter (the number of edges in the longest path between any two vertices), and degree of each vertex must be at most K. If it is possible then print all the possible edges otherwise print No.

Examples:

Input: N = 6, D = 3, K = 4
Output:
1 2
2 3
3 4
5 2
6 2

Input: N = 6, D = 2, K = 4
Output: N0



Approach: Let’s construct the tree with the following algorithm: If (d > n – 1), print “No” and terminate the program. Otherwise, let’s keep the array deg of the length n which will represent degrees of vertices.

The first step is to construct the diameter of the tree. Let the first (d + 1) vertices form it.
Let’s add d edges to the answer and increase degrees of vertices corresponding to these edges, and if some vertex has degree greater than k, print “No” and terminate the program.

The second (and the last) step is to attach the remaining (n – d – 1) vertices to the tree. Let’s call the vertex free if its degree is less than k. Also, let’s keep all free vertices forming the diameter in some data structure which allows us to take the vertex with the minimum maximal distance to any other vertex and remove such vertices. It can be done by, for example, set of pairs (distv, v), where distv is the maximum distance from the vertex v to any other vertex. Now let’s add all the vertices starting from the vertex (d + 1) (0-indexed) to the vertex n?1, let the current vertex be u. One can get the vertex with the minimum maximal distance to any other vertex, let it be v. Now increase the degree of vertices u and v, add the edge between them, and if v still be free, return it to the data structure, otherwise remove it. The same with the vertex u (it is obvious that its maximal distance to any other vertex will be equal to (distv + 1).

If at any step our data structure will be empty or the minimum maximal distance will equal d, the answer is “No”. Otherwise, we can print the answer.

Below is the implementation of the above approach:

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// C++ implementation of the approach
#include <bits/stdc++.h>
using namespace std;
  
// Function to Make a tree with n vertices, d as it's
// diameter and degree of each vertex is at most k
void Make_tree(int n, int d, int k)
{
    // If diameter > n - 1
    // impossible to build tree
    if (d > n - 1) {
        cout << "No";
        return;
    }
  
    // To store the degree of each vertex
    vector<int> deg(n);
  
    // To store the edge between vertices
    vector<pair<int, int> > ans;
  
    // To store maximum distance among
    // all the paths from a vertex
    set<pair<int, int> > q;
  
    // Make diameter of tree equals to d
    for (int i = 0; i < d; ++i) {
        ++deg[i];
        ++deg[i + 1];
        if (deg[i] > k || deg[i + 1] > k) {
            cout << "NO" << endl;
            return;
        }
  
        // Add an edge between them
        ans.push_back(make_pair(i, i + 1));
    }
  
    // Store maximum distance of each vertex
    // from other vertices
    for (int i = 1; i < d; ++i)
        q.insert(make_pair(max(i, d - i), i));
  
    // For next (n - d - 1) edges
    for (int i = d + 1; i < n; ++i) {
  
        // If the vertex already has the degree k
        while (!q.empty() && deg[q.begin()->second] == k)
            q.erase(q.begin());
  
        // If not possible
        if (q.empty() || q.begin()->first == d) {
            cout << "No";
            return;
        }
  
        // Increase the degree of vertices
        ++deg[i];
        ++deg[q.begin()->second];
  
        // Add an edge between them
        ans.push_back(make_pair(i, q.begin()->second));
  
        // Store the maximum distance of this vertex
        // from other vertices
        q.insert(make_pair(q.begin()->first + 1, i));
    }
  
    // Print all the edges of the built tree
    for (int i = 0; i < n - 1; ++i)
        cout << ans[i].first + 1 << " "
             << ans[i].second + 1 << endl;
}
  
// Driver code
int main()
{
    int n = 6, d = 3, k = 4;
    Make_tree(n, d, k);
  
    return 0;
}

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Output:

1 2
2 3
3 4
5 2
6 2


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