Maximize the number of uncolored vertices appearing along the path from root vertex and the colored vertices

Given a tree with N vertices numbered 1 through N with vertex 1 as root vertex and N – 1 edges. We have to color exactly k number of vertices and count the number of uncolored vertices between root vertex and every colored vertex. We have to include the root vertex in the count if it is not colored. The task to maximize the number of uncolored vertices occurring between the path from root vertex and the colored vertices.

Examples:

Input :

           1
         / |  \
       /   |    \
     2     3      4
          / \      \
         /   \      \
        5     6      7

k = 4 
Output : 7
Explanation:
If we color vertex 2, 5, 6 and 7, 
the number of uncolored vertices between the path
from root to colored vertices is maximum which is 7.

Input :

          1
         / \
        /   \
       2     3
      /
     /
    4

k = 1
Output : 2

Approach:

To solve the above-mentioned problem we observe that if a vertex is chosen to be uncolored then its parent must be chosen to be uncolored. Then we can calculate how many uncolored vertices we will get if we choose a certain path to the colored vertex. Simply calculate the difference between the number of vertices between root to each vertex and the number of vertices that occur below the current vertex. Take the largest k of all the difference and calculate the sum. Use nth_element stl to get an O(n) solution.

Below is the implementation of the above approach:



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// C++ program to Maximize the number
// of uncolored vertices occurring between
// the path from root vertex and the colored vertices
#include <bits/stdc++.h>
using namespace std;
  
// Comparator function
bool cmp(int a, int b)
{
    return a > b;
}
  
class graph {
    vector<vector<int> > g;
    vector<int> depth;
    vector<int> subtree;
    int* diff;
  
public:
    // Constructor
    graph(int n)
    {
        g = vector<vector<int> >(n + 1);
  
        depth = vector<int>(n + 1);
  
        subtree = vector<int>(n + 1);
  
        diff = new int[n + 1];
    }
  
    // Function to push edges
    void push(int a, int b)
    {
        g[a].push_back(b);
  
        g[b].push_back(a);
    }
  
    // function for dfs traversal
    int dfs(int v, int p)
    {
  
        // Store depth of vertices
        depth[v] = depth[p] + 1;
  
        subtree[v] = 1;
  
        for (auto i : g[v]) {
            if (i == p)
                continue;
  
            // Calculate number of vertices
            // in subtree of all vertices
            subtree[v] += dfs(i, v);
        }
  
        // Computing the difference
        diff[v] = depth[v] - subtree[v];
  
        return subtree[v];
    }
  
    // Function that print maximum number of
    // uncolored vertices occur between root vertex
    // and all colored vertices
    void solution(int n, int k)
    {
  
        // Computing first k largest difference
        nth_element(diff + 1, diff + k, diff + n + 1, cmp);
  
        int sum = 0;
  
        for (int i = 1; i <= k; i++) {
            sum += diff[i];
        }
  
        // Print the result
        cout << sum << "\n";
    }
};
  
// Driver Code
int main()
{
  
    int N = 7;
    int k = 4;
  
    // initialise graph
    graph g(N);
  
    g.push(1, 2);
    g.push(1, 3);
    g.push(1, 4);
    g.push(3, 5);
    g.push(3, 6);
    g.push(4, 7);
  
    g.dfs(1, 0);
  
    g.solution(N, k);
  
    return 0;
}

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Output:

7

Time Complexity: O(N)

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