# 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:

 `// C++ program to Maximize the number ` `// of uncolored vertices occurring between ` `// the path from root vertex and the colored vertices ` `#include ` `using` `namespace` `std; ` ` `  `// Comparator function ` `bool` `cmp(``int` `a, ``int` `b) ` `{ ` `    ``return` `a > b; ` `} ` ` `  `class` `graph { ` `    ``vector > g; ` `    ``vector<``int``> depth; ` `    ``vector<``int``> subtree; ` `    ``int``* diff; ` ` `  `public``: ` `    ``// Constructor ` `    ``graph(``int` `n) ` `    ``{ ` `        ``g = vector >(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; ` `} `

Output:

```7
```

Time Complexity: O(N)

Don’t stop now and take your learning to the next level. Learn all the important concepts of Data Structures and Algorithms with the help of the most trusted course: DSA Self Paced. Become industry ready at a student-friendly price.

My Personal Notes arrow_drop_up Check out this Author's contributed articles.

If you like GeeksforGeeks and would like to contribute, you can also write an article using contribute.geeksforgeeks.org or mail your article to contribute@geeksforgeeks.org. See your article appearing on the GeeksforGeeks main page and help other Geeks.

Please Improve this article if you find anything incorrect by clicking on the "Improve Article" button below.

Article Tags :
Practice Tags :

Be the First to upvote.

Please write to us at contribute@geeksforgeeks.org to report any issue with the above content.