# Maximum difference of count of black and white vertices in a path containing vertex V

• Last Updated : 22 Jun, 2021

Given a Tree with N vertices and N – 1 edge where the vertices are numbered from 0 to N – 1, and a vertex V present in the tree. It is given that each vertex in the tree has a color assigned to it which is either white or black and the respective colors of the vertices are represented by an array arr[]. The task is to find the maximum difference between the number of white-colored vertices and the number of black colored vertices from any possible subtree from the given tree that contains the given vertex V

Examples:

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```Input: V = 0,
arr[] = {'b', 'w', 'w', 'w', 'b',
'b', 'b', 'b', 'w'}
Tree:
0 b
/   \
/       \
1 w        2 w
/          / \
/          /   \
5 b      w 3     4 b
|          |     |
|          |     |
7 b      b 6     8 w
Output: 2
Explanation:
We can take the subtree
containing the vertex 0
which contains vertices
0, 1, 2, 3 such that
the difference between
the number of white
and the number of black vertices
is maximum which is equal to 2.

Input:
V = 2,
arr[] = {'b', 'b', 'w', 'b'}
Tree:
0 b
/  |  \
/   |   \
1    2    3
b    w     b
Output: 1 ```

Approach: The idea is to use the concept of dynamic programming to solve this problem.

• Firstly, make a vector for color array and for white color, push 1 and for black color, push -1.
• Make an array dp[] to calculate the maximum possible difference between the number of white and black vertices in some subtree containing the vertex V.
• Now, traverse through the tree using depth first search traversal and update the values in dp[] array.
• Finally, the minimum value present in the dp[] array is the required answer.

Below is the implementation of the above approach:

## C++

 `// C++ program to find maximum``// difference between count of``// black and white vertices in``// a path containing vertex V` `#include ``using` `namespace` `std;` `// Defining the tree class``class` `tree {``    ``vector<``int``> dp;``    ``vector > g;``    ``vector<``int``> c;` `public``:``    ``// Constructor``    ``tree(``int` `n)``    ``{``        ``dp = vector<``int``>(n);``        ``g = vector >(n);``        ``c = vector<``int``>(n);``    ``}` `    ``// Function for adding edges``    ``void` `addEdge(``int` `u, ``int` `v)``    ``{``        ``g[u].push_back(v);``        ``g[v].push_back(u);``    ``}` `    ``// Function to perform DFS``    ``// on the given tree``    ``void` `dfs(``int` `v, ``int` `p = -1)``    ``{``        ``dp[v] = c[v];` `        ``for` `(``auto` `i : g[v]) {``            ``if` `(i == p)``                ``continue``;` `            ``dfs(i, v);` `            ``// Returning calculated maximum``            ``// difference between white``            ``// and black for current vertex``            ``dp[v] += max(0, dp[i]);``        ``}``    ``}` `    ``// Function that prints the``    ``// maximum difference between``    ``// white and black vertices``    ``void` `maximumDifference(``int` `v,``                           ``char` `color[],``                           ``int` `n)``    ``{``        ``for` `(``int` `i = 0; i < n; i++) {` `            ``// Condition for white vertex``            ``if` `(color[i] == ``'w'``)``                ``c[i] = 1;` `            ``// Condition for black vertex``            ``else``                ``c[i] = -1;``        ``}` `        ``// Calling dfs function for vertex v``        ``dfs(v);` `        ``// Printing maximum difference between``        ``// white and black vertices``        ``cout << dp[v] << ``"\n"``;``    ``}``};` `// Driver code``int` `main()``{``    ``tree t(9);` `    ``t.addEdge(0, 1);``    ``t.addEdge(0, 2);``    ``t.addEdge(2, 3);``    ``t.addEdge(2, 4);``    ``t.addEdge(1, 5);``    ``t.addEdge(3, 6);``    ``t.addEdge(5, 7);``    ``t.addEdge(4, 8);` `    ``int` `V = 0;` `    ``char` `color[] = { ``'b'``, ``'w'``, ``'w'``,``                     ``'w'``, ``'b'``, ``'b'``,``                     ``'b'``, ``'b'``, ``'w'` `};` `    ``t.maximumDifference(V, color, 9);` `    ``return` `0;``}`

## Java

 `// Java program to find maximum``// difference between count of``// black and white vertices in``// a path containing vertex V``import` `java.util.*;` `// Defining the``// tree class``class` `GFG{``    ` `static` `int` `[]dp;``static` `Vector []g;``static` `int``[] c;` `// Constructor``GFG(``int` `n)``{``  ``dp = ``new` `int``[n];``  ``g =  ``new` `Vector[n];``  ` `  ``for` `(``int` `i = ``0``; i < g.length; i++)``    ``g[i] = ``new` `Vector();``  ` `  ``c = ``new` `int``[n];``}` `// Function for adding edges``void` `addEdge(``int` `u, ``int` `v)``{``  ``g[u].add(v);``  ``g[v].add(u);``}` `// Function to perform DFS``// on the given tree``static` `void` `dfs(``int` `v, ``int` `p)``{``  ``dp[v] = c[v];` `  ``for` `(``int` `i : g[v])``  ``{``    ``if` `(i == p)``      ``continue``;` `    ``dfs(i, v);` `    ``// Returning calculated maximum``    ``// difference between white``    ``// and black for current vertex``    ``dp[v] += Math.max(``0``, dp[i]);``  ``}``}` `// Function that prints the``// maximum difference between``// white and black vertices``void` `maximumDifference(``int` `v,``                       ``char` `color[],``                       ``int` `n)``{``  ``for` `(``int` `i = ``0``; i < n; i++)``  ``{``    ``// Condition for``    ``// white vertex``    ``if` `(color[i] == ``'w'``)``      ``c[i] = ``1``;` `    ``// Condition for``    ``// black vertex``    ``else``      ``c[i] = -``1``;``  ``}` `  ``// Calling dfs function``  ``// for vertex v``  ``dfs(v, -``1``);` `  ``// Printing maximum difference``  ``// between white and black vertices``  ``System.out.print(dp[v] + ``"\n"``);``}` `// Driver code``public` `static` `void` `main(String[] args)``{``  ``GFG t = ``new` `GFG(``9``);` `  ``t.addEdge(``0``, ``1``);``  ``t.addEdge(``0``, ``2``);``  ``t.addEdge(``2``, ``3``);``  ``t.addEdge(``2``, ``4``);``  ``t.addEdge(``1``, ``5``);``  ``t.addEdge(``3``, ``6``);``  ``t.addEdge(``5``, ``7``);``  ``t.addEdge(``4``, ``8``);` `  ``int` `V = ``0``;` `  ``char` `color[] = {``'b'``, ``'w'``, ``'w'``,``                  ``'w'``, ``'b'``, ``'b'``,``                  ``'b'``, ``'b'``, ``'w'``};``  ``t.maximumDifference(V, color, ``9``);``}``}` `// This code is contributed by 29AjayKumar`

## Python3

 `# Python3 program to find maximum``# difference between count of``# black and white vertices in``# a path containing vertex V` `# Function for adding edges``def` `addEdge(g, u, v):``    ` `    ``g[u].append(v)``    ``g[v].append(u)` `# Function to perform DFS``# on the given tree``def` `dfs(v, p, dp, c, g):``    ` `    ``dp[v] ``=` `c[v]``    ` `    ``for` `i ``in` `g[v]:``        ``if` `i ``=``=` `p:``            ``continue` `        ``dfs(i, v, dp, c, g)` `        ``# Returning calculated maximum``        ``# difference between white``        ``# and black for current vertex``        ``dp[v] ``+``=` `max``(``0``, dp[i])` `# Function that prints the``# maximum difference between``# white and black vertices``def` `maximumDifference(v, color, n, dp, c, g):``    ` `    ``for` `i ``in` `range``(n):``        ` `        ``# Condition for white vertex``        ``if``(color[i] ``=``=` `'w'``):``            ``c[i] ``=` `1` `        ``# Condition for black vertex``        ``else``:``            ``c[i] ``=` `-``1` `    ``# Calling dfs function for vertex v``    ``dfs(v, ``-``1``, dp, c, g)` `    ``# Printing maximum difference between``    ``# white and black vertices``    ``print``(dp[v])` `# Driver code``n ``=` `9``g ``=` `{}``dp ``=` `[``0``] ``*` `n``c ``=` `[``0``] ``*` `n` `for` `i ``in` `range``(``0``, n ``+` `1``):``    ``g[i] ``=` `[]``    ` `addEdge(g, ``0``, ``1``)``addEdge(g, ``0``, ``2``)``addEdge(g, ``2``, ``2``)``addEdge(g, ``2``, ``4``)``addEdge(g, ``1``, ``5``)``addEdge(g, ``3``, ``6``)``addEdge(g, ``5``, ``7``)``addEdge(g, ``4``, ``8``)` `V ``=` `0` `color ``=` `[ ``'b'``, ``'w'``, ``'w'``,``          ``'w'``, ``'b'``, ``'b'``,``          ``'b'``, ``'b'``, ``'w'` `]``          ` `maximumDifference(V, color, ``9``, dp, c, g)` `# This code is contributed by avanitrachhadiya2155`

## C#

 `// C# program to find maximum``// difference between count of``// black and white vertices in``// a path containing vertex V``using` `System;``using` `System.Collections.Generic;` `// Defining the``// tree class``class` `GFG{``    ` `static` `int` `[]dp;``static` `List<``int``> []g;``static` `int``[] c;` `// Constructor``GFG(``int` `n)``{``  ``dp = ``new` `int``[n];``  ``g = ``new` `List<``int``>[n];` `  ``for` `(``int` `i = 0; i < g.Length; i++)``    ``g[i] = ``new` `List<``int``>();` `  ``c = ``new` `int``[n];``}` `// Function for adding edges``void` `addEdge(``int` `u, ``int` `v)``{``  ``g[u].Add(v);``  ``g[v].Add(u);``}` `// Function to perform DFS``// on the given tree``static` `void` `dfs(``int` `v, ``int` `p)``{``  ``dp[v] = c[v];` `  ``foreach` `(``int` `i ``in` `g[v])``  ``{``    ``if` `(i == p)``      ``continue``;` `    ``dfs(i, v);` `    ``// Returning calculated maximum``    ``// difference between white``    ``// and black for current vertex``    ``dp[v] += Math.Max(0, dp[i]);``  ``}``}` `// Function that prints the``// maximum difference between``// white and black vertices``void` `maximumDifference(``int` `v,``                       ``char` `[]color,``                       ``int` `n)``{``  ``for` `(``int` `i = 0; i < n; i++)``  ``{``    ``// Condition for``    ``// white vertex``    ``if` `(color[i] == ``'w'``)``      ``c[i] = 1;` `    ``// Condition for``    ``// black vertex``    ``else``      ``c[i] = -1;``  ``}` `  ``// Calling dfs function``  ``// for vertex v``  ``dfs(v, -1);` `  ``// Printing maximum difference``  ``// between white and black vertices``  ``Console.Write(dp[v] + ``"\n"``);``}` `// Driver code``public` `static` `void` `Main(String[] args)``{``  ``GFG t = ``new` `GFG(9);` `  ``t.addEdge(0, 1);``  ``t.addEdge(0, 2);``  ``t.addEdge(2, 3);``  ``t.addEdge(2, 4);``  ``t.addEdge(1, 5);``  ``t.addEdge(3, 6);``  ``t.addEdge(5, 7);``  ``t.addEdge(4, 8);` `  ``int` `V = 0;` `  ``char` `[]color = {``'b'``, ``'w'``, ``'w'``,``                  ``'w'``, ``'b'``, ``'b'``,``                  ``'b'``, ``'b'``, ``'w'``};``  ``t.maximumDifference(V, color, 9);``}``}` `// This code is contributed by Rajput-Ji`

## Javascript

 ``
Output:
`2`

Time Complexity: O(N), where N is the number of vertices in the tree.

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