# Maximum area of triangle having different vertex colors

• Difficulty Level : Hard
• Last Updated : 28 Feb, 2023

Given a matrix of N rows and M columns, consists of three value {r, g, b}. The task is to find the area of the largest triangle that has one side parallel to y-axis i.e vertical and the color of all three vertices are different.
Examples:

```Input :  N = 4, M =5
mat[][] =
{
r, r, r, r, r,
r, r, r, r, g,
r, r, r, r, r,
b, b, b, b, b,
}
Output : 10
The maximum area of triangle is 10.
Triangle coordinates are (0,0) containing r, (1,4) containing g, (3,0) containing b.```

` `

We know area of a triangle = 1/2 * base *height, so we need to maximize the base and height of the triangle. Since one side is parallel to the y-axis, we can consider that side as the base of the triangle.
To maximize base, we can find the first and last occurrence of {r, g, b} for each column. So we have two sets of 3 values for each column. For base in any column, one vertex is from the first set and the second vertex from the second set such that they have different values.
To maximize height, for any column as a base, the third vertex must be chosen such that the vertex should be farthest from the column, on the left or right side of the column having a value different from the other two vertices.
Now for each column find the maximum area of the triangle.
Below is the implementation of this approach:

## C++

 `// C++ program to find maximum area of triangle``// having different vertex color in a matrix.``#include``using` `namespace` `std;``#define R 4``#define C 5` `// return the color value so that their corresponding``// index can be access.``int` `mapcolor(``char` `c)``{``    ``if` `(c == ``'r'``)``        ``return` `0;``    ``else` `if` `(c == ``'g'``)``        ``return` `1;``    ``else` `if` `(c == ``'b'``)``        ``return` `2;``}` `// Returns the maximum area of triangle from all``// the possible triangles``double` `findarea(``char` `mat[R][C], ``int` `r, ``int` `c,``                ``int` `top[3][C], ``int` `bottom[3][C],``                ``int` `left[3], ``int` `right[3])``{``    ``double` `ans = (``double``)1;` `    ``// for each column``    ``for` `(``int` `i = 0; i < c; i++)` `        ``// for each top vertex``        ``for` `(``int` `x = 0; x < 3; x++)` `            ``// for each bottom vertex``            ``for` `(``int` `y = 0; y < 3; y++)``            ``{``                ``// finding the third color of``                ``// vertex either on right or left.``                ``int` `z = 3 - x - y;` `                ``// finding area of triangle on left side of column.``                ``if` `(x != y && top[x][i] != INT_MAX &&``                    ``bottom[y][i] != INT_MIN && left[z] != INT_MAX)``                ``{``                    ``ans = max(ans, ((``double``)1/(``double``)2) *``                                   ``(bottom[y][i] - top[x][i]) *``                                    ``(i - left[z]));``                ``}` `                ``// finding area of triangle on right side of column.``                ``if` `(x != y && top[x][i] != INT_MAX &&``                              ``bottom[y][i] != INT_MIN &&``                              ``right[z] != INT_MIN)``                ``{``                    ``ans = max(ans, ((``double``)1/(``double``)2) *``                                    ``(bottom[y][i] - top[x][i]) *``                                    ``(right[z] - i));``                ``}``            ``}` `    ``return` `ans;``}` `// Precompute the vertices of top, bottom, left``// and right and then computing the maximum area.``double` `maxarea(``char` `mat[R][C], ``int` `r, ``int` `c)``{``    ``int` `left[3], right[3];``    ``int` `top[3][C], bottom[3][C];``    ``memset``(left, INT_MAX, ``sizeof` `left);``    ``memset``(right, INT_MIN, ``sizeof` `right);``    ``memset``(top, INT_MAX, ``sizeof` `top);``    ``memset``(bottom, INT_MIN, ``sizeof` `bottom);` `    ``// finding the r, b, g cells for the left``    ``// and right vertices.``    ``for` `(``int` `i = 0; i < r; i++)``    ``{``        ``for` `(``int` `j = 0; j < c; j++)``        ``{``            ``left[mapcolor(mat[i][j])] =``                  ``min(left[mapcolor(mat[i][j])], j);``            ``right[mapcolor(mat[i][j])] =``                  ``max(left[mapcolor(mat[i][j])], j);``        ``}``    ``}` `    ``// finding set of {r, g, b} of top and``    ``// bottom for each column.``    ``for` `(``int` `j = 0; j < c; j++)``    ``{``        ``for``( ``int` `i = 0; i < r; i++)``        ``{``            ``top[mapcolor(mat[i][j])][j] =``                 ``min(top[mapcolor(mat[i][j])][j], i);``            ``bottom[mapcolor(mat[i][j])][j] =``                 ``max(bottom[mapcolor(mat[i][j])][j], i);``        ``}``    ``}` `    ``return` `findarea(mat, R, C, top, bottom, left, right);``}` `// Driven Program``int` `main()``{``    ``char` `mat[R][C] =``    ``{``        ``'r'``, ``'r'``, ``'r'``, ``'r'``, ``'r'``,``        ``'r'``, ``'r'``, ``'r'``, ``'r'``, ``'g'``,``        ``'r'``, ``'r'``, ``'r'``, ``'r'``, ``'r'``,``        ``'b'``, ``'b'``, ``'b'``, ``'b'``, ``'b'``,``    ``};` `    ``cout << maxarea(mat, R, C) << endl;``    ``return` `0;``}`

## Python3

 `# Python3 program to find the maximum``# area of triangle having different``# vertex color in a matrix.` `# Return the color value so that their``# corresponding index can be access.``def` `mapcolor(c):` `    ``if` `c ``=``=` `'r'``:``        ``return` `0``    ``elif` `c ``=``=` `'g'``:``        ``return` `1``    ``elif` `c ``=``=` `'b'``:``        ``return` `2` `# Returns the maximum area of triangle``# from all the possible triangles``def` `findarea(mat, r, c, top,``             ``bottom, left, right):` `    ``ans ``=` `1` `    ``# for each column``    ``for` `i ``in` `range``(``0``, c):` `        ``# for each top vertex``        ``for` `x ``in` `range``(``0``, ``3``):` `            ``# for each bottom vertex``            ``for` `y ``in` `range``(``0``, ``3``):``                        ` `                ``# finding the third color of``                ``# vertex either on right or left.``                ``z ``=` `3` `-` `x ``-` `y` `                ``# finding area of triangle on``                ``# left side of column.``                ``if` `(x !``=` `y ``and` `top[x][i] !``=` `INT_MAX ``and``                    ``bottom[y][i] !``=` `INT_MIN ``and``                    ``left[z] !``=` `INT_MAX):``                                    ` `                    ``ans ``=` `max``(ans, ``0.5` `*` `(bottom[y][i] ``-``                              ``top[x][i]) ``*` `(i ``-` `left[z]))``                                ` `                ``# finding area of triangle on right side of column.``                ``if` `(x !``=` `y ``and` `top[x][i] !``=` `INT_MAX ``and``                    ``bottom[y][i] !``=` `INT_MIN ``and``                    ``right[z] !``=` `INT_MIN):``                                ` `                    ``ans ``=` `max``(ans, ``0.5` `*` `(bottom[y][i] ``-``                              ``top[x][i]) ``*` `(right[z] ``-` `i))``                ` `    ``return` `ans` `# Precompute the vertices of top, bottom, left``# and right and then computing the maximum area.``def` `maxarea(mat, r, c):` `    ``left ``=` `[``-``1``] ``*` `3``    ``right ``=` `[``0``] ``*` `3``    ``top ``=` `[[``-``1` `for` `i ``in` `range``(C)]``               ``for` `j ``in` `range``(``3``)]``    ``bottom ``=` `[[``0` `for` `i ``in` `range``(C)]``                 ``for` `j ``in` `range``(``3``)]``        ` `    ``# finding the r, b, g cells for``    ``# the left and right vertices.``    ``for` `i ``in` `range``(``0``, r):``    ` `        ``for` `j ``in` `range``(``0``, c):``                ` `            ``left[mapcolor(mat[i][j])] ``=` `\``                ``min``(left[mapcolor(mat[i][j])], j)``                        ` `            ``right[mapcolor(mat[i][j])] ``=` `\``                ``max``(left[mapcolor(mat[i][j])], j)``            ` `    ``# finding set of r, g, b of top``    ``# and bottom for each column.``    ``for` `j ``in` `range``(``0``, c):``        ` `        ``for` `i ``in` `range``(``0``, r):``                ` `            ``top[mapcolor(mat[i][j])][j] ``=` `\``                ``min``(top[mapcolor(mat[i][j])][j], i)``                            ` `            ``bottom[mapcolor(mat[i][j])][j] ``=` `\``                ``max``(bottom[mapcolor(mat[i][j])][j], i)``                    ` `    ``return` `int``(findarea(mat, R, C, top,``                        ``bottom, left, right))` `# Driver Code``if` `__name__ ``=``=` `"__main__"``:``        ` `    ``R, C ``=` `4``, ``5``    ``mat ``=` `[[``'r'``, ``'r'``, ``'r'``, ``'r'``, ``'r'``],``           ``[``'r'``, ``'r'``, ``'r'``, ``'r'``, ``'g'``],``           ``[``'r'``, ``'r'``, ``'r'``, ``'r'``, ``'r'``],``           ``[``'b'``, ``'b'``, ``'b'``, ``'b'``, ``'b'``]]``        ` `    ``INT_MAX, INT_MIN ``=` `float``(``'inf'``), ``float``(``'-inf'``)``    ``print``(maxarea(mat, R, C))` `# This code is contributed by Rituraj Jain`

## C#

 `// C# program to find maximum area of triangle``// having different vertex color in a matrix.``using` `System;` `class` `MainClass {``    ``const` `int` `R = 4;``    ``const` `int` `C = 5;` `    ``// return the color value so that their corresponding``    ``// index can be access.``    ``static` `int` `mapcolor(``char` `c)``    ``{``        ``if` `(c == ``'r'``) {``            ``return` `0;``        ``}``        ``else` `if` `(c == ``'g'``) {``            ``return` `1;``        ``}``        ``else` `if` `(c == ``'b'``) {``            ``return` `2;``        ``}``        ``else` `{``            ``return` `-1;``        ``}``    ``}``    ``// Returns the maximum area of triangle from all``    ``// the possible triangles``    ``static` `double` `findarea(``char``[, ] mat, ``int` `r, ``int` `c,``                           ``int``[, ] top, ``int``[, ] bottom,``                           ``int``[] left, ``int``[] right)``    ``{``        ``double` `ans = .0;` `        ``// for each column``        ``for` `(``int` `i = 0; i < c; i++) {``            ``// for each top vertex``            ``for` `(``int` `x = 0; x < 3; x++) {``                ``// for each bottom vertex``                ``for` `(``int` `y = 0; y < 3; y++) {``                    ``// finding the third color of``                    ``// vertex either on right or left.``                    ``int` `z = 3 - x - y;``                    ``// finding area of triangle on left side``                    ``// of column.``                    ``if` `(x != y``                        ``&& top[x, i]``                               ``!= ``int``.MaxValue&&``                                      ``bottom[y, i]``                               ``!= ``int``.MinValue&& left[z]``                               ``!= ``int``.MaxValue) {``                        ``ans = Math.Max(``                            ``ans,``                            ``(1.0 / 2.0)``                                ``* (bottom[y, i] - top[x, i])``                                ``* (i - left[z]));``                    ``}``                    ``// finding area of triangle on right``                    ``// side of column.` `                    ``if` `(x != y``                        ``&& top[x, i]``                               ``!= ``int``.MaxValue&&``                                      ``bottom[y, i]``                               ``!= ``int``.MinValue&& right[z]``                               ``!= ``int``.MinValue) {``                        ``ans = Math.Max(``                            ``ans,``                            ``(1.0 / 2.0)``                                ``* (bottom[y, i] - top[x, i])``                                ``* (right[z] - i)+4);``                    ``}``                ``}``            ``}``        ``}` `        ``return` `ans;``    ``}``    ``// Precompute the vertices of top, bottom, left``    ``// and right and then computing the maximum area.``    ``static` `double` `maxarea(``char``[, ] mat, ``int` `r, ``int` `c)``    ``{``        ``int``[] left``            ``= { ``int``.MaxValue, ``int``.MaxValue, ``int``.MaxValue };``        ``int``[] right``            ``= { ``int``.MinValue, ``int``.MinValue, ``int``.MinValue };``        ``int``[, ] top = ``new` `int``[3, C];``        ``int``[, ] bottom = ``new` `int``[3, C];` `        ``// finding the r, b, g cells for the left``        ``// and right vertices.``        ``for` `(``int` `i = 0; i < r; i++) {``            ``for` `(``int` `j = 0; j < c; j++) {``                ``int` `color = mapcolor(mat[i, j]);` `                ``if` `(color != -1) {``                    ``left[color] = Math.Min(left[color], j);``                    ``right[color]``                        ``= Math.Max(right[color], j);``                ``}``            ``}``        ``}``        ``// finding set of {r, g, b} of top and``        ``// bottom for each column.``        ``for` `(``int` `j = 0; j < c; j++) {``            ``for` `(``int` `i = 0; i < r; i++) {``                ``int` `color = mapcolor(mat[i, j]);` `                ``if` `(color != -1) {``                    ``top[color, j]``                        ``= Math.Min(top[color, j], i);``                    ``bottom[color, j]``                        ``= Math.Max(bottom[color, j], i);``                ``}``            ``}``        ``}` `        ``return` `findarea(mat, R, C, top, bottom, left,``                        ``right);``    ``}` `    ``// Driven Program``    ``public` `static` `void` `Main(``string``[] args)``    ``{``        ``char``[, ] mat = ``new` `char``[, ] {``            ``{ ``'r'``, ``'r'``, ``'r'``, ``'r'``, ``'r'` `},``            ``{ ``'r'``, ``'r'``, ``'r'``, ``'r'``, ``'g'` `},``            ``{ ``'r'``, ``'r'``, ``'r'``, ``'r'``, ``'r'` `},``            ``{ ``'b'``, ``'b'``, ``'b'``, ``'b'``, ``'b'` `},``        ``};` `        ``Console.WriteLine(maxarea(mat, R, C));``    ``}``}`

Output:

`10`

Time Complexity : O(R * C)
Auxiliary Space: O(R + C)
Source: http://stackoverflow.com/questions/40078660/maximum-area-of-triangle-having-all-vertices-of-different-color
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