Étant donné une matrice de N lignes et M Les colonnes se composent de trois valeurs {r g b}. La tâche consiste à trouver l'aire du plus grand triangle dont un côté est parallèle à l'axe y, c'est-à-dire vertical et la couleur des trois sommets est différente.
Exemples :
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 (00) containing r (14) containing g (30) containing b.
Nous connaissons l'aire d'un triangle = 1/2 * base * hauteur, nous devons donc maximiser la base et la hauteur du triangle. Puisqu’un côté est parallèle à l’axe des y, nous pouvons considérer ce côté comme la base du triangle.
Pour maximiser la base, nous pouvons trouver la première et la dernière occurrence de {r g b} pour chaque colonne. Nous avons donc deux ensembles de 3 valeurs pour chaque colonne. Pour la base de n'importe quelle colonne, un sommet appartient au premier ensemble et le deuxième sommet au deuxième ensemble, de sorte qu'ils ont des valeurs différentes.
Pour maximiser la hauteur de n'importe quelle colonne comme base, le troisième sommet doit être choisi de telle sorte que le sommet soit le plus éloigné de la colonne sur le côté gauche ou droit de la colonne ayant une valeur différente des deux autres sommets.
Maintenant, pour chaque colonne, trouvez l’aire maximale du triangle.
Ci-dessous la mise en œuvre de cette approche :
C++
// C++ program to find maximum area of triangle // having different vertex color in a matrix. #include
import java.util.Arrays; public class Main { static int R = 4; static int C = 5; static char[][] mat = { {'r' 'r' 'r' 'r' 'r'} {'r' 'r' 'r' 'r' 'g'} {'r' 'r' 'r' 'r' 'r'} {'b' 'b' 'b' 'b' 'b'} }; public static void main(String[] args) { System.out.println(maxArea(mat R C)); } // Returns the color value so that their corresponding index can be accessed. 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 = 10; // 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] != Integer.MAX_VALUE && bottom[y][i] != Integer.MIN_VALUE && left[z] != Integer.MAX_VALUE) { ans = Math.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 && top[x][i] != Integer.MAX_VALUE && bottom[y][i] != Integer.MIN_VALUE && right[z] != Integer.MIN_VALUE) { ans = Math.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. static double maxArea(char[][] mat int r int c) { int[] left = new int[3]; Arrays.fill(left Integer.MAX_VALUE); int[] right = new int[3]; Arrays.fill(right Integer.MIN_VALUE); int[][] top = new int[3][c]; for (int[] row : top) Arrays.fill(row Integer.MAX_VALUE); int[][] bottom = new int[3][c]; for (int[] row : bottom) Arrays.fill(row Integer.MIN_VALUE); // 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]); 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]); 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); } }
# 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# 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)); } }
// Javascript program to find maximum area of triangle // having different vertex color in a matrix. // return the color value so that their corresponding // index can be accessed. function mapcolor(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 function findarea(mat r c top bottom left right) { let ans = 10; // for each column for (let i = 0; i < c; i++) { // for each top vertex for (let x = 0; x < 3; x++) { // for each bottom vertex for (let y = 0; y < 3; y++) { // finding the third color of // vertex either on right or left. let z = 3 - x - y; // finding area of triangle on left side of column. if (x != y && top[x][i] != Number.MAX_SAFE_INTEGER && bottom[y][i] != Number.MIN_SAFE_INTEGER && left[z] != Number.MAX_SAFE_INTEGER) { ans = Math.max(ans (1/2) * (bottom[y][i] - top[x][i]) * (i - left[z])); } // finding area of triangle on right side of column. if (x != y && top[x][i] != Number.MAX_SAFE_INTEGER && bottom[y][i] != Number.MIN_SAFE_INTEGER && right[z] != Number.MIN_SAFE_INTEGER) { ans = Math.max(ans (1/2) * (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. function maxarea(mat r c) { let left = [Number.MAX_SAFE_INTEGER Number.MAX_SAFE_INTEGER Number.MAX_SAFE_INTEGER]; let right = [Number.MIN_SAFE_INTEGER Number.MIN_SAFE_INTEGER Number.MIN_SAFE_INTEGER]; let top = Array.from({length: 3} () => Array(c).fill(Number.MAX_SAFE_INTEGER)); let bottom = Array.from({length: 3} () => Array(c).fill(Number.MIN_SAFE_INTEGER)); // finding the r b g cells for the left // and right vertices. for (let i = 0; i < r; i++) { for (let j = 0; j < c; j++) { let color = mapcolor(mat[i][j]); 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 (let j = 0; j < c; j++) { for (let i = 0; i < r; i++) { let color = mapcolor(mat[i][j]); 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 const R = 4; const C = 5; const mat = [ ['r' 'r' 'r' 'r' 'r'] ['r' 'r' 'r' 'r' 'g'] ['r' 'r' 'r' 'r' 'r'] ['b' 'b' 'b' 'b' 'b'] ]; console.log(maxarea(mat R C)); // akashish__
Sortir:
10
Complexité temporelle : O(R*C)
Espace auxiliaire : O(R+C)
Source: https://stackoverflow.com/questions/40078660/maximum-area-of-triangle-having-all-vertices-of-différent-color