Dados N puntos en un espacio bidimensional, necesitamos encontrar tres puntos tales que el triángulo formado al elegir estos puntos no contenga ningún otro punto en su interior. No todos los puntos dados estarán en la misma recta, por lo que siempre existirá una solución.
Ejemplos:
In above diagram possible triangle with no point
inside can be formed by choosing these triplets
[(0 0) (2 0) (1 1)]
[(0 0) (1 1) (0 2)]
[(1 1) (2 0) (2 2)]
[(1 1) (0 2) (2 2)]
So any of the above triplets can be the final answer.
La solución se basa en el hecho de que si existen triángulos sin puntos dentro, entonces podemos formar un triángulo con cualquier punto aleatorio entre todos los puntos.
Podemos resolver este problema buscando los tres puntos uno por uno. El primer punto se puede elegir al azar. Después de elegir el primer punto, necesitamos dos puntos cuya pendiente sea diferente y ningún punto debe estar dentro del triángulo de estos tres puntos. Podemos hacer esto eligiendo el segundo punto como el punto más cercano al primero y el tercer punto como el segundo punto más cercano con pendiente diferente. Para hacer esto, primero iteramos sobre todos los puntos y elegimos el punto más cercano al primero y lo designamos como segundo punto del triángulo requerido. Luego iteramos una vez más para encontrar el punto que tenga una pendiente diferente y que tenga la distancia más pequeña, que será el tercer punto de nuestro triángulo.
que es hibernarC++
// C/C++ program to find triangle with no point inside #include
// Java program to find triangle // with no point inside import java.io.*; class GFG { // method to get square of distance between // (x1 y1) and (x2 y2) static int getDistance(int x1 int y1 int x2 int y2) { return (x2 - x1)*(x2 - x1) + (y2 - y1)*(y2 - y1); } // Method prints points which make triangle with no // point inside static void triangleWithNoPointInside(int points[][] int N) { // any point can be chosen as first point of triangle int first = 0; int second =0; int third =0; int minD = Integer.MAX_VALUE; // choose nearest point as second point of triangle for (int i = 0; i < N; i++) { if (i == first) continue; // Get distance from first point and choose // nearest one int d = getDistance(points[i][0] points[i][1] points[first][0] points[first][1]); if (minD > d) { minD = d; second = i; } } // Pick third point by finding the second closest // point with different slope. minD = Integer.MAX_VALUE; for (int i = 0; i < N; i++) { // if already chosen point then skip them if (i == first || i == second) continue; // get distance from first point int d = getDistance(points[i][0] points[i][1] points[first][0] points[first][1]); /* the slope of the third point with the first point should not be equal to the slope of second point with first point (otherwise they'll be collinear) and among all such points we choose point with the smallest distance */ // here cross multiplication is compared instead // of division comparison if ((points[i][0] - points[first][0]) * (points[second][1] - points[first][1]) != (points[second][0] - points[first][0]) * (points[i][1] - points[first][1]) && minD > d) { minD = d; third = i; } } System.out.println(points[first][0] + ' ' + points[first][1]); System.out.println(points[second][0]+ ' ' + points[second][1]) ; System.out.println(points[third][0] +' ' + points[third][1]); } // Driver code public static void main (String[] args) { int points[][] = {{0 0} {0 2} {2 0} {2 2} {1 1}}; int N = points.length; triangleWithNoPointInside(points N); } } // This article is contributed by vt_m.
# Python3 program to find triangle # with no point inside import sys # method to get square of distance between # (x1 y1) and (x2 y2) def getDistance(x1 y1 x2 y2): return (x2 - x1) * (x2 - x1) + (y2 - y1) * (y2 - y1) # Method prints points which make triangle # with no point inside def triangleWithNoPointInside(points N): # any point can be chosen as # first point of triangle first = 0 second = 0 third = 0 minD = sys.maxsize # choose nearest point as # second point of triangle for i in range(0 N): if i == first: continue # Get distance from first point and choose # nearest one d = getDistance(points[i][0] points[i][1] points[first][0] points[first][1]) if minD > d: minD = d second = i # Pick third point by finding the second closest # point with different slope. minD = sys.maxsize for i in range (0 N): # if already chosen point then skip them if i == first or i == second: continue # get distance from first point d = getDistance(points[i][0] points[i][1] points[first][0] points[first][1]) ''' the slope of the third point with the first point should not be equal to the slope of second point with first point (otherwise they'll be collinear) and among all such points we choose point with the smallest distance ''' # here cross multiplication is compared instead # of division comparison if ((points[i][0] - points[first][0]) * (points[second][1] - points[first][1]) != (points[second][0] - points[first][0]) * (points[i][1] - points[first][1]) and minD > d) : minD = d third = i print(points[first][0] ' ' points[first][1]) print(points[second][0] ' ' points[second][1]) print(points[third][0] ' ' points[third][1]) # Driver code points = [[0 0] [0 2] [2 0] [2 2] [1 1]] N = len(points) triangleWithNoPointInside(points N) # This code is contributed by Gowtham Yuvaraj
using System; // C# program to find triangle // with no point inside public class GFG { // method to get square of distance between // (x1 y1) and (x2 y2) public static int getDistance(int x1 int y1 int x2 int y2) { return (x2 - x1) * (x2 - x1) + (y2 - y1) * (y2 - y1); } // Method prints points which make triangle with no // point inside public static void triangleWithNoPointInside(int[][] points int N) { // any point can be chosen as first point of triangle int first = 0; int second = 0; int third = 0; int minD = int.MaxValue; // choose nearest point as second point of triangle for (int i = 0; i < N; i++) { if (i == first) { continue; } // Get distance from first point and choose // nearest one int d = getDistance(points[i][0] points[i][1] points[first][0] points[first][1]); if (minD > d) { minD = d; second = i; } } // Pick third point by finding the second closest // point with different slope. minD = int.MaxValue; for (int i = 0; i < N; i++) { // if already chosen point then skip them if (i == first || i == second) { continue; } // get distance from first point int d = getDistance(points[i][0] points[i][1] points[first][0] points[first][1]); /* the slope of the third point with the first point should not be equal to the slope of second point with first point (otherwise they'll be collinear) and among all such points we choose point with the smallest distance */ // here cross multiplication is compared instead // of division comparison if ((points[i][0] - points[first][0]) * (points[second][1] - points[first][1]) != (points[second][0] - points[first][0]) * (points[i][1] - points[first][1]) && minD > d) { minD = d; third = i; } } Console.WriteLine(points[first][0] + ' ' + points[first][1]); Console.WriteLine(points[second][0] + ' ' + points[second][1]); Console.WriteLine(points[third][0] + ' ' + points[third][1]); } // Driver code public static void Main(string[] args) { int[][] points = new int[][] { new int[] {0 0} new int[] {0 2} new int[] {2 0} new int[] {2 2} new int[] {1 1} }; int N = points.Length; triangleWithNoPointInside(points N); } } // This code is contributed by Shrikant13
<script> // javascript program to find triangle // with no point inside // method to get square of distance between // (x1 y1) and (x2 y2) function getDistance(x1 y1 x2 y2) { return (x2 - x1) * (x2 - x1) + (y2 - y1) * (y2 - y1); } // Method prints points which make triangle with no // point inside function triangleWithNoPointInside(points N) { // any point can be chosen as first point of triangle var first = 0; var second = 0; var third = 0; var minD = Number.MAX_VALUE; // choose nearest point as second point of triangle for (i = 0; i < N; i++) { if (i == first) continue; // Get distance from first point and choose // nearest one var d = getDistance(points[i][0] points[i][1] points[first][0] points[first][1]); if (minD > d) { minD = d; second = i; } } // Pick third point by finding the second closest // point with different slope. minD = Number.MAX_VALUE; for (i = 0; i < N; i++) { // if already chosen point then skip them if (i == first || i == second) continue; // get distance from first point var d = getDistance(points[i][0] points[i][1] points[first][0] points[first][1]); /* * the slope of the third point with the first point should not be equal to the * slope of second point with first point (otherwise they'll be collinear) and * among all such points we choose point with the smallest distance */ // here cross multiplication is compared instead // of division comparison if ((points[i][0] - points[first][0]) * (points[second][1] - points[first][1]) != (points[second][0] - points[first][0]) * (points[i][1] - points[first][1]) && minD > d) { minD = d; third = i; } } document.write(points[first][0] + ' ' + points[first][1]+'
'); document.write(points[second][0] + ' ' + points[second][1]+'
'); document.write(points[third][0] + ' ' + points[third][1]+'
'); } // Driver code var points = [ [ 0 0 ] [ 0 2 ] [ 2 0 ] [ 2 2 ] [ 1 1 ] ]; var N = points.length; triangleWithNoPointInside(points N); // This code contributed by umadevi9616 </script>
Producción:
0 0
1 1
0 2
Complejidad del tiempo: En)
Espacio Auxiliar: O(1)
Este artículo es una contribución de Utkarsh Trivedi .
Enfoque n.º 2: uso de la fuerza bruta
Este código itera sobre todos los triángulos posibles que se pueden formar a partir del conjunto de puntos dado y comprueba si hay algún otro punto dentro de cada triángulo. Si se encuentra un triángulo donde no hay ningún punto dentro, el código devuelve el triángulo. De lo contrario, devuelve Ninguno.
Algoritmo
1. Repita todos los triángulos posibles con vértices de los puntos dados.
2. Para cada triángulo, verifica si alguno de los puntos restantes se encuentra dentro del triángulo.
3. Si no hay puntos dentro de ningún triángulo, devuelva las coordenadas del primer triángulo encontrado.
#include
import java.util.ArrayList; import java.util.List; public class Main { static double area(int x1 int y1 int x2 int y2 int x3 int y3) { return Math.abs((x1*(y2-y3) + x2*(y3-y1) + x3*(y1-y2))/2.0); } static boolean isInsideTriangle(int x1 int y1 int x2 int y2 int x3 int y3 int x int y) { double A = area(x1 y1 x2 y2 x3 y3); double A1 = area(x y x2 y2 x3 y3); double A2 = area(x1 y1 x y x3 y3); double A3 = area(x1 y1 x2 y2 x y); return A == A1 + A2 + A3; } static List<List<Integer>> noPointInsideTriangle(List<List<Integer>> points) { for (int i = 0; i < points.size(); i++) { for (int j = i+1; j < points.size(); j++) { for (int k = j+1; k < points.size(); k++) { boolean inside = false; for (int l = 0; l < points.size(); l++) { if (l != i && l != j && l != k) { if (isInsideTriangle(points.get(i).get(0) points.get(i).get(1) points.get(j).get(0) points.get(j).get(1) points.get(k).get(0) points.get(k).get(1) points.get(l).get(0) points.get(l).get(1))) { inside = true; break; } } } if (!inside) { List<List<Integer>> result = new ArrayList<>(); result.add(points.get(i)); result.add(points.get(j)); result.add(points.get(k)); return result; } } } } return null; } public static void main(String[] args) { List<List<Integer>> points = new ArrayList<>(); points.add(new ArrayList<Integer>(){{add(0); add(0);}}); points.add(new ArrayList<Integer>(){{add(0); add(2);}}); points.add(new ArrayList<Integer>(){{add(2); add(0);}}); points.add(new ArrayList<Integer>(){{add(2); add(2);}}); points.add(new ArrayList<Integer>(){{add(1); add(1);}}); List<List<Integer>> result = noPointInsideTriangle(points); if (result != null) { System.out.println(result); } else { System.out.println('No triangle found.'); } } }
def area(x1 y1 x2 y2 x3 y3): return abs((x1*(y2-y3) + x2*(y3-y1) + x3*(y1-y2))/2.0) def is_inside_triangle(x1 y1 x2 y2 x3 y3 x y): A = area(x1 y1 x2 y2 x3 y3) A1 = area(x y x2 y2 x3 y3) A2 = area(x1 y1 x y x3 y3) A3 = area(x1 y1 x2 y2 x y) return A == A1 + A2 + A3 def no_point_inside_triangle(points): for i in range(len(points)): for j in range(i+1 len(points)): for k in range(j+1 len(points)): inside = False for l in range(len(points)): if l != i and l != j and l != k: if is_inside_triangle(points[i][0] points[i][1] points[j][0] points[j][1] points[k][0] points[k][1] points[l][0] points[l][1]): inside = True break if not inside: return [points[i] points[j] points[k]] return None # Example usage points = [[0 0] [0 2] [2 0] [2 2] [1 1]] print(no_point_inside_triangle(points))
using System; using System.Collections.Generic; class Program { // Function to calculate the area of a triangle given its three vertices static double Area(double x1 double y1 double x2 double y2 double x3 double y3) { return Math.Abs((x1 * (y2 - y3) + x2 * (y3 - y1) + x3 * (y1 - y2)) / 2.0); } // Function to check if a point (x y) is inside a triangle defined by its vertices (x1 y1) (x2 y2) and (x3 y3) static bool IsInsideTriangle(double x1 double y1 double x2 double y2 double x3 double y3 double x double y) { double A = Area(x1 y1 x2 y2 x3 y3); double A1 = Area(x y x2 y2 x3 y3); double A2 = Area(x1 y1 x y x3 y3); double A3 = Area(x1 y1 x2 y2 x y); return Math.Abs(A - (A1 + A2 + A3)) < 1e-9; // Use a small epsilon value for comparison due to floating-point precision } // Function to find three points from a list that do not form a triangle with any other point inside static List<double[]> NoPointInsideTriangle(List<double[]> points) { for (int i = 0; i < points.Count; ++i) { for (int j = i + 1; j < points.Count; ++j) { for (int k = j + 1; k < points.Count; ++k) { bool inside = false; for (int l = 0; l < points.Count; ++l) { if (l != i && l != j && l != k) { if (IsInsideTriangle(points[i][0] points[i][1] points[j][0] points[j][1] points[k][0] points[k][1] points[l][0] points[l][1])) { inside = true; break; } } } if (!inside) { List<double[]> result = new List<double[]> { points[i] points[j] points[k] }; return result; } } } } return new List<double[]>(); // Return an empty list if no such set of points is found } static void Main(string[] args) { List<double[]> points = new List<double[]> { new double[] {0 0} new double[] {0 2} new double[] {2 0} new double[] {2 2} new double[] {1 1} }; List<double[]> result = NoPointInsideTriangle(points); if (result.Count > 0) { Console.WriteLine('Points that do not form a triangle with any other point inside:'); foreach (var point in result) { Console.WriteLine($'({point[0]} {point[1]})'); } } else { Console.WriteLine('No such set of points found.'); } } }
// JavaScript equivalent of the Python code above function area(x1 y1 x2 y2 x3 y3) { return Math.abs((x1 * (y2 - y3) + x2 * (y3 - y1) + x3 * (y1 - y2)) / 2.0); } function is_inside_triangle(x1 y1 x2 y2 x3 y3 x y) { const A = area(x1 y1 x2 y2 x3 y3); const A1 = area(x y x2 y2 x3 y3); const A2 = area(x1 y1 x y x3 y3); const A3 = area(x1 y1 x2 y2 x y); return A === A1 + A2 + A3; } function no_point_inside_triangle(points) { for (let i = 0; i < points.length; i++) { for (let j = i + 1; j < points.length; j++) { for (let k = j + 1; k < points.length; k++) { let inside = false; for (let l = 0; l < points.length; l++) { if (l !== i && l !== j && l !== k) { if (is_inside_triangle(points[i][0] points[i][1] points[j][0] points[j][1] points[k][0] points[k][1] points[l][0] points[l][1])) { inside = true; break; } } } if (!inside) { return [points[i] points[j] points[k]]; } } } } return null; } // Example usage const points = [[0 0] [0 2] [2 0] [2 2] [1 1]]; console.log(no_point_inside_triangle(points));
Producción
[[0 0] [0 2] [1 1]]
Complejidad del tiempo: O (n^4) donde n es el número de puntos. Esto se debe a que necesitamos recorrer todos los triángulos posibles, que es n, elegir 3 y luego verificar si cada uno de los puntos restantes se encuentra dentro del triángulo que es O (n).
Complejidad del espacio: O(1) ya que solo almacenamos unas pocas variables a la vez.
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