Med tanke på N punkter i det 2-dimensionella rummet måste vi hitta tre punkter så att triangeln som skapas genom att välja dessa punkter inte bör innehålla några andra punkter inuti. Alla givna punkter kommer inte att ligga på samma linje så lösningen kommer alltid att finnas.
Exempel:
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.
Lösningen är baserad på det faktum att om det finns triangel(ar) utan punkter inuti så kan vi bilda en triangel med valfri slumpmässig punkt bland alla punkter.
Vi kan lösa detta problem genom att söka på alla tre punkterna en efter en. Den första punkten kan väljas slumpmässigt. Efter att ha valt den första punkten behöver vi två punkter så att deras lutning ska vara annorlunda och ingen punkt ska ligga inuti triangeln av dessa tre punkter. Vi kan göra detta genom att välja den andra punkten som närmaste punkt till första och tredje punkt som näst närmaste punkt med olika lutning. För att göra detta itererar vi först över alla punkter och väljer den punkt som är närmast den första och betecknar den som andra punkt i den önskade triangeln. Sedan itererar vi en gång till för att hitta punkt som har olika lutning och har det minsta avståndet som kommer att vara den tredje punkten i vår triangel.
dubbel till sträng javaC++
// 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>
Produktion:
0 0
1 1
0 2
Tidskomplexitet: På)
Hjälputrymme: O(1)
Denna artikel är bidragit av Utkarsh Trivedi .
Tillvägagångssätt #2: Använd brute kraft
Denna kod itererar över alla möjliga trianglar som kan bildas från den givna uppsättningen punkter och kontrollerar om någon annan punkt ligger inuti varje triangel. Om en triangel hittas där ingen punkt ligger inuti koden returneras triangeln. Annars returnerar den Ingen.
Algoritm
1. Iterera genom alla möjliga trianglar med hörn från de givna punkterna.
2. Kontrollera för varje triangel om någon av de återstående punkterna ligger inuti triangeln.
3. Om inga punkter ligger inuti någon triangel, returnera koordinaterna för den första triangeln som hittas.
#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));
Produktion
[[0 0] [0 2] [1 1]]
Tidskomplexitet: O(n^4) där n är antalet punkter. Det beror på att vi behöver iterera igenom alla möjliga trianglar som är n välj 3 och sedan kontrollera om var och en av de återstående punkterna ligger innanför triangeln som är O(n).
Rymdkomplexitet: O(1) eftersom vi bara lagrar ett fåtal variabler åt gången.
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