Given n 2-D points points[], the task is to find the perimeter of the convex hull for the set of points. A convex hull for a set of points is the smallest convex polygon that contains all the points. Examples:
Input: points[] = {{0, 3}, {2, 2}, {1, 1}, {2, 1}, {3, 0}, {0, 0}, {3, 3}} Output: 12 Input: points[] = {{0, 2}, {2, 1}, {3, 1}, {3, 7}} Output: 15.067
Approach: Monotone chain algorithm constructs the convex hull in O(n * log(n)) time. We have to sort the points first and then calculate the upper and lower hulls in O(n) time. The points will be sorted with respect to x-coordinates (with respect to y-coordinates in case of a tie in x-coordinates), we will then find the left most point and then try to rotate in clockwise direction and find the next point and then repeat the step until we reach the rightmost point and then again rotate in the clockwise direction and find the lower hull. We will then find the perimeter of the convex hull using the points on the convex hull which can be done in O(n) time as the points are already sorted in clockwise order. Below is the implementation of the above approach:
CPP
// C++ implementation of the approach#include <bits/stdc++.h>#define llu long long intusing namespace std;struct Point { llu x, y; bool operator<(Point p) { return x < p.x || (x == p.x && y < p.y); }};// Cross product of two vectors OA and OB// returns positive for counter clockwise// turn and negative for clockwise turnllu cross_product(Point O, Point A, Point B){ return (A.x - O.x) * (B.y - O.y) - (A.y - O.y) * (B.x - O.x);}// Returns a list of points on the convex hull// in counter-clockwise ordervector<Point> convex_hull(vector<Point> A){ int n = A.size(), k = 0; if (n <= 3) return A; vector<Point> ans(2 * n); // Sort points lexicographically sort(A.begin(), A.end()); // Build lower hull for (int i = 0; i < n; ++i) { // If the point at K-1 position is not a part // of hull as vector from ans[k-2] to ans[k-1] // and ans[k-2] to A[i] has a clockwise turn while (k >= 2 && cross_product(ans[k - 2], ans[k - 1], A[i]) <= 0) k--; ans[k++] = A[i]; } // Build upper hull for (size_t i = n - 1, t = k + 1; i > 0; --i) { // If the point at K-1 position is not a part // of hull as vector from ans[k-2] to ans[k-1] // and ans[k-2] to A[i] has a clockwise turn while (k >= t && cross_product(ans[k - 2], ans[k - 1], A[i - 1]) <= 0) k--; ans[k++] = A[i - 1]; } // Resize the array to desired size ans.resize(k - 1); return ans;}// Function to return the distance between two pointsdouble dist(Point a, Point b){ return sqrt((a.x - b.x) * (a.x - b.x) + (a.y - b.y) * (a.y - b.y));}// Function to return the perimeter of the convex hulldouble perimeter(vector<Point> ans){ double perimeter = 0.0; // Find the distance between adjacent points for (int i = 0; i < ans.size() - 1; i++) { perimeter += dist(ans[i], ans[i + 1]); } // Add the distance between first and last point perimeter += dist(ans[0], ans[ans.size() - 1]); return perimeter;}// Driver codeint main(){ vector<Point> points; // Add points points.push_back({ 0, 3 }); points.push_back({ 2, 2 }); points.push_back({ 1, 1 }); points.push_back({ 2, 1 }); points.push_back({ 3, 0 }); points.push_back({ 0, 0 }); points.push_back({ 3, 3 }); // Find the convex hull vector<Point> ans = convex_hull(points); // Find the perimeter of convex polygon cout << perimeter(ans); return 0;} |
Java
import java.util.*;import java.lang.*;import java.io.*;class Point { int x, y; Point(int x, int y) { this.x = x; this.y = y; }}class Main { static int cross_product(Point o, Point a, Point b) { return (a.x - o.x) * (b.y - o.y) - (a.y - o.y) * (b.x - o.x); } static List<Point> convex_hull(List<Point> points) { int n = points.size(); if (n <= 3) return points; Collections.sort(points, new Comparator<Point>() { @Override public int compare(Point p1, Point p2) { if (p1.x == p2.x) return p1.y - p2.y; return p1.x - p2.x; } }); Point[] ans = new Point[2 * n]; int k = 0; // Build lower hull for (int i = 0; i < n; i++) { while (k >= 2 && cross_product(ans[k-2], ans[k-1], points.get(i)) <= 0) k--; ans[k++] = points.get(i); } // Build upper hull int t = k + 1; for (int i = n-2; i >= 0; i--) { while (k >= t && cross_product(ans[k-2], ans[k-1], points.get(i)) <= 0) k--; ans[k++] = points.get(i); } List<Point> result = new ArrayList<>(); for (int i = 0; i < k-1; i++) result.add(ans[i]); return result; } static double dist(Point a, Point b) { return Math.sqrt(Math.pow(a.x - b.x, 2) + Math.pow(a.y - b.y, 2)); } static double perimeter(List<Point> points) { int n = points.size(); double perimeter = 0.0; for (int i = 0; i < n-1; i++) perimeter += dist(points.get(i), points.get(i+1)); perimeter += dist(points.get(0), points.get(n-1)); return perimeter; } public static void main(String[] args) throws java.lang.Exception { List<Point> points = new ArrayList<>(); points.add(new Point(0, 3)); points.add(new Point(2, 2)); points.add(new Point(1, 1)); points.add(new Point(2, 1)); points.add(new Point(3, 0)); points.add(new Point(0, 0)); points.add(new Point(3, 3)); List<Point> convex_points = convex_hull(points); System.out.println(perimeter(convex_points)); }} |
Python3
from typing import Listfrom math import sqrtclass Point: def __init__(self, x: int, y: int): self.x = x self.y = ydef cross_product(o: Point, a: Point, b: Point) -> int: return (a.x - o.x) * (b.y - o.y) - (a.y - o.y) * (b.x - o.x)def convex_hull(points: List[Point]) -> List[Point]: n = len(points) if n <= 3: return points points.sort(key=lambda p: (p.x, p.y)) ans = [None] * (2 * n) # Build lower hull k = 0 for i in range(n): while k >= 2 and cross_product(ans[k-2], ans[k-1], points[i]) <= 0: k -= 1 ans[k] = points[i] k += 1 # Build upper hull t = k + 1 for i in range(n-2, -1, -1): while k >= t and cross_product(ans[k-2], ans[k-1], points[i]) <= 0: k -= 1 ans[k] = points[i] k += 1 ans = ans[:k-1] return ansdef dist(a: Point, b: Point) -> float: return sqrt((a.x - b.x) ** 2 + (a.y - b.y) ** 2)def perimeter(points: List[Point]) -> float: n = len(points) perimeter = 0.0 for i in range(n-1): perimeter += dist(points[i], points[i+1]) perimeter += dist(points[0], points[n-1]) return perimeter# Driver codepoints = [Point(0, 3), Point(2, 2), Point(1, 1), Point(2, 1), Point(3, 0), Point(0, 0), Point(3, 3)]convex_points = convex_hull(points)print(perimeter(convex_points)) |
Javascript
class Point { constructor(x, y) { this.x = x; this.y = y; }}function crossProduct(o, a, b) { return (a.x - o.x) * (b.y - o.y) - (a.y - o.y) * (b.x - o.x);}function convexHull(points) { let n = points.length; if (n <= 3) { return points; } points.sort((a, b) => a.x - b.x || a.y - b.y); let ans = new Array(2 * n).fill(null); // Build lower hull let k = 0; for (let i = 0; i < n; i++) { while (k >= 2 && crossProduct(ans[k-2], ans[k-1], points[i]) <= 0) { k--; } ans[k] = points[i]; k++; } // Build upper hull let t = k + 1; for (let i = n - 2; i >= 0; i--) { while (k >= t && crossProduct(ans[k-2], ans[k-1], points[i]) <= 0) { k--; } ans[k] = points[i]; k++; } ans = ans.slice(0, k-1); return ans;}function dist(a, b) { return Math.sqrt((a.x - b.x) ** 2 + (a.y - b.y) ** 2);}function perimeter(points) { let n = points.length; let perimeter = 0.0; for (let i = 0; i < n - 1; i++) { perimeter += dist(points[i], points[i+1]); } perimeter += dist(points[0], points[n-1]); return perimeter;}// Driver codelet points = [new Point(0, 3), new Point(2, 2), new Point(1, 1), new Point(2, 1), new Point(3, 0), new Point(0, 0), new Point(3, 3)];let convex_points = convexHull(points);console.log(perimeter(convex_points)); |
C#
using System;using System.Collections.Generic;using System.Linq;class Point { public int x, y; public Point(int x, int y) { this.x = x; this.y = y; }}class MainClass { // Cross product of two vectors OA and OB // returns positive for counter clockwise // turn and negative for clockwise turn static int cross_product(Point o, Point a, Point b) { return (a.x - o.x) * (b.y - o.y) - (a.y - o.y) * (b.x - o.x); } // Returns a list of points on the convex hull // in counter-clockwise order static List<Point> convex_hull(List<Point> points) { int n = points.Count; if (n <= 3) return points; points.Sort((p1, p2) => { if (p1.x == p2.x) return p1.y - p2.y; return p1.x - p2.x; }); Point[] ans = new Point[2 * n]; int k = 0; // Build lower hull for (int i = 0; i < n; i++) { // If the point at K-1 position is not a part // of hull as vector from ans[k-2] to ans[k-1] // and ans[k-2] to A[i] has a clockwise turn while (k >= 2 && cross_product(ans[k-2], ans[k-1], points[i]) <= 0) k--; ans[k++] = points[i]; } // Build upper hull int t = k + 1; for (int i = n-2; i >= 0; i--) { // If the point at K-1 position is not a part // of hull as vector from ans[k-2] to ans[k-1] // and ans[k-2] to A[i] has a clockwise turn while (k >= t && cross_product(ans[k-2], ans[k-1], points[i]) <= 0) k--; ans[k++] = points[i]; } List<Point> result = new List<Point>(); for (int i = 0; i < k-1; i++) result.Add(ans[i]); return result; } // Function to return the distance between two points static double dist(Point a, Point b) { return Math.Sqrt(Math.Pow(a.x - b.x, 2) + Math.Pow(a.y - b.y, 2)); } // Function to return the perimeter of the convex hull static double perimeter(List<Point> points) { int n = points.Count; double perimeter = 0.0; for (int i = 0; i < n-1; i++) perimeter += dist(points[i], points[i+1]); perimeter += dist(points[0], points[n-1]); return perimeter; } // Driver code public static void Main(string[] args) { List<Point> points = new List<Point>(); // Add points points.Add(new Point(0, 3)); points.Add(new Point(2, 2)); points.Add(new Point(1, 1)); points.Add(new Point(2, 1)); points.Add(new Point(3, 0)); points.Add(new Point(0, 0)); points.Add(new Point(3, 3)); // Find the convex hull List<Point> convex_points = convex_hull(points); // Find the perimeter of convex polygon Console.WriteLine(perimeter(convex_points)); }} |
Output:
12
Time Complexity : O(nlogn), where n is the number of points in the input vector.
Auxiliary Space Complexity : O(n), where n is the number of points in the input vector.
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