Convex Hull | Monotone chain algorithm

• Difficulty Level : Hard
• Last Updated : 27 Jul, 2021

Given a set of points, the task is to find the convex hull of the given points. The convex hull 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:
(0, 0)
(3, 0)
(3, 3)
(0, 3)

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 the clockwise direction and find the lower hull.
Below is the implementation of the above approach:

CPP

 // C++ implementation of the approach#include #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 convex_hull(vector A){    int n = A.size(), k = 0;      if (n <= 3)        return A;      vector 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;}  // Driver codeint main(){    vector 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 ans = convex_hull(points);      // Print the convex hull    for (int i = 0; i < ans.size(); i++)        cout << "(" << ans[i].x << ", "              << ans[i].y << ")" << endl;      return 0;}
Output:
(0, 0)
(3, 0)
(3, 3)
(0, 3)

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