凸包|单调链算法

给定一组点，任务是找到给定点的凸包。凸包是包含所有点的最小凸多边形。
请先检查这篇文章：凸包|设置1(贾维斯算法或包装)

例子：

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)

方法：单调链算法在O(n * log(n))时间内构造凸包。我们必须先对这些点进行排序，然后以O(n)的时间计算上下船体。这些点将相对于x坐标进行排序(如果x坐标为平局，则相对于y坐标)，然后我们将找到最左边的点，然后尝试沿顺时针方向旋转并找到下一个点然后重复该步骤，直到到达最右边，然后再次沿顺时针方向旋转，找到下部船体。

下面是上述方法的实现：

// C++ implementation of the approach
#include 
#define llu long long int
using 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 turn
llu 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
vector 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 code
int 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;
}

输出：

(0, 0)
(3, 0)
(3, 3)
(0, 3)

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