Monotone chain

// Implementation of Andrew's monotone chain 2D convex hull algorithm.
// Asymptotic complexity: O(n log n).
// Practical performance: 0.5-1.0 seconds for n=1000000 on a 1GHz machine.
#include <algorithm>
#include <vector>
using namespace std;

typedef double coord_t;         // coordinate type
typedef double coord2_t;  // must be big enough to hold 2*max(|coordinate|)^2

struct Point {
    coord_t x, y;

    bool operator <(const Point &p) const {
        return x < p.x || (x == p.x && y < p.y);
    }
};

// 3D cross product of OA and OB vectors, (i.e z-component of their "2D" cross product, but remember that it is not defined in "2D").
// Returns a positive value, if OAB makes a counter-clockwise turn,
// negative for clockwise turn, and zero if the points are collinear.
coord2_t cross(const Point &O, const Point &A, const 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.
// Note: the last point in the returned list is the same as the first one.
vector<Point> convex_hull(vector<Point> P)
{
    size_t n = P.size(), k = 0;
    if (n <= 3) return P;
    vector<Point> H(2*n);

    // Sort points lexicographically
    sort(P.begin(), P.end());

    // Build lower hull
    for (size_t i = 0; i < n; ++i) {
        while (k >= 2 && cross(H[k-2], H[k-1], P[i]) <= 0) k--;
        H[k++] = P[i];
    }

    // Build upper hull
    for (size_t i = n-1, t = k+1; i > 0; --i) {
        while (k >= t && cross(H[k-2], H[k-1], P[i-1]) <= 0) k--;
        H[k++] = P[i-1];
    }

    H.resize(k-1);
    return H;
}