使用C++写分治法求解凸包问题

好的，我可以为您解答这个问题。

对于凸包问题，我们可以使用分治法来解决。具体步骤如下：

1. 将所有点按照 x 坐标进行排序；
2. 将点集分成左右两组，分别求解每一组的凸包；
3. 合并左右两组的凸包。

接下来是具体的 C++ 代码实现：

#include <iostream>
#include <vector>
#include <algorithm>
using namespace std;

struct Point {
    int x;
    int y;
};

// 按照 x 坐标排序
bool cmp(Point a, Point b) {
    return a.x < b.x;
}

// 叉积计算
int crossProduct(Point a, Point b, Point c) {
    return (b.x - a.x) * (c.y - a.y) - (c.x - a.x) * (b.y - a.y);
}

// 求解凸包
vector<Point> convexHull(vector<Point> points) {
    int n = points.size();
    sort(points.begin(), points.end(), cmp);

    vector<Point> hull;
    for (int i = 0; i < n; i++) {
        while (hull.size() >= 2 && crossProduct(hull[hull.size() - 2], hull.back(), points[i]) < 0) {
            hull.pop_back();
        }
        hull.push_back(points[i]);
    }

    int t = hull.size();
    for (int i = n - 2; i >= 0; i--) {
        while (hull.size() > t && crossProduct(hull[hull.size() - 2], hull.back(), points[i]) < 0) {
            hull.pop_back();
        }
        hull.push_back(points[i]);
    }

    hull.pop_back();
    return hull;
}

// 合并凸包
vector<Point> mergeConvexHull(vector<Point> left, vector<Point> right) {
    int leftmost = 0, rightmost = 0;
    for (int i = 1; i < left.size(); i++) {
        if (left[i].x < left[leftmost].x) {
            leftmost = i;
        }
    }
    for (int i = 1; i < right.size(); i++) {
        if (right[i].x > right[rightmost].x) {
            rightmost = i;
        }
    }

    vector<Point> hull;
    hull.push_back(left[leftmost]);
    hull.push_back(right[rightmost]);

    int l = leftmost, r = rightmost;
    while (l < left.size() - 1 || r > 0) {
        if (l == left.size() - 1) {
            hull.push_back(right[--r]);
        } else if (r == 0) {
            hull.push_back(left[++l]);
        } else {
            int cp = crossProduct(left[l], right[r], right[r - 1]);
            if (cp > 0) {
                hull.push_back(right[--r]);
            } else {
                hull.push_back(left[++l]);
            }
        }
    }

    return hull;
}

// 分治求解凸包
vector<Point> divideAndConquerConvexHull(vector<Point> points) {
    if (points.size() <= 1) {
        return points;
    }

    int mid = points.size() / 2;
    vector<Point> left(points.begin(), points.begin() + mid);
    vector<Point> right(points.begin() + mid, points.end());

    vector<Point> hullLeft = divideAndConquerConvexHull(left);
    vector<Point> hullRight = divideAndConquerConvexHull(right);

    return mergeConvexHull(hullLeft, hullRight);
}

int main() {
    vector<Point> points = {{0, 0}, {1, 1}, {2, 2}, {3, 3}, {2, 4}, {3, 5}, {4, 3}, {5, 2}, {4, 0}, {3, 1}};
    vector<Point> hull = divideAndConquerConvexHull(points);

    for (int i = 0; i < hull.size(); i++) {
        cout << "(" << hull[i].x << ", " << hull[i].y << ")" << endl;
    }

    return 0;
}

以上代码实现了分治法求解凸包问题，其中 `Point` 结构体表示一个二维点，`cmp` 函数用于按照 x 坐标排序，`crossProduct` 函数用于计算叉积，`convexHull` 函数用于求解单组点集的凸包，`mergeConvexHull` 函数用于合并左右两组点集的凸包，`divideAndConquerConvexHull` 函数用于利用分治法求解凸包问题。