poj 1514 Metal Cutting (dfs+多边形切割)

1514 -- Metal Cutting

　　一道类似于半平面交的题。

　　题意相当简单，给出一块矩形以及最后被切出来的的多边形各个顶点的位置。每次切割必须从一端切到另一端，问切出多边形最少要切多长的距离。

　　因为最短的切割距离肯定是没有多余的切割痕迹的，而且对于多边形的每一条边，都需要至少经过一次，也就是这些是必要切割。又因为最多就只有8条切割的痕迹，所以我们可以枚举每条痕迹的先后次序，然后模拟切割即刻。复杂度O(n!*n)。

代码如下：

 #include <cstdio>
 #include <cstring>
 #include <cmath>
 #include <vector>
 #include <iostream>
 #include <algorithm>

 using namespace std;

 const double EPS = 1e-;
 const double PI = acos(-1.0);
 template <class T> T sqr(T x) { return x * x;}
 struct Point {
     double x, y;
     Point() {}
     Point(double x, double y) : x(x), y(y) {}
 } ;
 typedef Point Vec;
 Vec operator + (Vec a, Vec b) { return Vec(a.x + b.x, a.y + b.y);}
 Vec operator - (Vec a, Vec b) { return Vec(a.x - b.x, a.y - b.y);}
 Vec operator * (Vec a, double p) { return Vec(a.x * p, a.y * p);}
 Vec operator / (Vec a, double p) { return Vec(a.x / p, a.y / p);}
 inline int sgn(double x) { return (x > EPS) - (x < -EPS);}
 bool operator < (Point a, Point b) { return sgn(a.x - b.x) <  || sgn(a.x - b.x) ==  && a.y < b.y;}
 bool operator == (Point a, Point b) { return sgn(a.x - b.x) ==  && sgn(a.y - b.y) == ;}

 inline double dotDet(Vec a, Vec b) { return a.x * b.x + a.y * b.y;}
 inline double crossDet(Vec a, Vec b) { return a.x * b.y - a.y * b.x;}
 inline double dotDet(Point o, Point a, Point b) { return dotDet(a - o, b - o);}
 inline double crossDet(Point o, Point a, Point b) { return crossDet(a - o, b - o);}
 inline double vecLen(Vec x) { return sqrt(dotDet(x, x));}
 inline Vec vecUnit(Vec x) { return x / vecLen(x);}
 inline Vec normal(Vec x) { return Vec(-x.y, x.x) / vecLen(x);}
 inline bool onSeg(Point x, Point a, Point b) { return sgn(crossDet(x, a, b)) ==  && sgn(dotDet(x, a, b)) < ;}

 int segIntersect(Point a, Point c, Point b, Point d) {
     Vec v1 = b - a, v2 = c - b, v3 = d - c, v4 = a - d;
     int a_bc = sgn(crossDet(v1, v2));
     int b_cd = sgn(crossDet(v2, v3));
     int c_da = sgn(crossDet(v3, v4));
     int d_ab = sgn(crossDet(v4, v1));
 //    cout << a_bc << ' ' << b_cd << ' ' << c_da << ' ' << d_ab << endl;
     if (a_bc * c_da >  && b_cd * d_ab > ) return ;
     if (onSeg(b, a, c) && c_da) return ;
     if (onSeg(c, b, d) && d_ab) return ;
     if (onSeg(d, c, a) && a_bc) return ;
     if (onSeg(a, d, b) && b_cd) return ;
     return ;
 }

 Point lineIntersect(Point P, Vec v, Point Q, Vec w) {
     Vec u = P - Q;
     double t = crossDet(w, u) / crossDet(v, w);
     return P + v * t;
 }

 struct Poly {
     vector<Point> pt;
     Poly() { pt.clear();}
     ~Poly() {}
     Poly(vector<Point> &pt) : pt(pt) {}
     Point operator [] (int x) const { return pt[x];}
     int size() { return pt.size();}
 } ;

 Poly cutPoly(Poly &poly, Point a, Point b) {
     Poly ret = Poly();
     int n = poly.size();
     for (int i = ; i < n; i++) {
         Point c = poly[i], d = poly[(i + ) % n];
         if (sgn(crossDet(a, b, c)) >= ) ret.pt.push_back(c);
         if (sgn(crossDet(b - a, c - d)) != ) {
             Point ip = lineIntersect(a, b - a, c, d - c);
             if (onSeg(ip, c, d)) ret.pt.push_back(ip);
         }
     }
     return ret;
 }

 const double dir[][] = { {0.0, 0.0}, {1.0, 0.0}, {1.0, 1.0}, {0.0, 1.0}};
 const double FINF = 1e100;
 bool vis[];
 Point rec[];
 double minLen;

 void dfs(int p, int n, Poly &poly, double len) {
 //    for (int i = 0; i < n; i++) cout << vis[i]; cout << endl;
     if (p >= n) {
         minLen = min(len, minLen);
         return ;
     }
     for (int i = ; i < n; i++) {
 //        cout << i << endl;
         if (vis[i]) continue;
         vis[i] = true;
         Vec d = vecUnit(rec[(i + ) % n] - rec[i]) * 500.0;
 //        cout << rec[(i + 1) % n].x << ' ' << rec[(i + 1) % n].y << " !!! " << rec[i].x << ' ' << rec[i].y << endl;
 //        cout << d.x << ' ' << d.y << endl;
         Point s = rec[(i + ) % n] + d, t = rec[i] - d;
         vector<Point> ip;
         ip.clear();
         for (int j = , sz = poly.size(); j < sz; j++) {
             if (segIntersect(s, t, poly[j], poly[(j + ) % sz])) {
                 ip.push_back(lineIntersect(s, t - s, poly[j], poly[(j + ) % sz] - poly[j]));
 //                cout << "has one" << endl;
             }
         }
         sort(ip.begin(), ip.end());
         int cnt = (int) (unique(ip.begin(), ip.end()) - ip.begin());
         if (cnt != ) {
             puts("shit!!");
             while () {}
         }
         Poly tmp = cutPoly(poly, s, t);
         dfs(p + , n, tmp, len + vecLen(ip[] - ip[]));
         vis[i] = false;
     }
 }

 int main() {
 //    freopen("in", "r", stdin);
     double x, y;
     int n;
     while (cin >> x >> y) {
         Poly tmp = Poly();
         for (int i = ; i < ; i++) tmp.pt.push_back(Point(dir[i][] * x, dir[i][] * y));
         memset(vis, false, sizeof(vis));
         cin >> n;
         for (int i = ; i < n; i++) cin >> rec[i].x >> rec[i].y;
         minLen = FINF;
         dfs(, n, tmp, 0.0);
         printf("Minimum total length = %.3f\n", minLen);
     }
     return ;
 }

——written by Lyon