Gym 101986D Making Perimeter of the Convex Hull Shortest（凸包+极角排序）

  1 //      ——By DD_BOND
  2 
  3 //#include<bits/stdc++.h>
  4 //#include<unordered_map>
  5 //#include<unordered_set>
  6 #include<functional>
  7 #include<algorithm>
  8 #include<iostream>
  9 //#include<ext/rope>
 10 #include<iomanip>
 11 #include<climits>
 12 #include<cstring>
 13 #include<cstdlib>
 14 #include<cstddef>
 15 #include<cstdio>
 16 #include<memory>
 17 #include<vector>
 18 #include<cctype>
 19 #include<string>
 20 #include<cmath>
 21 #include<queue>
 22 #include<deque>
 23 #include<ctime>
 24 #include<stack>
 25 #include<map>
 26 #include<set>
 27 
 28 #define fi first
 29 #define se second
 30 #define pb push_back
 31 #define MP make_pair
 32 
 33 #pragma GCC optimize(3)
 34 #pragma GCC target("sse,sse2,sse3,ssse3,sse4,popcnt,abm,mmx,avx,tune=native")
 35 
 36 using namespace std;
 37 
 38 typedef long double db;
 39 typedef long long ll;
 40 typedef pair<db,db> Pd;
 41 typedef pair<int,int> P;
 42 typedef pair<ll,ll> Pll;
 43 
 44 const db eps=1e-8;
 45 const int MAXN=1e6+10;
 46 const db pi=acos(-1.0);
 47 const ll INF=0x3f3f3f3f3f3f3f3f;
 48 
 49 inline int dcmp(db x){
 50     if(fabs(x)<eps) return 0;
 51     return (x>0? 1: -1);
 52 }
 53 
 54 inline db Sqrt(db x){
 55     return x>0? sqrt(x): 0;
 56 }
 57 
 58 inline db sqr(db x){ return x*x; }
 59 
 60 struct Point{
 61     db x,y,ang;
 62     Point(){ x=0,y=0; }
 63     Point(db _x,db _y):x(_x),y(_y){}
 64     void input(){
 65         double _x,_y;
 66         scanf("%lf%lf",&_x,&_y);
 67         x=_x,y=_y;
 68     }
 69     bool operator ==(const Point &b)const{
 70         return (dcmp(x-b.x)==0&&dcmp(y-b.y)==0);
 71     }
 72     bool operator !=(const Point &b)const{
 73         return !((dcmp(x-b.x)==0&&dcmp(y-b.y)==0));
 74     }
 75     bool operator <(const Point &b)const{
 76         return (dcmp(x-b.x)==0? dcmp(y-b.y)<0 : x<b.x);
 77     }
 78     Point operator +(const Point &b)const{
 79         return Point(x+b.x,y+b.y);
 80     }
 81     Point operator -(const Point &b)const{
 82         return Point(x-b.x,y-b.y);
 83     }
 84     Point operator *(db a){
 85         return Point(x*a,y*a);
 86     }
 87     Point operator /(db a){
 88         return Point(x/a,y/a);
 89     }
 90     db len2(){  //长度平方
 91         return sqr(x)+sqr(y);
 92     }
 93     db len(){   //长度
 94         return Sqrt(len2());
 95     }
 96     db polar(){ //向量的极角
 97         return atan2(y,x);   //返回与x轴正向夹角(-pi~pi]
 98     }
 99 };
100 
101 inline db cross(Point a,Point b){   //叉积
102     return a.x*b.y-a.y*b.x;
103 }
104 
105 inline db dot(Point a,Point b){ //点积
106     return a.x*b.x+a.y*b.y;
107 }
108 
109 inline db dis(Point a,Point b){ //两点的距离
110     Point p=b-a;    return p.len();
111 }
112 
113 Point centre_of_polygon(Point *p,int n){    //三角形重心加面积权值的平均求多边形的重心
114     db sum=0,sumx=0,sumy=0;
115     Point p1=p[0],p2=p[1],p3;
116     for(int i=2;i<n;i++){
117         p3=p[i];
118         db area=cross(p2-p1,p3-p2)/2;
119         sum+=area;
120         sumx+=(p1.x+p2.x+p3.x)*area;
121         sumy+=(p1.y+p2.y+p3.y)*area;
122         p2=p3;
123     }
124     return Point(sumx/(3*sum),sumy/(3*sum));
125 }
126 
127 Point tmp[MAXN],ins[MAXN];
128 
129 int convex_hull(Point *p,int n,Point *ch){  //求凸包
130     int m=0;
131     sort(p,p+n);
132     for(int i=0;i<n;i++){
133         while(m>1&&dcmp(cross(tmp[m-1]-tmp[m-2],p[i]-tmp[m-1]))<0) m--;
134         tmp[m++]=p[i];
135     }
136     int k=m;
137     for(int i=n-2;i>=0;i--){
138         while(m>k&&dcmp(cross(tmp[m-1]-tmp[m-2],p[i]-tmp[m-1]))<0) m--;
139         tmp[m++]=p[i];
140     }
141     if(n>1) m--;
142     for(int i=0;i<m;i++)    ch[i]=tmp[i];
143     return m;
144 }
145 
146 db ans;
147 pair<db,int>rec[MAXN];
148 Point point[MAXN],convex[MAXN],o;
149 vector<Point>side[MAXN],in[MAXN],st;
150 
151 bool cmp(Point a,Point b){
152     db dx=(a-o).polar(),dy=(b-o).polar();
153     if(dcmp(dx-dy)==0)  return dis(a,o)>dis(b,o);
154     return dx<dy;
155 }
156 
157 int main(void){
158     int n,m;  scanf("%d",&n);
159     for(int i=0;i<n;i++)    point[i].input();
160 
161     m=convex_hull(point,n,convex);
162     o=centre_of_polygon(convex,m);
163 
164     sort(point,point+n,cmp);
165     sort(convex,convex+m,cmp);
166 
167     for(int i=0;i<n;i++){
168         point[i].ang=(point[i]-o).polar();
169         point[i+n]=point[i];
170         point[i+n].ang+=2*pi;
171     }
172     for(int i=0;i<m;i++){
173         convex[i].ang=(convex[i]-o).polar();
174         convex[i+m]=convex[i];
175         convex[i+m].ang+=2*pi;
176     }
177 
178     for(int i=0,j=0;i<m;i++){
179         while(dcmp(convex[i].ang-point[j].ang)>0)   j++;
180         while(dcmp(point[j].ang-convex[i+1].ang)<0){
181             if(point[j]!=convex[i]) side[i].pb(point[j]);
182             j++;
183         }
184     }
185 
186     // a point on convex
187     for(int i=0;i<m;i++){
188         int l=(i==0? m-1: i-1),p=0;
189         tmp[p++]=convex[l];
190         for(int j=0;j<side[l].size();j++){
191             while(p>1&&dcmp(cross(tmp[p-1]-tmp[p-2],side[l][j]-tmp[p-1]))<0)    p--;
192             tmp[p++]=side[l][j];
193         }
194         for(int j=0;j<side[i].size();j++){
195             while(p>1&&dcmp(cross(tmp[p-1]-tmp[p-2],side[i][j]-tmp[p-1]))<0)    p--;
196             tmp[p++]=side[i][j];
197         }
198         while(p>1&&dcmp(cross(tmp[p-1]-tmp[p-2],convex[i+1]-tmp[p-1]))<0)    p--;
199         tmp[p++]=convex[i+1];
200 
201         db sum=dis(convex[l],convex[i])+dis(convex[i],convex[i+1]);
202         for(int j=0;j<p-1;j++)  sum-=dis(tmp[j],tmp[j+1]);
203         rec[i]=MP(sum,i);
204     }
205     sort(rec,rec+m,greater<pair<db,int> >());
206     if(abs(rec[0].se-rec[1].se)!=1&&abs(rec[0].se-rec[1].se)!=m-1)  ans=max(ans,rec[0].fi+rec[1].fi);
207     if(abs(rec[0].se-rec[2].se)!=1&&abs(rec[0].se-rec[2].se)!=m-1)  ans=max(ans,rec[0].fi+rec[2].fi);
208     if(abs(rec[1].se-rec[2].se)!=1&&abs(rec[1].se-rec[2].se)!=m-1)  ans=max(ans,rec[1].fi+rec[2].fi);
209 
210     // two consecutive point on convex
211     for(int i=0;i<m;i++){
212         int l=(i==0? m-1: i-1),p=0;
213         tmp[p++]=convex[l];
214         for(int j=0;j<side[l].size();j++){
215             while(p>1&&dcmp(cross(tmp[p-1]-tmp[p-2],side[l][j]-tmp[p-1]))<0)    p--;
216             tmp[p++]=side[l][j];
217         }
218         for(int j=0;j<side[i].size();j++){
219             while(p>1&&dcmp(cross(tmp[p-1]-tmp[p-2],side[i][j]-tmp[p-1]))<0)    p--;
220             tmp[p++]=side[i][j];
221         }
222         for(int j=0;j<side[(i+1)%m].size();j++){
223             while(p>1&&dcmp(cross(tmp[p-1]-tmp[p-2],side[(i+1)%m][j]-tmp[p-1]))<0)    p--;
224             tmp[p++]=side[(i+1)%m][j];
225         }
226         while(p>1&&dcmp(cross(tmp[p-1]-tmp[p-2],convex[i+2]-tmp[p-1]))<0)    p--;
227         tmp[p++]=convex[i+2];
228 
229         db sum=dis(convex[l],convex[i])+dis(convex[i],convex[i+1])+dis(convex[i+1],convex[i+2]);
230         for(int j=0;j<p-1;j++)  sum-=dis(tmp[j],tmp[j+1]);
231         ans=max(ans,sum);
232     }
233 
234     // a point on convex and a point inside convex
235     for(int i=0;i<m;i++){
236         int l=(i==0? m-1: i-1),p=0;
237         tmp[p++]=convex[l];
238         if(i==0)    tmp[p-1].ang-=2*pi;
239         for(int j=0;j<side[l].size();j++){
240             st.pb(side[l][j]);
241             if(i==0)    st.back().ang-=2*pi;
242             while(p>1&&dcmp(cross(tmp[p-1]-tmp[p-2],side[l][j]-tmp[p-1]))<0)    p--;
243             tmp[p++]=side[l][j];
244         }
245         for(int j=0;j<side[i].size();j++){
246             st.pb(side[i][j]);
247             while(p>1&&dcmp(cross(tmp[p-1]-tmp[p-2],side[i][j]-tmp[p-1]))<0)    p--;
248             tmp[p++]=side[i][j];
249         }
250         while(p>1&&dcmp(cross(tmp[p-1]-tmp[p-2],convex[i+1]-tmp[p-1]))<0)    p--;
251         tmp[p++]=convex[i+1];
252 
253         for(int j=0,k=0;j<p-1;j++){
254             while(k<st.size()&&dcmp(tmp[j].ang-st[k].ang)>0)   k++;
255             while(k<st.size()&&dcmp(st[k].ang-tmp[j+1].ang)<0){
256                 if(tmp[j]!=st[k])   in[j].pb(st[k]);
257                 k++;
258             }
259         }
260         db sum=dis(convex[l],convex[i])+dis(convex[i],convex[i+1]);
261         for(int j=0;j<p-1;j++)  sum-=dis(tmp[j],tmp[j+1]);
262         for(int j=1;j<p-1;j++){
263             int t=0;
264             ins[t++]=tmp[j-1];
265             for(int k=0;k<in[j-1].size();k++){
266                 while(t>1&&dcmp(cross(ins[t-1]-ins[t-2],in[j-1][k]-ins[t-1]))<0)    t--;
267                 ins[t++]=in[j-1][k];
268             }
269             for(int k=0;k<in[j].size();k++){
270                 while(t>1&&dcmp(cross(ins[t-1]-ins[t-2],in[j][k]-ins[t-1]))<0)  t--;
271                 ins[t++]=in[j][k];
272             }
273             while(t>1&&dcmp(cross(ins[t-1]-ins[t-2],tmp[j+1]-ins[t-1]))<0)  t--;
274             ins[t++]=tmp[j+1];
275 
276             db now=sum+dis(tmp[j-1],tmp[j])+dis(tmp[j],tmp[j+1]);
277             for(int k=0;k<t-1;k++)  now-=dis(ins[k],ins[k+1]);
278             ans=max(ans,now);
279         }
280         st.clear();
281         for(int j=0;j<p;j++)    in[j].clear();
282     }
283     printf("%.12f\n",(double)ans);
284     return 0;
285 }