UVALive 6859

题目大意是给定一些点，在坐标网格上，用另外的点围成一个多边形包围这些点，并使多边形的周长尽量小，同时注意边的斜率只能是90度或者45度

思路

      凸包

为了保证周长尽量小，我们选用的点就是和给定的点的相邻点，用Graham扫描法扫描这些点，可以求出一个大体的凸包

最后处理这些边的长度时，如果是90度或者45度直接加和，否则则对这个边进行分解，再加和。#include <cstdio>
#include <cstring>
#include <vector>
#include <algorithm>
#include <iostream>
#include <cmath>
using namespace std;

int N;
const int maxn = 100000 + 10;
const double eps = 1e-10;
double add(double a, double b){
if(abs(a + b) < eps * (abs(a) + abs(b))) return 0;
return a + b;
}

struct Point{
double x, y;
Point(){}
Point(double x, double y) : x(x), y(y) {}
Point operator + (const Point & p){
return Point(add(x, p.x), add(y, p.y));
}
Point operator - (const Point & p){
return Point(add(x,-p.x), add(y, -p.y));
}
Point operator * (double d){
return Point(x * d, y * d);
}
double dot(const Point & p){
return add(x * p.x, y * p.y);
}
double det(const Point & p){
return add(x * p.y, -y * p.x);
}
};

bool cmp_x(const Point & p, const Point &q){
if(p.x != q.x) return (p.x - q.x) < 0;
return p.y - q.y < 0;
}

vector<Point> convex_hull(Point * ps, int n){
sort(ps, ps + n, cmp_x);
int k = 0;
vector<Point> qs(n * 2);
for(int i = 0; i < n; ++i){
while(k > 1 && (qs[k - 1] - qs[k - 2]).det(ps[i] - qs[k - 1]) <= 0) k--;
qs[k++] = ps[i];
}
for(int i = n - 2, t = k; i >= 0; i--){
while(k > t && (qs[k - 1] - qs[k - 2]).det(ps[i] - qs[k - 1]) <= 0) k--;
qs[k++] = ps[i];
}
qs.resize(k - 1);
return qs;
}

#define sqr(x) ((x)*(x))
double dist(Point & p, Point & q){
return sqrt(sqr(p.x - q.x) + sqr(p.y - q.y));
}

Point ps[5 * maxn];

double solve(int cnt){
vector<Point> qs = convex_hull(ps, cnt);
double res = 0.000;
int len = qs.size();
for(int i = 0; i < len; ++i){
Point tp = qs[i], tq = qs[(i + 1) % len];
double ans = 0;
if(tp.x == tq.x || tp.y == tq.y){
ans = dist(tp, tq);
}else{
if(fabs(tp.y - tq.y) == fabs(tp.x - tq.x)) ans = dist(tp, tq);
else {
double dy = fabs(tp.y - tq.y), dx = fabs(tp.x - tq.x);
ans += min(dy, dx) * sqrt(2) + fabs(dy - dx);
}
}
res += ans;
}
return res;
}

int main()
{
while(scanf("%d", &N) == 1){
int cnt = 0;
for(int i = 0; i < N; ++i){
int x, y;
scanf("%d %d", &x, &y);
ps[cnt].x = x - 1;
ps[cnt++].y = y;
ps[cnt].x = x + 1;
ps[cnt++].y = y;
ps[cnt].x = x;
ps[cnt++].y = y + 1;
ps[cnt].x = x;
ps[cnt++].y = y - 1;
}
printf("%.4f\n", solve(cnt));
}
return 0;
}