Most Distant Point from the Sea, Japan 2007（二分，半平面交）

Description

给定一个多边形，要求在其内部找到一个点使得该点到边界的最短距离最长，求出该距离。

Solution

考虑二分距离d" role="presentation">d$d$，剩下的问题只有怎么判断d" role="presentation">d$d$可行：
将所有的边向内推进d" role="presentation">d$d$，求半平面交即可。

Source

/****************************
 * Au: Hany01
 * Prob: [LA3890] [POJ3525] Most Distant Point from the Sea
 * Date: Feb 13th, 2018
 * Email: hany01@foxmail.com
****************************/

#include
#include
#include
#include
#include
#include
#include
#include
#include
#include

using namespace std;

typedef long long LL;
typedef pair PII;
typedef vector VI;
#define rep(i , j) for (int i = 0 , i##_end_ = j; i < i##_end_ ; ++ i)
#define For(i , j , k) for (int i = (j) , i##_end_ = (k) ; i <= i##_end_ ; ++ i)
#define Fordown(i , j , k) for (int i = (j) , i##_end_ = (k) ; i >= i##_end_ ; -- i)
#define Set(a , b) memset(a , b , sizeof(a))
#define SZ(a) ((int)(a.size()))
#define ALL(a) a.begin(), a.end()
#define pb(a) push_back(a)
#define mp(a, b) make_pair(a, b)
#define INF (0x3f3f3f3f)
#define INF1 (2139062143)
#define Mod (1000000007)
#define y1 wozenmezhemecaia 
#ifdef hany01
#define debug(...) fprintf(stderr , __VA_ARGS__)
#else
#define debug(...)
#endif

inline void File() {
#ifdef hany01 
    freopen("la3890.in" , "r" , stdin);
    freopen("la3890.out" , "w" , stdout);
#endif
}

template inline bool chkmax(T &a, T b) { return a < b ? a = b, 1 : 0; }
template inline bool chkmin(T &a, T b) { return b < a ? a = b, 1 : 0; }

inline int read() {
    register char c_; register int _ , __;
    for (_ = 0 , __ = 1 , c_ = getchar() ; !isdigit(c_) ; c_ = getchar()) if (c_ == '-')  __ = -1;
    for ( ; isdigit(c_) ; c_ = getchar()) _ = (_ << 1) + (_ << 3) + (c_ ^ 48);
    return _ * __;
}

const double eps = 1e-11;

inline int dcmp(double x) { if (fabs(x) < eps) return 0; return x < 0 ? -1 : 1; }

struct Point
{
    double x, y;
    Point(double x = 0, double y = 0): x(x), y(y) {}
};
typedef Point Vector;

Vector operator + (Vector A, Vector B) { return Vector(A.x + B.x, A.y + B.y); }
Vector operator - (Vector A, Vector B) { return Vector(A.x - B.x, A.y - B.y); }
Vector operator * (Vector A, double p) { return Vector(A.x * p, A.y * p); }
Vector operator / (Vector A, double p) { return Vector(A.x / p, A.y / p); }
double Dot(Vector A, Vector B) { return A.x * B.x + A.y * B.y; }
double Cross(Vector A, Vector B) { return A.x * B.y - A.y * B.x; }
double Length(Vector A) { return sqrt(Dot(A, A)); }
double Angle(Vector v) { return atan2(v.y, v.x); }
Vector Normal(Vector v) { return Vector(-v.y, v.x) / Length(v); }

struct Line
{
    Point P; Vector v; double ang;
    Line(Point P = Point(0, 0), Vector v = Vector(0, 0)): P(P), v(v) { ang = Angle(v); }
};
bool operator < (const Line& A, const Line& B) { return A.ang < B.ang; }

inline bool OnLeft(Line L, Point p) { return dcmp(Cross(L.v, p - L.P)) > 0; }

inline Point LineIntersection(Line A, Line B) {
    Vector u = A.P - B.P;
    double t = Cross(B.v, u) / Cross(A.v, B.v);
    return A.P + A.v * t;
}

const int maxn = 1000;

inline int HalfplaneIntersection(Line* L, int n)
{
    int head, tail;
    Point p[maxn];
    Line q[maxn];
    q[head = tail = 0] = L[1];
    For(i, 2, n) {
        while (head < tail && !OnLeft(L[i], p[tail - 1])) -- tail;
        while (head < tail && !OnLeft(L[i], p[head])) ++ head;
        q[++ tail] = L[i];
        if (!dcmp(Cross(q[tail].v, q[tail - 1].v))) {
            -- tail;
            if (OnLeft(q[tail], L[i].P)) q[tail] = L[i];
        }
        if (head < tail) p[tail - 1] = LineIntersection(q[tail], q[tail - 1]);
    }
    while (head < tail && !OnLeft(q[head], p[tail - 1])) -- tail;
    if (head + 1 >= tail) return 0;
    return 1;
}

int n;
Point p[maxn];
Line L[maxn], L_[maxn];
Vector dt[maxn];

int main()
{
    File();
    while (scanf("%d", &n) != EOF && n) {
        For(i, 1, n) scanf("%lf%lf", &p[i].x, &p[i].y);
        p[n + 1] = p[1];
        For(i, 1, n) L[i] = Line(p[i], p[i + 1] - p[i]), dt[i] = Normal(p[i + 1] - p[i]);
        double l = 0, r = 20000.0, mid;
        while (r - l > eps) {
            mid = (l + r) * 0.5;
            For(i, 1, n) L_[i] = L[i], L_[i].P = L_[i].P + dt[i] * mid;
            sort(L_ + 1, L_ + 1 + n);
            if (HalfplaneIntersection(L_, n)) l = mid; else r = mid;
        }
        printf("%.6lf\n", l);
    }
    return 0;
}