题解 UVA1077 【The Sky is the Limit】

题目链接：Link

Solution

这题坑了我很长时间。。。血篇题解纪念一下。。。

---

如果枚举每一条线段并计算它露出的部分会非常麻烦，可以考虑把这个地图分割成一些薯条竖条，使得每个区域内露出的部分无交点，显然按照线段交点的x坐标分割比较好。

于是，我们得出了如下代码：

#include<cstdio>
#include<cmath>
#include<algorithm>
using namespace std;
struct Point
{
    double x,y;
    Point(double _x=0,double _y=0):x(_x),y(_y) { }
    inline bool operator<(const Point &b) { return x<b.x||(x==b.x&&y<b.y); }
};
typedef Point Vector;
inline Vector operator+(const Vector &a,const Vector &b) { return Vector(a.x+b.x,a.y+b.y); }
inline Vector operator-(const Vector &a,const Vector &b) { return Vector(a.x-b.x,a.y-b.y); }
inline Vector operator*(const Vector &a,double b) { return Vector(a.x*b,a.y*b); }
inline Vector operator/(const Vector &a,double b) { return Vector(a.x/b,a.y/b); }
inline double Dot(const Vector &a,const Vector &b) { return a.x*b.x+a.y*b.y; }
inline double Cross(const Vector &a,const Vector &b) { return a.x*b.y-a.y*b.x; }
inline double Length(const Vector &a) { return sqrt(Dot(a,a)); }
const double eps=1e-6;
inline int dcmp(double x) { return fabs(x)<eps?0:(x>0?1:-1); }
inline bool SIT(const Point &a1,const Point &a2,const Point &b1,const Point &b2)
{
    double c1=Cross(a2-a1,b1-a1),c2=Cross(a2-a1,b2-a1),
           c3=Cross(b2-b1,a1-b1),c4=Cross(b2-b1,a2-b1);
    return dcmp(c1)*dcmp(c2)<0&&dcmp(c3)*dcmp(c4)<0;
}
inline Point LIT(const Point &P,const Vector &v,const Point &Q,const Vector &w)
{
    Vector u=P-Q;
    double t=Cross(w,u)/Cross(v,w);
    return P+v*t;
}

const int maxn=105;
int n,kase;
double x[maxn*3+maxn*maxn];
Point L[maxn][2][2];
int main()
{
#ifdef local
    freopen("pro.in","r",stdin);
#endif
    while(scanf("%d",&n)==1&&n)
    {
        int c=0; double res=0;
        for(int i=0;i<n;i++)
        {
            double X,H,B;
            scanf("%lf%lf%lf",&X,&H,&B);
            L[i][0][0]=Point(X-B*0.5,0);
            L[i][0][1]=L[i][1][0]=Point(X,H);
            L[i][1][1]=Point(X+B*0.5,0);
            x[c++]=X-B*0.5; x[c++]=X; x[c++]=X+B*0.5;
        }
        for(int i=0;i<n;i++) for(int a=0;a<2;a++)
            for(int j=i+1;j<n;j++) for(int b=0;b<2;b++)
            {
                Point &P1=L[i][a][0],&P2=L[i][a][1],&P3=L[j][b][0],&P4=L[j][b][1];
                if(SIT(P1,P2,P3,P4)) x[c++]=LIT(P1,P2-P1,P3,P4-P3).x;
            }
        sort(x,x+c);
        c=unique(x,x+c)-x;
        Point l=Point(x[0],0);
        for(int i=1;i<c;i++)
        {
            Point r=Point(x[i],0);
            double lx=x[i-1],rx=x[i];
            for(int j=0;j<n;j++) for(int a=0;a<2;a++)
            {
                Point &P1=L[j][a][0],&P2=L[j][a][1];
                if(P1.x<=lx&&lx<=P2.x&&P1.x<=rx&&rx<=P2.x)
                {
                    Vector v=P2-P1;
                    Point tr=P1+v*(rx-P1.x)/(P2.x-P1.x);
                    if(tr.y>=r.y) r=tr;
                }
            }
            l.y=max(l.y,0.0);
            res+=Length(r-l);
            l=r;
        }
        printf("Case %d: %.0lf

",++kase,res);
    }
    return 0;
}

发现，咦，好像挂掉了。。。然后发现当山之间有间隔时地平线会被算上。。。

赶紧改一下

#include<cstdio>
#include<cmath>
#include<algorithm>
using namespace std;
struct Point
{
    double x,y;
    Point(double _x=0,double _y=0):x(_x),y(_y) { }
    inline bool operator<(const Point &b) { return x<b.x||(x==b.x&&y<b.y); }
};
typedef Point Vector;
inline Vector operator+(const Vector &a,const Vector &b) { return Vector(a.x+b.x,a.y+b.y); }
inline Vector operator-(const Vector &a,const Vector &b) { return Vector(a.x-b.x,a.y-b.y); }
inline Vector operator*(const Vector &a,double b) { return Vector(a.x*b,a.y*b); }
inline Vector operator/(const Vector &a,double b) { return Vector(a.x/b,a.y/b); }
inline double Dot(const Vector &a,const Vector &b) { return a.x*b.x+a.y*b.y; }
inline double Cross(const Vector &a,const Vector &b) { return a.x*b.y-a.y*b.x; }
inline double Length(const Vector &a) { return sqrt(Dot(a,a)); }
const double eps=1e-6;
inline int dcmp(double x) { return fabs(x)<eps?0:(x>0?1:-1); }
inline bool SIT(const Point &a1,const Point &a2,const Point &b1,const Point &b2)
{
    double c1=Cross(a2-a1,b1-a1),c2=Cross(a2-a1,b2-a1),
           c3=Cross(b2-b1,a1-b1),c4=Cross(b2-b1,a2-b1);
    return dcmp(c1)*dcmp(c2)<0&&dcmp(c3)*dcmp(c4)<0;
}
inline Point LIT(const Point &P,const Vector &v,const Point &Q,const Vector &w)
{
    Vector u=P-Q;
    double t=Cross(w,u)/Cross(v,w);
    return P+v*t;
}

const int maxn=105;
int n,kase;
double x[maxn*3+maxn*maxn];
Point L[maxn][2][2];
int main()
{
#ifdef local
    freopen("pro.in","r",stdin);
#endif
    while(scanf("%d",&n)==1&&n)
    {
        int c=0; double res=0;
        for(int i=0;i<n;i++)
        {
            double X,H,B;
            scanf("%lf%lf%lf",&X,&H,&B);
            L[i][0][0]=Point(X-B*0.5,0);
            L[i][0][1]=L[i][1][0]=Point(X,H);
            L[i][1][1]=Point(X+B*0.5,0);
            x[c++]=X-B*0.5; x[c++]=X; x[c++]=X+B*0.5;
        }
        for(int i=0;i<n;i++) for(int a=0;a<2;a++)
            for(int j=i+1;j<n;j++) for(int b=0;b<2;b++)
            {
                Point &P1=L[i][a][0],&P2=L[i][a][1],&P3=L[j][b][0],&P4=L[j][b][1];
                if(SIT(P1,P2,P3,P4)) x[c++]=LIT(P1,P2-P1,P3,P4-P3).x;
            }
        sort(x,x+c);
        c=unique(x,x+c)-x;
        Point l=Point(x[0],0);
        for(int i=1;i<c;i++)
        {
            Point r=Point(x[i],0);
            double lx=x[i-1],rx=x[i];
            for(int j=0;j<n;j++) for(int a=0;a<2;a++)
            {
                Point &P1=L[j][a][0],&P2=L[j][a][1];
                if(P1.x<=lx&&lx<=P2.x&&P1.x<=rx&&rx<=P2.x)
                {
                    Vector v=P2-P1;
                    Point tr=P1+v*(rx-P1.x)/(P2.x-P1.x);
                    if(tr.y>=r.y) r=tr;
                }
            }
            l.y=max(l.y,0.0);
            if(dcmp(r.y)) res+=Length(r-l);
            l=r;
        }
        printf("Case %d: %.0lf

",++kase,res);
    }
    return 0;
}

咦，怎么第一组数据又过不去了？？？好像右山坡没了。。。

正解：

#include<cstdio>
#include<cmath>
#include<algorithm>
using namespace std;
struct Point
{
    double x,y;
    Point(double _x=0,double _y=0):x(_x),y(_y) { }
    inline bool operator<(const Point &b) { return x<b.x||(x==b.x&&y<b.y); }
};
typedef Point Vector;
inline Vector operator+(const Vector &a,const Vector &b) { return Vector(a.x+b.x,a.y+b.y); }
inline Vector operator-(const Vector &a,const Vector &b) { return Vector(a.x-b.x,a.y-b.y); }
inline Vector operator*(const Vector &a,double b) { return Vector(a.x*b,a.y*b); }
inline Vector operator/(const Vector &a,double b) { return Vector(a.x/b,a.y/b); }
inline double Dot(const Vector &a,const Vector &b) { return a.x*b.x+a.y*b.y; }
inline double Cross(const Vector &a,const Vector &b) { return a.x*b.y-a.y*b.x; }
inline double Length(const Vector &a) { return sqrt(Dot(a,a)); }
const double eps=1e-6;
inline int dcmp(double x) { return fabs(x)<eps?0:(x>0?1:-1); }
inline bool SIT(const Point &a1,const Point &a2,const Point &b1,const Point &b2)
{
    double c1=Cross(a2-a1,b1-a1),c2=Cross(a2-a1,b2-a1),
           c3=Cross(b2-b1,a1-b1),c4=Cross(b2-b1,a2-b1);
    return dcmp(c1)*dcmp(c2)<0&&dcmp(c3)*dcmp(c4)<0;
}
inline Point LIT(const Point &P,const Vector &v,const Point &Q,const Vector &w)
{
    Vector u=P-Q;
    double t=Cross(w,u)/Cross(v,w);
    return P+v*t;
}

const int maxn=105;
int n,kase;
double x[maxn*3+maxn*maxn];
Point L[maxn][2][2];
int main()
{
#ifdef local
    freopen("pro.in","r",stdin);
#endif
    while(scanf("%d",&n)==1&&n)
    {
        int c=0; double res=0;
        for(int i=0;i<n;i++)
        {
            double X,H,B;
            scanf("%lf%lf%lf",&X,&H,&B);
            L[i][0][0]=Point(X-B*0.5,0);
            L[i][0][1]=L[i][1][0]=Point(X,H);
            L[i][1][1]=Point(X+B*0.5,0);
            x[c++]=X-B*0.5; x[c++]=X; x[c++]=X+B*0.5;
        }
        for(int i=0;i<n;i++) for(int a=0;a<2;a++)
            for(int j=i+1;j<n;j++) for(int b=0;b<2;b++)
            {
                Point &P1=L[i][a][0],&P2=L[i][a][1],&P3=L[j][b][0],&P4=L[j][b][1];
                if(SIT(P1,P2,P3,P4)) x[c++]=LIT(P1,P2-P1,P3,P4-P3).x;
            }
        sort(x,x+c);
        c=unique(x,x+c)-x;
        Point l=Point(x[0],0);
        for(int i=1;i<c;i++)
        {
            Point r=Point(x[i],0);
            double lx=x[i-1],rx=x[i];
            for(int j=0;j<n;j++) for(int a=0;a<2;a++)
            {
                Point &P1=L[j][a][0],&P2=L[j][a][1];
                if(P1.x<=lx&&lx<=P2.x&&P1.x<=rx&&rx<=P2.x)
                {
                    Vector v=P2-P1;
                    Point tr=P1+v*(rx-P1.x)/(P2.x-P1.x);
                    if(tr.y>=r.y) r=tr;
                }
            }
            l.y=max(l.y,0.0);
            if(dcmp(l.y)||dcmp(r.y)) res+=Length(r-l);
            l=r;
        }
        printf("Case %d: %.0lf

",++kase,res);
    }
    return 0;
}

各种各样的y打成x、j打成i之类的错误就不想提了。。。一把辛酸泪啊。。。

#include<cstdio>
#include<cmath>
#include<algorithm>
#include<cassert>
using namespace std;
struct Point
{
    double x,y;
    Point(double _x=0,double _y=0):x(_x),y(_y) { }
    inline bool operator<(const Point &b) { return x<b.x||(x==b.x&&y<b.y); }
};
typedef Point Vector;
inline Vector operator+(const Vector &a,const Vector &b) { return Vector(a.x+b.x,a.y+b.y); }
inline Vector operator-(const Vector &a,const Vector &b) { return Vector(a.x-b.x,a.y-b.y); }
inline Vector operator*(const Vector &a,double b) { return Vector(a.x*b,a.y*b); }
inline Vector operator/(const Vector &a,double b) { return Vector(a.x/b,a.y/b); }
inline double Dot(const Vector &a,const Vector &b) { return a.x*b.x+a.y*b.y; }
inline double Cross(const Vector &a,const Vector &b) { return a.x*b.y-a.y*b.x; }
inline double Length(const Vector &a) { return sqrt(Dot(a,a)); }
const double eps=1e-6;
inline int dcmp(double x) { return fabs(x)<eps?0:(x>0?1:-1); }
inline bool SIT(const Point &a1,const Point &a2,const Point &b1,const Point &b2)
{
    double c1=Cross(a2-a1,b1-a1),c2=Cross(a2-a1,b2-a1),
           c3=Cross(b2-b1,a1-b1),c4=Cross(b2-b1,a2-b1);
    return dcmp(c1)*dcmp(c2)<0&&dcmp(c3)*dcmp(c4)<0;
}
inline Point LIT(const Point &P,const Vector &v,const Point &Q,const Vector &w)
{
    Vector u=P-Q;
    double t=Cross(w,u)/Cross(v,w);
    return P+v*t;
}

const int maxn=105;
int n,kase;
double x[maxn*3+maxn*maxn];
Point L[maxn][2][2];
int main()
{
#ifdef local
    freopen("pro.in","r",stdin);
#endif
    while(scanf("%d",&n)==1&&n)
    {
        int c=0; double res=0;
        for(int i=0;i<n;i++)
        {
            double X,H,B;
            scanf("%lf%lf%lf",&X,&H,&B);
            L[i][0][0]=Point(X-B*0.5,0);
            L[i][0][1]=L[i][1][0]=Point(X,H);
            L[i][1][1]=Point(X+B*0.5,0);
            x[c++]=X-B*0.5; x[c++]=X; x[c++]=X+B*0.5;
        }
#ifdef debug
        printf("Points:
");
        for(int i=0;i<n;i++)
            printf("(%g,%g)
(%g,%g)
(%g,%g)
"
                ,L[i][0][0].x,L[i][0][0].y
                ,L[i][0][1].x,L[i][0][1].y
                ,L[i][1][1].x,L[i][1][1].y);
        printf("
");
#endif
        for(int i=0;i<n;i++) for(int a=0;a<2;a++)
            for(int j=i+1;j<n;j++) for(int b=0;b<2;b++)
            {
                Point &P1=L[i][a][0],&P2=L[i][a][1],&P3=L[j][b][0],&P4=L[j][b][1];
                if(SIT(P1,P2,P3,P4))
                {
                    x[c++]=LIT(P1,P2-P1,P3,P4-P3).x;
                    double nowx=x[c-1];
                    assert(P1.x<=P2.x);
                    assert(P3.x<=P4.x);
                    assert(P1.x<=nowx&&nowx<=P2.x);
                    assert(P3.x<=nowx&&nowx<=P4.x);
                }
            }
        sort(x,x+c);
        c=unique(x,x+c)-x;
        Point l=Point(x[0],0);
#ifdef debug
        printf("x:
");
        for(int i=0;i<c;i++) printf("%.5lf
",x[i]);
        printf("
");
#endif
        for(int i=1;i<c;i++)
        {
            Point r=Point(x[i],0);
            double lx=x[i-1],rx=x[i];
            for(int j=0;j<n;j++) for(int a=0;a<2;a++)
            {
                Point &P1=L[j][a][0],&P2=L[j][a][1];
                assert(P1.x<=P2.x);
                if(P1.x<=lx&&lx<=P2.x&&P1.x<=rx&&rx<=P2.x)
                {
                    Vector v=P2-P1;
                    Point tr=P1+v*(rx-P1.x)/(P2.x-P1.x);
                    if(tr.y>=r.y) r=tr;
                }
            }
            l.y=max(l.y,0.0);
            if(dcmp(l.y)||dcmp(r.y))
            {
                res+=Length(r-l);
#ifdef debug
                printf("i=%d res+=%.5lf l=(%.5lf,%.5lf) r=(%.5lf,%.5lf)
"
                    ,i,Length(r-l)
                    ,l.x,l.y
                    ,r.x,r.y);
                assert(dcmp(r.x-x[i])==0);
                assert(dcmp(r.y+1));
#endif
            }
#ifdef debug
            printf("after i=%d res is %.5lf
",i,res);
            printf("
");
#endif
            l=r;
        }
        printf("Case %d: %.0lf

",++kase,res);
    }
    return 0;
}
/*
1
100 50 100
3
20  30  35
37 24   29
60 20 13
0
*/