AtCoder ABC 130F Minimum Bounding Box

题目链接：https://atcoder.jp/contests/abc130/tasks/abc130_f

题目大意

　　给定地图上 N 个点的坐标和移动方向，它们会以每秒 1 个单位的速度移动，设 Ans(t) 为在 t 时刻，$(x_{max} - x_{min}) * (y_{max} - y_{min})$的值，求 Ans(t) 的最小值。(最小值可能不是一个整数)

分析

　　稍加思考可以发现，不是所有点的所有坐标都对答案有影响，很多点完全可以忽略不计，下面以 Y 坐标为例，讨论影响$(y_{max} - y_{min})$的值。

　　首先我们把 N 个点分为 3 类，向下移动的，向上移动的和水平方向移动的，具体如下图所示：

　　其中，UU 代表往下移动的点中 Y 坐标最大值，UD 代表往下移动的点中 Y 坐标最小值，其余同理。

　　我们发现，影响$(y_{max} - y_{min})$就是这 6 个值，当它们某两个值重合时就有可能改变答案。

　　X 坐标方向上也是同理，只要旋转一下即可。

　　当我们把 X 和 Y 坐标上对应的 6 个值都算出来的时候，把它们两两组合，暴力枚举所有可能时刻，就能求出最终答案。

代码如下

#include <bits/stdc++.h>
using namespace std;
 
#define INIT() ios::sync_with_stdio(false);cin.tie(0);cout.tie(0);
#define Rep(i,n) for (int i = 0; i < (n); ++i)
#define For(i,s,t) for (int i = (s); i <= (t); ++i)
#define rFor(i,t,s) for (int i = (t); i >= (s); --i)
#define ForLL(i, s, t) for (LL i = LL(s); i <= LL(t); ++i)
#define rForLL(i, t, s) for (LL i = LL(t); i >= LL(s); --i)
#define foreach(i,c) for (__typeof(c.begin()) i = c.begin(); i != c.end(); ++i)
#define rforeach(i,c) for (__typeof(c.rbegin()) i = c.rbegin(); i != c.rend(); ++i)
 
#define pr(x) cout << #x << " = " << x << "  "
#define prln(x) cout << #x << " = " << x << endl
 
#define LOWBIT(x) ((x)&(-x))
 
#define ALL(x) x.begin(),x.end()
#define INS(x) inserter(x,x.begin())
#define UNIQUE(x) x.erase(unique(x.begin(), x.end()), x.end())
#define REMOVE(x, c) x.erase(remove(x.begin(), x.end(), c), x.end()); // 删去 x 中所有 c 
#define TOLOWER(x) transform(x.begin(), x.end(), x.begin(),::tolower);
#define TOUPPER(x) transform(x.begin(), x.end(), x.begin(),::toupper);
 
#define ms0(a) memset(a,0,sizeof(a))
#define msI(a) memset(a,inf,sizeof(a))
#define msM(a) memset(a,-1,sizeof(a))

#define MP make_pair
#define PB push_back
#define ft first
#define sd second
 
template<typename T1, typename T2>
istream &operator>>(istream &in, pair<T1, T2> &p) {
    in >> p.first >> p.second;
    return in;
}
 
template<typename T>
istream &operator>>(istream &in, vector<T> &v) {
    for (auto &x: v)
        in >> x;
    return in;
}
 
template<typename T1, typename T2>
ostream &operator<<(ostream &out, const std::pair<T1, T2> &p) {
    out << "[" << p.first << ", " << p.second << "]" << "
";
    return out;
}

inline int gc(){
    static const int BUF = 1e7;
    static char buf[BUF], *bg = buf + BUF, *ed = bg;
    
    if(bg == ed) fread(bg = buf, 1, BUF, stdin);
    return *bg++;
} 

inline int ri(){
    int x = 0, f = 1, c = gc();
    for(; c<48||c>57; f = c=='-'?-1:f, c=gc());
    for(; c>47&&c<58; x = x*10 + c - 48, c=gc());
    return x*f;
}

template<class T>
inline string toString(T x) {
    ostringstream sout;
    sout << x;
    return sout.str();
}

inline int toInt(string s) {
    int v;
    istringstream sin(s);
    sin >> v;
    return v;
}

//min <= aim <= max
template<typename T>
inline bool BETWEEN(const T aim, const T min, const T max) {
    return min <= aim && aim <= max;
}
 
typedef long long LL;
typedef unsigned long long uLL;
typedef pair< double, double > PDD;
typedef pair< int, int > PII;
typedef pair< int, PII > PIPII;
typedef pair< string, int > PSI;
typedef pair< int, PSI > PIPSI;
typedef set< int > SI;
typedef set< PII > SPII;
typedef vector< int > VI;
typedef vector< double > VD;
typedef vector< VI > VVI;
typedef vector< SI > VSI;
typedef vector< PII > VPII;
typedef map< int, int > MII;
typedef map< int, string > MIS;
typedef map< int, PII > MIPII;
typedef map< PII, int > MPIII;
typedef map< string, int > MSI;
typedef map< string, string > MSS;
typedef map< PII, string > MPIIS;
typedef map< PII, PII > MPIIPII;
typedef multimap< int, int > MMII;
typedef multimap< string, int > MMSI;
//typedef unordered_map< int, int > uMII;
typedef pair< LL, LL > PLL;
typedef vector< LL > VL;
typedef vector< VL > VVL;
typedef priority_queue< int > PQIMax;
typedef priority_queue< int, VI, greater< int > > PQIMin;
const double EPS = 1e-8;
const LL inf = 0x7fffffff;
const LL infLL = 0x7fffffffffffffffLL;
const LL mod = 1e9 + 7;
const int maxN = 1e5 + 7;
const LL ONE = 1;
const LL evenBits = 0xaaaaaaaaaaaaaaaa;
const LL oddBits = 0x5555555555555555;

#define UU A[0]
#define UD A[1]
#define MU A[2]
#define MD A[3]
#define DU A[4]
#define DD A[5]

struct State{
    double A[6] = {-inf, inf, -inf, inf, -inf, inf};
    double _max = -inf, _min = inf;
    
    State operator+ (const double &x) const {
        State ret = *this;
        if(fabs(ret.UU) < 1e9) ret.UU -= x;
        if(fabs(ret.UD) < 1e9) ret.UD -= x;
        if(fabs(ret.DU) < 1e9) ret.DU += x;
        if(fabs(ret.DD) < 1e9) ret.DD += x;
        
        Rep(i, 6) {
            if(fabs(ret.A[i]) < 1e9) {
                ret._max = max(ret._max, ret.A[i]);
                ret._min = min(ret._min, ret.A[i]);
            }
        }
        return ret;
    }
};

int N;
State X, Y;
double ans = infLL; 
set< double > T; // 记录关键时间节点 

int main(){
    //freopen("MyOutput.txt","w",stdout);
    //freopen("input.txt","r",stdin);
    //INIT();
    cin >> N;
    Rep(i, N) {
        double x, y;
        string d;
        cin >> x >> y >> d;
        
        if(d[0] == 'U') {
            Y.DU = max(Y.DU, y);
            Y.DD = min(Y.DD, y);
            
            X.MU = max(X.MU, x);
            X.MD = min(X.MD, x);
        }
        if(d[0] == 'D') {
            Y.UU = max(Y.UU, y);
            Y.UD = min(Y.UD, y);
            
            X.MU = max(X.MU, x);
            X.MD = min(X.MD, x);
        }
        if(d[0] == 'R') {
            X.DU = max(X.DU, x);
            X.DD = min(X.DD, x);
            
            Y.MU = max(Y.MU, y);
            Y.MD = min(Y.MD, y);
        }
        if(d[0] == 'L') {
            X.UU = max(X.UU, x);
            X.UD = min(X.UD, x);
            
            Y.MU = max(Y.MU, y);
            Y.MD = min(Y.MD, y);
        }
    }
    
    Rep(i, 6) {
        Rep(j, 6) {
            T.insert(fabs(X.A[i] - X.A[j]));
            T.insert(fabs(X.A[i] - X.A[j]) / 2);
            T.insert(fabs(Y.A[i] - Y.A[j]));
            T.insert(fabs(Y.A[i] - Y.A[j]) / 2);
            T.insert(fabs(X.A[i] - Y.A[j]));
            T.insert(fabs(X.A[i] - Y.A[j]) / 2);
        }
    }
    
    foreach(i, T) {
        if(*i > 1e9) break;
        State tmpX = X + *i; 
        State tmpY = Y + *i; 
        
        ans = min(ans, (tmpX._max - tmpX._min) * (tmpY._max - tmpY._min));
    }
        
    printf("%.10f
", ans);
    return 0;
}