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

  1 #include <bits/stdc++.h>
  2 using namespace std;
  3  
  4 #define INIT() ios::sync_with_stdio(false);cin.tie(0);cout.tie(0);
  5 #define Rep(i,n) for (int i = 0; i < (n); ++i)
  6 #define For(i,s,t) for (int i = (s); i <= (t); ++i)
  7 #define rFor(i,t,s) for (int i = (t); i >= (s); --i)
  8 #define ForLL(i, s, t) for (LL i = LL(s); i <= LL(t); ++i)
  9 #define rForLL(i, t, s) for (LL i = LL(t); i >= LL(s); --i)
 10 #define foreach(i,c) for (__typeof(c.begin()) i = c.begin(); i != c.end(); ++i)
 11 #define rforeach(i,c) for (__typeof(c.rbegin()) i = c.rbegin(); i != c.rend(); ++i)
 12  
 13 #define pr(x) cout << #x << " = " << x << "  "
 14 #define prln(x) cout << #x << " = " << x << endl
 15  
 16 #define LOWBIT(x) ((x)&(-x))
 17  
 18 #define ALL(x) x.begin(),x.end()
 19 #define INS(x) inserter(x,x.begin())
 20 #define UNIQUE(x) x.erase(unique(x.begin(), x.end()), x.end())
 21 #define REMOVE(x, c) x.erase(remove(x.begin(), x.end(), c), x.end()); // 删去 x 中所有 c 
 22 #define TOLOWER(x) transform(x.begin(), x.end(), x.begin(),::tolower);
 23 #define TOUPPER(x) transform(x.begin(), x.end(), x.begin(),::toupper);
 24  
 25 #define ms0(a) memset(a,0,sizeof(a))
 26 #define msI(a) memset(a,inf,sizeof(a))
 27 #define msM(a) memset(a,-1,sizeof(a))
 28 
 29 #define MP make_pair
 30 #define PB push_back
 31 #define ft first
 32 #define sd second
 33  
 34 template<typename T1, typename T2>
 35 istream &operator>>(istream &in, pair<T1, T2> &p) {
 36     in >> p.first >> p.second;
 37     return in;
 38 }
 39  
 40 template<typename T>
 41 istream &operator>>(istream &in, vector<T> &v) {
 42     for (auto &x: v)
 43         in >> x;
 44     return in;
 45 }
 46  
 47 template<typename T1, typename T2>
 48 ostream &operator<<(ostream &out, const std::pair<T1, T2> &p) {
 49     out << "[" << p.first << ", " << p.second << "]" << "
";
 50     return out;
 51 }
 52 
 53 inline int gc(){
 54     static const int BUF = 1e7;
 55     static char buf[BUF], *bg = buf + BUF, *ed = bg;
 56     
 57     if(bg == ed) fread(bg = buf, 1, BUF, stdin);
 58     return *bg++;
 59 } 
 60 
 61 inline int ri(){
 62     int x = 0, f = 1, c = gc();
 63     for(; c<48||c>57; f = c=='-'?-1:f, c=gc());
 64     for(; c>47&&c<58; x = x*10 + c - 48, c=gc());
 65     return x*f;
 66 }
 67 
 68 template<class T>
 69 inline string toString(T x) {
 70     ostringstream sout;
 71     sout << x;
 72     return sout.str();
 73 }
 74 
 75 inline int toInt(string s) {
 76     int v;
 77     istringstream sin(s);
 78     sin >> v;
 79     return v;
 80 }
 81 
 82 //min <= aim <= max
 83 template<typename T>
 84 inline bool BETWEEN(const T aim, const T min, const T max) {
 85     return min <= aim && aim <= max;
 86 }
 87  
 88 typedef long long LL;
 89 typedef unsigned long long uLL;
 90 typedef pair< double, double > PDD;
 91 typedef pair< int, int > PII;
 92 typedef pair< int, PII > PIPII;
 93 typedef pair< string, int > PSI;
 94 typedef pair< int, PSI > PIPSI;
 95 typedef set< int > SI;
 96 typedef set< PII > SPII;
 97 typedef vector< int > VI;
 98 typedef vector< double > VD;
 99 typedef vector< VI > VVI;
100 typedef vector< SI > VSI;
101 typedef vector< PII > VPII;
102 typedef map< int, int > MII;
103 typedef map< int, string > MIS;
104 typedef map< int, PII > MIPII;
105 typedef map< PII, int > MPIII;
106 typedef map< string, int > MSI;
107 typedef map< string, string > MSS;
108 typedef map< PII, string > MPIIS;
109 typedef map< PII, PII > MPIIPII;
110 typedef multimap< int, int > MMII;
111 typedef multimap< string, int > MMSI;
112 //typedef unordered_map< int, int > uMII;
113 typedef pair< LL, LL > PLL;
114 typedef vector< LL > VL;
115 typedef vector< VL > VVL;
116 typedef priority_queue< int > PQIMax;
117 typedef priority_queue< int, VI, greater< int > > PQIMin;
118 const double EPS = 1e-8;
119 const LL inf = 0x7fffffff;
120 const LL infLL = 0x7fffffffffffffffLL;
121 const LL mod = 1e9 + 7;
122 const int maxN = 1e5 + 7;
123 const LL ONE = 1;
124 const LL evenBits = 0xaaaaaaaaaaaaaaaa;
125 const LL oddBits = 0x5555555555555555;
126 
127 #define UU A[0]
128 #define UD A[1]
129 #define MU A[2]
130 #define MD A[3]
131 #define DU A[4]
132 #define DD A[5]
133 
134 struct State{
135     double A[6] = {-inf, inf, -inf, inf, -inf, inf};
136     double _max = -inf, _min = inf;
137     
138     State operator+ (const double &x) const {
139         State ret = *this;
140         if(fabs(ret.UU) < 1e9) ret.UU -= x;
141         if(fabs(ret.UD) < 1e9) ret.UD -= x;
142         if(fabs(ret.DU) < 1e9) ret.DU += x;
143         if(fabs(ret.DD) < 1e9) ret.DD += x;
144         
145         Rep(i, 6) {
146             if(fabs(ret.A[i]) < 1e9) {
147                 ret._max = max(ret._max, ret.A[i]);
148                 ret._min = min(ret._min, ret.A[i]);
149             }
150         }
151         return ret;
152     }
153 };
154 
155 int N;
156 State X, Y;
157 double ans = infLL; 
158 set< double > T; // 记录关键时间节点 
159 
160 int main(){
161     //freopen("MyOutput.txt","w",stdout);
162     //freopen("input.txt","r",stdin);
163     //INIT();
164     cin >> N;
165     Rep(i, N) {
166         double x, y;
167         string d;
168         cin >> x >> y >> d;
169         
170         if(d[0] == 'U') {
171             Y.DU = max(Y.DU, y);
172             Y.DD = min(Y.DD, y);
173             
174             X.MU = max(X.MU, x);
175             X.MD = min(X.MD, x);
176         }
177         if(d[0] == 'D') {
178             Y.UU = max(Y.UU, y);
179             Y.UD = min(Y.UD, y);
180             
181             X.MU = max(X.MU, x);
182             X.MD = min(X.MD, x);
183         }
184         if(d[0] == 'R') {
185             X.DU = max(X.DU, x);
186             X.DD = min(X.DD, x);
187             
188             Y.MU = max(Y.MU, y);
189             Y.MD = min(Y.MD, y);
190         }
191         if(d[0] == 'L') {
192             X.UU = max(X.UU, x);
193             X.UD = min(X.UD, x);
194             
195             Y.MU = max(Y.MU, y);
196             Y.MD = min(Y.MD, y);
197         }
198     }
199     
200     Rep(i, 6) {
201         Rep(j, 6) {
202             T.insert(fabs(X.A[i] - X.A[j]));
203             T.insert(fabs(X.A[i] - X.A[j]) / 2);
204             T.insert(fabs(Y.A[i] - Y.A[j]));
205             T.insert(fabs(Y.A[i] - Y.A[j]) / 2);
206             T.insert(fabs(X.A[i] - Y.A[j]));
207             T.insert(fabs(X.A[i] - Y.A[j]) / 2);
208         }
209     }
210     
211     foreach(i, T) {
212         if(*i > 1e9) break;
213         State tmpX = X + *i; 
214         State tmpY = Y + *i; 
215         
216         ans = min(ans, (tmpX._max - tmpX._min) * (tmpY._max - tmpY._min));
217     }
218         
219     printf("%.10f
", ans);
220     return 0;
221 }

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