AtCoder ABC 131F Must Be Rectangular!

Posted 2022-02-26 zaq19970105

题目链接：https://atcoder.jp/contests/abc131/tasks/abc131_f

转自博客：https://blog.csdn.net/qq_37656398/article/details/93496476

题目大意

　　给定 N 个点，若有如下图所示的三个点（黑点）就可以构造岀一个新的点（红点）。求可构造的点的最多的个数。

分析

　　首先，如果两个点的某一个坐标相等，我们定义这两个点是联通的。

　　其次，如果红点能够被构造，它必然是由处于同一连通块中的点所构造的。

　　于是，只要分别计算每个联通块能构造多少点就可以了。

　　对于每个连通块，把 x 坐标和 y 坐标单独弄一个集合，然后就在两个集合中不断分别取两个进行构造，直到不能构造为止，图示如下：

　　我们发现，构造的过程就是把构造前的二部图构造成完全二部图，每多一条边就表明构造了一个点。

　　于是把所有连通块所对应的完全二部图边数都加起来再减去 N 就能得到能构造的点的数目了。

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 << "]" << "\\n";
 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 int N, K, M;
128 LL ans;
129 
130 int f[maxN << 1], sz[maxN << 1][2];
131 
132 int Find(int x)
133     while (x != f[x]) x = f[x] = f[f[x]];
134     return x;
135  
136 
137 int main()
138     //freopen("MyOutput.txt","w",stdout);
139     //freopen("input.txt","r",stdin);
140     //INIT();
141     cin >> N;
142     Rep(i, maxN << 1) f[i] = i;
143     
144     Rep(i, N) 
145         int x, y;
146         cin >> x >> y;
147         y += maxN;
148         f[Find(x)] = Find(y);
149     
150     Rep(i, maxN << 1) ++sz[Find(i)][i / maxN];
151     Rep(i, maxN << 1) ans += (LL)sz[i][0] * sz[i][1];
152     cout << ans - N << endl;
153     return 0;
154 

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