POJ 2553 The Bottom of a Graph

深藏阁楼爱情的钟 2021-11-01 06:18 180阅读 0赞

题目链接：[https://vjudge.net/problem/POJ-2553][https_vjudge.net_problem_POJ-2553]

### 题目大意 ###

给定有向图 G，对于 G 中一个点 v，如果 v 所能到达的点都能到达 v，那么称 v 为一个 sink 点，求 G 中所有的 sink 点。

### 分析 ###

Tarjan 算法求出度为 0 的 SCC 的模板题。

### 代码如下 ###

![ContractedBlock.gif][] ![ExpandedBlockStart.gif][]

1 #include <cmath>
      2 #include <ctime>
      3 #include <iostream>
      4 #include <string>
      5 #include <vector>
      6 #include <cstdio>
      7 #include <cstdlib>
      8 #include <cstring>
      9 #include <queue>
     10 #include <map>
     11 #include <set>
     12 #include <algorithm>
     13 #include <cctype>
     14 #include <stack>
     15 #include <deque>
     16 #include <list>
     17 #include <sstream>
     18 #include <cassert>
     19 using namespace std;
     20  
     21 #define INIT() ios::sync_with_stdio(false);cin.tie(0);cout.tie(0);
     22 #define Rep(i,n) for (int i = 0; i < (n); ++i)
     23 #define For(i,s,t) for (int i = (s); i <= (t); ++i)
     24 #define rFor(i,t,s) for (int i = (t); i >= (s); --i)
     25 #define ForLL(i, s, t) for (LL i = LL(s); i <= LL(t); ++i)
     26 #define rForLL(i, t, s) for (LL i = LL(t); i >= LL(s); --i)
     27 #define foreach(i,c) for (__typeof(c.begin()) i = c.begin(); i != c.end(); ++i)
     28 #define rforeach(i,c) for (__typeof(c.rbegin()) i = c.rbegin(); i != c.rend(); ++i)
     29  
     30 #define pr(x) cout << #x << " = " << x << "  "
     31 #define prln(x) cout << #x << " = " << x << endl
     32  
     33 #define LOWBIT(x) ((x)&(-x))
     34  
     35 #define ALL(x) x.begin(),x.end()
     36 #define INS(x) inserter(x,x.begin())
     37 #define UNIQUE(x) x.erase(unique(x.begin(), x.end()), x.end())
     38 #define REMOVE(x, c) x.erase(remove(x.begin(), x.end(), c), x.end()); // 删去 x 中所有 c 
     39 #define TOLOWER(x) transform(x.begin(), x.end(), x.begin(),::tolower);
     40 #define TOUPPER(x) transform(x.begin(), x.end(), x.begin(),::toupper);
     41  
     42 #define ms0(a) memset(a,0,sizeof(a))
     43 #define msI(a) memset(a,inf,sizeof(a))
     44 #define msM(a) memset(a,-1,sizeof(a))
     45 
     46 #define MP make_pair
     47 #define PB push_back
     48 #define ft first
     49 #define sd second
     50  
     51 template<typename T1, typename T2>
     52 istream &operator>>(istream &in, pair<T1, T2> &p) {
     53     in >> p.first >> p.second;
     54     return in;
     55 }
     56  
     57 template<typename T>
     58 istream &operator>>(istream &in, vector<T> &v) {
     59     for (auto &x: v)
     60         in >> x;
     61     return in;
     62 }
     63  
     64 template<typename T1, typename T2>
     65 ostream &operator<<(ostream &out, const std::pair<T1, T2> &p) {
     66     out << "[" << p.first << ", " << p.second << "]" << "\n";
     67     return out;
     68 }
     69 
     70 inline int gc(){
     71     static const int BUF = 1e7;
     72     static char buf[BUF], *bg = buf + BUF, *ed = bg;
     73     
     74     if(bg == ed) fread(bg = buf, 1, BUF, stdin);
     75     return *bg++;
     76 } 
     77 
     78 inline int ri(){
     79     int x = 0, f = 1, c = gc();
     80     for(; c<48||c>57; f = c=='-'?-1:f, c=gc());
     81     for(; c>47&&c<58; x = x*10 + c - 48, c=gc());
     82     return x*f;
     83 }
     84 
     85 template<class T>
     86 inline string toString(T x) {
     87     ostringstream sout;
     88     sout << x;
     89     return sout.str();
     90 }
     91 
     92 inline int toInt(string s) {
     93     int v;
     94     istringstream sin(s);
     95     sin >> v;
     96     return v;
     97 }
     98 
     99 //min <= aim <= max
    100 template<typename T>
    101 inline bool BETWEEN(const T aim, const T min, const T max) {
    102     return min <= aim && aim <= max;
    103 }
    104  
    105 typedef long long LL;
    106 typedef unsigned long long uLL;
    107 typedef pair< double, double > PDD;
    108 typedef pair< int, int > PII;
    109 typedef pair< int, PII > PIPII;
    110 typedef pair< string, int > PSI;
    111 typedef pair< int, PSI > PIPSI;
    112 typedef set< int > SI;
    113 typedef set< PII > SPII;
    114 typedef vector< int > VI;
    115 typedef vector< double > VD;
    116 typedef vector< VI > VVI;
    117 typedef vector< SI > VSI;
    118 typedef vector< PII > VPII;
    119 typedef map< int, int > MII;
    120 typedef map< LL, int > MLLI;
    121 typedef map< int, string > MIS;
    122 typedef map< int, PII > MIPII;
    123 typedef map< PII, int > MPIII;
    124 typedef map< string, int > MSI;
    125 typedef map< string, string > MSS;
    126 typedef map< PII, string > MPIIS;
    127 typedef map< PII, PII > MPIIPII;
    128 typedef multimap< int, int > MMII;
    129 typedef multimap< string, int > MMSI;
    130 //typedef unordered_map< int, int > uMII;
    131 typedef pair< LL, LL > PLL;
    132 typedef vector< LL > VL;
    133 typedef vector< VL > VVL;
    134 typedef priority_queue< int > PQIMax;
    135 typedef priority_queue< int, VI, greater< int > > PQIMin;
    136 const double EPS = 1e-8;
    137 const LL inf = 0x3fffffff;
    138 const LL infLL = 0x3fffffffffffffffLL;
    139 const LL mod = 20100713;
    140 const int maxN = 5e3 + 7;
    141 const LL ONE = 1;
    142 const LL evenBits = 0xaaaaaaaaaaaaaaaa;
    143 const LL oddBits = 0x5555555555555555;
    144 
    145 struct Edge{
    146     int from, to;
    147     
    148     Edge() {}
    149     Edge(int x, int y) : from(x), to(y) {}
    150 };
    151 
    152 istream& operator>> (istream& in, Edge &x) {
    153     in >> x.from >> x.to;
    154     return in;
    155 }
    156 
    157 template<typename T>
    158 ostream &operator<<(ostream &out, const vector<T> &v) {
    159     Rep(i, v.size()) {
    160         if(i) out << " ";
    161         out << v[i];
    162     }
    163     out << endl;
    164     return out;
    165 }
    166 
    167 int N, M;
    168 VI V[maxN], ans;
    169 vector< Edge > E;
    170 
    171 void addEdge(Edge &x) {
    172     V[x.from].PB(E.size());
    173     E.PB(x);
    174 }
    175 
    176 stack< int > sk; // 递归处理 SCC (强连通分量)
    177 bool insk[maxN]; // 是否在栈中 
    178 
    179 int scc[maxN], sccid; // 存每个点对应SCC的编号 
    180 int in[maxN], out[maxN]; // SCC的入度与出度 
    181 
    182 int Time;
    183 int tp[maxN]; // timestamp，时间戳
    184 int facr[maxN]; // The farthest ancestor that can be reached，每个节点最远的返回的祖先 
    185 // S ：当前节点号
    186 // 划分SCC并缩点 
    187 void Tarjan(int S) {
    188     tp[S] = facr[S] = ++Time;
    189     sk.push(S);
    190     insk[S] = 1;
    191     
    192     Rep(i, V[S].size()) {
    193         Edge &e = E[V[S][i]];
    194         
    195         if(!tp[e.to]) {
    196             Tarjan(e.to);
    197             facr[S] = min(facr[S], facr[e.to]);
    198         }
    199         else if(insk[e.to]) facr[S] = min(facr[S], tp[e.to]); // 必须要保证在栈中，不然不能保证是一块SCC 
    200     }
    201     
    202     if(facr[S] == tp[S]) {
    203         ++sccid;
    204         while(!sk.empty()) {
    205             int tmp = sk.top(); sk.pop();
    206             
    207             insk[tmp] = 0;
    208             scc[tmp] = sccid;
    209             if(tmp == S) break;
    210         } 
    211     }
    212 }
    213 
    214 void init() {
    215     For(i, 1, N) V[i].clear();
    216     E.clear();
    217     ans.clear();
    218     
    219     while(!sk.empty()) sk.pop();
    220     ms0(insk);
    221     
    222     ms0(scc); sccid = 0;
    223     //ms0(in); 
    224     ms0(out);
    225     
    226     Time = 0;
    227     ms0(tp);
    228     ms0(facr);
    229 }
    230 
    231 int main(){
    232     //freopen("MyOutput.txt","w",stdout);
    233     //freopen("input.txt","r",stdin);
    234     INIT();
    235     while(cin >> N && N) {
    236         init();
    237         cin >> M;
    238         For(i, 1, M) {
    239             Edge t;
    240             cin >> t;
    241             addEdge(t);
    242         }
    243         
    244         For(i, 1, N) if(!tp[i]) Tarjan(i);
    245         
    246         For(i, 1, N) {
    247             Rep(j, V[i].size()) {
    248                 Edge &e = E[V[i][j]];
    249                 if(scc[e.from] != scc[e.to]) ++out[scc[e.from]];
    250             }
    251         }
    252         
    253         For(i, 1, N) if(!out[scc[i]]) ans.PB(i); 
    254         
    255         sort(ALL(ans));
    256         cout << ans;
    257     }
    258     return 0;
    259 }

转载于:https://www.cnblogs.com/zaq19970105/p/11268803.html

[https_vjudge.net_problem_POJ-2553]: https://vjudge.net/problem/POJ-2553
[ContractedBlock.gif]: https://images.cnblogs.com/OutliningIndicators/ContractedBlock.gif
[ExpandedBlockStart.gif]: /images/20211101/1000d1e9021b49bc9364e0372e13772a.png