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Codeforces Round #369 Div2 D. Directed Roads

競技プログラミング Codeforces

問題

Problem - D - Codeforces

N頂点N辺の有向グラフがある。頂点 i からは a_i への辺が出ている。好きなだけ辺の向きを変えることができるとき、有向閉路を持たないグラフは何通りできるか。

解法

長さ d の閉路の辺の組み合わせは  2^d-2 通りである。それ以外の辺はどちらの向きでも良い。

コード

import java.io.IOException;
import java.io.InputStream;
import java.io.OutputStream;
import java.io.PrintWriter;
import java.util.ArrayDeque;
import java.util.ArrayList;
import java.util.Collections;
import java.util.NoSuchElementException;

/*
                   _ooOoo_
                  o8888888o
                  88" . "88
                  (| -_- |)
                  O\  =  /O
               ____/`---'\____
             .'  \\|     |//  `.
            /  \\|||  :  |||//  \
           /  _||||| -:- |||||-  \
           |   | \\\  -  /// |   |
           | \_|  ''\---/''  |   |
           \  .-\__  `-`  ___/-. /
         ___`. .'  /--.--\  `. . __
      ."" '<  `.___\_<|>_/___.'  >'"".
     | | :  `- \`.;`\ _ /`;.`/ - ` : | |
     \  \ `-.   \_ __\ /__ _/   .-` /  /
======`-.____`-.___\_____/___.-`____.-'======
                   `=---='
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
            pass System Test!
*/

public class D {
  private static class Task {
    final long MOD = (int) 1e9 + 7;
    long modPow(long x, long e) {
      long ret = 1;
      long cur = x;
      while (e > 0) {
        if ((e & 1) != 0) ret = (ret * cur) % MOD;
        cur = (cur * cur) % MOD;
        e /= 2;
      }
      return ret;
    }

    void solve(FastScanner in, PrintWriter out) throws Exception {
      int N = in.nextInt();
      int[] to = new int[N];
      for (int i = 0; i < N; i++) {
        to[i] = in.nextInt() - 1;
      }
      ArrayList<ArrayList<Integer>> G = new ArrayList<>();
      for (int i = 0; i < N; i++) G.add(new ArrayList<>());
      for (int i = 0; i < N; i++) {
        G.get(i).add(to[i]);
      }

      StronglyConnectedComponents scc = new StronglyConnectedComponents();
      int[] cmp = scc.decompose(G);
      int size = scc.k;

      int[] cnt = new int[size];
      for (int c : cmp) cnt[c]++;

      long ans = 1;
      for (int c : cnt) {
        if (c > 1) ans = (ans * (modPow(2, c) - 2 + MOD)) % MOD;
        else ans = (ans * 2) % MOD;
      }
      out.println(ans);
    }
    class StronglyConnectedComponents {
      int k;
      int[] decompose(ArrayList<ArrayList<Integer>> G) {
        ArrayList<Integer> vs = new ArrayList<>();
        int V = G.size();
        int[] cmp = new int[V];

        ArrayList<ArrayList<Integer>> rG = new ArrayList<>(V);
        for (int i = 0; i < V; i++) rG.add(new ArrayList<Integer>());
        for (int i = 0; i < V; i++) for (int v : G.get(i)) rG.get(v).add(i);
        boolean[] used = new boolean[V];

        ArrayDeque<Integer> stack = new ArrayDeque<>();
        boolean[] added = new boolean[V];
        for (int i = 0; i < V; i++)
          if (!used[i]) {
            stack.addFirst(i);
            while (!stack.isEmpty()) {
              int v = stack.peekFirst();
              used[v] = true;
              boolean pushed = false;
              for (int j = G.get(v).size() - 1; j >= 0; j--) {
                int u = G.get(v).get(j);
                if (!used[u]) {
                  stack.addFirst(u);
                  pushed = true;
                }
              }
              if (!pushed) {
                stack.pollFirst();
                if (!added[v]) {
                  vs.add(v);
                  added[v] = true;
                }
              }
            }
          }

        used = new boolean[V];
        k = 0;
        Collections.reverse(vs);
        for (int i : vs)
          if (!used[i]) {
            stack.push(i);
            used[i] = true;
            cmp[i] = k;
            while (!stack.isEmpty()) {
              int v = stack.peek();
              boolean pushed = false;
              for (int u : rG.get(v))
                if (!used[u]) {
                  used[u] = true;
                  cmp[u] = k;
                  stack.push(u);
                  pushed = true;
                }
              if (!pushed) stack.pop();
            }
            k++;
          }
        return cmp;
      }
    }
  }

  /**
   * ここから下はテンプレートです。
   */
  public static void main(String[] args) throws Exception {
    OutputStream outputStream = System.out;
    FastScanner in = new FastScanner();
    PrintWriter out = new PrintWriter(outputStream);
    Task solver = new Task();
    solver.solve(in, out);
    out.close();
  }
  private static class FastScanner {
    private final InputStream in = System.in;
    private final byte[] buffer = new byte[1024];
    private int ptr = 0;
    private int bufferLength = 0;

    private boolean hasNextByte() {
      if (ptr < bufferLength) {
        return true;
      } else {
        ptr = 0;
        try {
          bufferLength = in.read(buffer);
        } catch (IOException e) {
          e.printStackTrace();
        }
        if (bufferLength <= 0) {
          return false;
        }
      }
      return true;
    }

    private int readByte() {
      if (hasNextByte()) return buffer[ptr++];
      else return -1;
    }

    private static boolean isPrintableChar(int c) {
      return 33 <= c && c <= 126;
    }

    private void skipUnprintable() {
      while (hasNextByte() && !isPrintableChar(buffer[ptr])) ptr++;
    }

    boolean hasNext() {
      skipUnprintable();
      return hasNextByte();
    }

    public String next() {
      if (!hasNext()) throw new NoSuchElementException();
      StringBuilder sb = new StringBuilder();
      int b = readByte();
      while (isPrintableChar(b)) {
        sb.appendCodePoint(b);
        b = readByte();
      }
      return sb.toString();
    }

    long nextLong() {
      if (!hasNext()) throw new NoSuchElementException();
      long n = 0;
      boolean minus = false;
      int b = readByte();
      if (b == '-') {
        minus = true;
        b = readByte();
      }
      if (b < '0' || '9' < b) {
        throw new NumberFormatException();
      }
      while (true) {
        if ('0' <= b && b <= '9') {
          n *= 10;
          n += b - '0';
        } else if (b == -1 || !isPrintableChar(b)) {
          return minus ? -n : n;
        } else {
          throw new NumberFormatException();
        }
        b = readByte();
      }
    }

    double nextDouble() {
      return Double.parseDouble(next());
    }

    double[] nextDoubleArray(int n) {
      double[] array = new double[n];
      for (int i = 0; i < n; i++) {
        array[i] = nextDouble();
      }
      return array;
    }

    double[][] nextDoubleMap(int n, int m) {
      double[][] map = new double[n][];
      for (int i = 0; i < n; i++) {
        map[i] = nextDoubleArray(m);
      }
      return map;
    }

    public int nextInt() {
      return (int) nextLong();
    }

    public int[] nextIntArray(int n) {
      int[] array = new int[n];
      for (int i = 0; i < n; i++) array[i] = nextInt();
      return array;
    }

    public long[] nextLongArray(int n) {
      long[] array = new long[n];
      for (int i = 0; i < n; i++) array[i] = nextLong();
      return array;
    }

    public String[] nextStringArray(int n) {
      String[] array = new String[n];
      for (int i = 0; i < n; i++) array[i] = next();
      return array;
    }

    public char[][] nextCharMap(int n) {
      char[][] array = new char[n][];
      for (int i = 0; i < n; i++) array[i] = next().toCharArray();
      return array;
    }
  }
}