Codeforces Round #369 Div2 D. Directed Roads
問題
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 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; } } }