Codeforces Round #363 (Div. 2) DE
復習シリーズ
D. Fix a Tree
N 頂点の有向グラフがあり、頂点 i から頂点 a[i] に向かう辺があります。できるだけ少ない辺を張り替えてある1つの頂点に向かう有向木にしてください。
解法
連結成分内に閉路が含まれているとき、閉路は必ず先端にあるので、この内一つを終点に張り替えれば良いです。
コード
import java.io.IOException; import java.io.InputStream; import java.io.OutputStream; import java.io.PrintWriter; import java.util.*; /* _ooOoo_ o8888888o 88" . "88 (| -_- |) O\ = /O ____/`---'\____ .' \\| |// `. / \\||| : |||// \ / _||||| -:- |||||- \ | | \\\ - /// | | | \_| ''\---/'' | | \ .-\__ `-` ___/-. / ___`. .' /--.--\ `. . __ ."" '< `.___\_<|>_/___.' >'"". | | : `- \`.;`\ _ /`;.`/ - ` : | | \ \ `-. \_ __\ /__ _/ .-` / / ======`-.____`-.___\_____/___.-`____.-'====== `=---=' ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ pass System Test! */ public class D2 { private static class Task { void solve(FastScanner in, PrintWriter out) throws Exception { int N = in.nextInt(); int[] parent = new int[N]; for (int i = 0; i < N; i++) { parent[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(parent[i]).add(i); } int[] d = decompose(G); for (int i = 0; i < N; i++) { G.get(i).add(parent[i]); } ArrayList<ArrayList<Integer>> cycles = new ArrayList<>(); boolean[] vis = new boolean[N]; ArrayDeque<Integer> deque = new ArrayDeque<>(); for (int i = 0; i < N; i++) { if (vis[i]) continue; ArrayList<Integer> cycle = new ArrayList<>(); deque.add(i); vis[i] = true; cycle.add(i); while (!deque.isEmpty()) { int v = deque.poll(); for (int w : G.get(v)) { if (vis[w]) continue; vis[w] = true; deque.add(w); if (d[cycle.get(0)] < d[w]) continue; if (d[cycle.get(0)] > d[w]) cycle.clear(); cycle.add(w); } } cycles.add(cycle); } int single = -1; for (ArrayList<Integer> cycle : cycles) if (cycle.size() == 1) single = cycle.get(0); if (single == -1) single = cycles.get(0).get(0); int cost = 0; for (ArrayList<Integer> cycle : cycles) { int v = cycle.get(0); if (parent[v] != single) cost++; parent[v] = single; } out.println(cost); for (int i = 0; i < N; i++) { if (i > 0) out.print(" "); out.print(parent[i] + 1); } } public 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]; int 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; } public int[][] nextIntMap(int n, int m) { int[][] map = new int[n][]; for (int i = 0; i < n; i++) { map[i] = nextIntArray(m); } return map; } } }
E. LRU
サイズ K のメモリと、N 個の要素があります。最初はメモリは空で、要素を選択してメモリに加える操作を行います。要素 i が選択される確率は p[i] です。メモリに K 個の要素が含まれていて、K+1個目が選択されたとき、メモリ内の最も古く選択された要素が削除されます。操作を無限に行ったとき、要素 i がメモリに含まれている確率を求めてください。
解法
コード
import java.io.IOException; import java.io.InputStream; import java.io.OutputStream; import java.io.PrintWriter; import java.util.NoSuchElementException; /* _ooOoo_ o8888888o 88" . "88 (| -_- |) O\ = /O ____/`---'\____ .' \\| |// `. / \\||| : |||// \ / _||||| -:- |||||- \ | | \\\ - /// | | | \_| ''\---/'' | | \ .-\__ `-` ___/-. / ___`. .' /--.--\ `. . __ ."" '< `.___\_<|>_/___.' >'"". | | : `- \`.;`\ _ /`;.`/ - ` : | | \ \ `-. \_ __\ /__ _/ .-` / / ======`-.____`-.___\_____/___.-`____.-'====== `=---=' ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ pass System Test! */ public class E2 { private static class Task { double[] solve(double[] p, int K) { int N = p.length; double[] dp = new double[1 << N]; dp[0] = 1.0; for (int mask = 0; mask < (1 << N); mask++) { double remain = 0.0; for (int i = 0; i < N; i++) { if ((mask & (1 << i)) != 0) continue; remain += p[i]; } for (int i = 0; i < N; i++) { if ((mask & (1 << i)) != 0) continue; int next = mask + (1 << i); if (Integer.bitCount(next) > K) continue; dp[next] += dp[mask] * (p[i] / remain); } } double[] ans = new double[N]; for (int mask = 0; mask < (1 << N); mask++) { if (Integer.bitCount(mask) != K) continue; for (int i = 0; i < N; i++) { if ((mask & (1 << i)) != 0) { ans[i] += dp[mask]; } } } return ans; } void solve(FastScanner in, PrintWriter out) throws Exception { int N = in.nextInt(); int K = in.nextInt(); double[] p = in.nextDoubleArray(N); int zero = 0; for (double q : p) if (q == 0.0) zero++; double[] q = new double[N - zero]; int cur = 0; for (int i = 0; i < N; i++) { if (p[i] != 0.0) { q[cur] = p[i]; cur++; } } double[] ans = solve(q, Math.min(K, q.length)); cur = 0; for (int i = 0; i < N; i++) { if (i > 0) out.print(" "); if (p[i] == 0.0) { out.print(0.0); } else { out.print(ans[cur]); cur++; } } } } /** * 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; } public int[][] nextIntMap(int n, int m) { int[][] map = new int[n][]; for (int i = 0; i < n; i++) { map[i] = nextIntArray(m); } return map; } } }