yukicoder No.409 ダイエット - 宇宙ツイッタラーXの憂鬱

問題

嘘解法

ドーナッツの増加量の上限が 1,000,000 であることから、B が正の値であるときストレス値が 1000 を超えないことが分かる。（1000 を超える時、どのドーナツよりもコストが高くなってしまう）

よってストレス値をキーに持った DP をすると $O(N \sqrt{max(D)}$ で計算できる。これはオンラインジャッジに祈りを捧げれば通る。

嘘解法コード

import java.io.IOException;
import java.io.InputStream;
import java.io.OutputStream;
import java.io.PrintWriter;
import java.util.Arrays;
import java.util.NoSuchElementException;

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

public class yukicoder409_uso {
  private static class Task {
    final long INF = (long) 1e18;
    void solve(FastScanner in, PrintWriter out) {
      int N = in.nextInt();
      int A = in.nextInt();
      int B = in.nextInt();
      int W = in.nextInt();
      int[] D = in.nextIntArray(N);

      if (B == 0) {
        out.println(W - A * N);
        return;
      }

      final int width = 1001;

      long[][] dps = new long[2][width + 1];
      Arrays.fill(dps[0], INF);
      dps[0][0] = 0;
      for (int i = 0; i < N; i++) {
        long[] dp = dps[i % 2];
        long[] next = dps[(i + 1) % 2];
        Arrays.fill(next, INF);
        for (int j = 0; j < width; j++) {
          next[0] = Math.min(next[0], dp[j] + D[i]);
          next[j + 1] = Math.min(next[j + 1], dp[j] + B * (j + 1) - A);
        }
      }

      long ans = INF;
      for (long d : dps[N % 2]) ans = Math.min(ans, d);
      out.println(ans + W);
    }
  }

  /**
   * ここから下はテンプレートです。
   */
  public static void main(String[] args) {
    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;
    }
  }
}

解法

yukicoder 「No409 ダイエット」 - 競技プログラミングをするんだよ

Convex Hull Trick

コード

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

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

public class yukicoder409 {
  private static class Task {
    class ConvexHullTrick {
      ArrayList<long[]> ls = new ArrayList<>();
      ConvexHullTrick() {
      }

      boolean check(long[] l1, long[] l2, long[] l3) {
        return (l2[0] - l1[0]) * (l3[1] - l2[1]) >=
          (l2[1] - l1[1]) * (l3[0] - l2[0]);
      }

      void add(long a, long b) {
        long[] l = new long[]{a, b};
        while (ls.size() >= 2 && check(ls.get(ls.size() - 2), ls.get(ls.size() - 1), l)) ls.remove(ls.size() - 1);
        ls.add(l);
      }

      long f(int k, long x) {
        long[] l = ls.get(k);
        return l[0] * x + l[1];
      }

      long query(long x) {
        int low = -1;
        int high = ls.size() - 1;
        while (high - low > 1) {
          int mid = (high + low) / 2;
          if (f(mid, x) >= f(mid + 1, x)) {
            low = mid;
          } else {
            high = mid;
          }
        }
        return f(high, x);
      }
    }

    final long INF = (long) 1e18;

    void solve(FastScanner in, PrintWriter out) {
      int N = in.nextInt();
      int A = in.nextInt();
      long B = in.nextInt();
      int W = in.nextInt();
      int[] D = in.nextIntArray(N);

      if (B == 0) {
        out.println(W - A * N);
        return;
      }

      long[] dp = new long[N + 1];
      Arrays.fill(dp, INF);
      dp[0] = W;
      ConvexHullTrick deq = new ConvexHullTrick();
      deq.add(0, dp[0]);
      for (long i = 0; i < N; i++) {
        dp[(int) i + 1] = D[(int) i] - i * A + (B * i * (i + 1)) / 2 + deq.query(i);
        long a = -(i + 1) * B;
        long b = dp[(int) i + 1] + (i + 1) * A + (B * (i + 1) * i) / 2;
        deq.add(a, b);
      }
      long res = INF;
      for (int i = 0; i <= N; i++)
        res = Math.min(res, dp[i] - A * (N - i) + B * (N - i) * (N - i + 1) / 2);
      out.println(res);

    }
  }

  /**
   * ここから下はテンプレートです。
   */
  public static void main(String[] args) {
    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;
    }
  }
}