Another Chocolate Maniac

题目描述

Bob really LOVES chocolate. He thinks he never gets enough. Imagine his joy when his parents told him that they would buy him many rectangular chocolate pieces for his birthday. A piece of chocolate is a $2 \times 1$ or $1 \times 2$ rectangle. Bob’s parents also bought him a nice birthday cake, which can be imagined as a matrix having $M$ rows and $N$ columns. Some positions on the cake are occupied by candles, the others are empty. Bob’s parents asked their son to place as many chocolate pieces as he can on the empty squares on the cake, in such a manner that no two chocolate pieces overlap. However, he would like to keep the chocolate pieces to himself. That’s why, he wants to place only a minimal amount of them on the cake and keep the rest. In order not to make Mon and Dad suspicious, Bob wants to place the chocolate pieces in such a way, that no other piece may be placed on the cake (that is, there won’t exist any two adjacent empty squares). Find the minimal number of pieces which need to be placed on the cake, so that they do not overlap and no extra piece may be added.

题意概述

给定一个$M \times N$的网格，其中有些格子是满的。要求用$2 \times 1$和$1 \times 2$两种方块填充空格子，使得不存在两个相邻的空格子。求至少需要多少方块。

数据范围：$1 \le M \le 70, \ 1 \le N \le 7$。

算法分析

令$f_{i, j, k}$表示处理到第$i$行，第$(i-1)$行的状态为$j$且第$i$行的状态为$k$的方案数。转移时要注意判断上下两行是否有相邻的空格子。

代码

#include <cmath>
#include <cctype>
#include <cstdio>
#include <cstring>
#include <iostream>
#include <algorithm>

using namespace std;

namespace std {
  template <typename T> void maxify(T &a, T b) { b > a && (a = b); }
  template <typename T> void minify(T &a, T b) { b < a && (a = b); }
}

struct IOManager {
  template <typename T> inline bool read(T &x) { char c = '\n'; bool flag = false; x = 0; while (~ c && ! isdigit(c = getchar()) && c != '-') ; c == '-' && (flag = true, c = getchar()); if (! ~ c) return false; while (isdigit(c)) x = (x << 3) + (x << 1) + c - '0', c = getchar(); return (flag && (x = -x), true); }
  inline bool read(char &c) { c = '\n'; while (~ c && ! (isprint(c = getchar()) && c != ' ')) ; return ~ c; }
  inline int read(char s[]) { char c = '\n'; int len = 0; while (~ c && ! (isprint(c = getchar()) && c != ' ')) ; if (! ~ c) return 0; while (isprint(c) && c != ' ') s[len ++] = c, c = getchar(); return (s[len] = '\0', len); }
  template <typename T> inline IOManager operator > (T &x) { read(x); return *this; }
  template <typename T> inline void write(T x) { if (x < 0) putchar('-'), x = -x; if (x > 9) write(x / 10); putchar(x % 10 + '0'); }
  inline void write(char c) { putchar(c); }
  inline void write(char s[]) { int pos = 0; while (s[pos] != '\0') putchar(s[pos ++]); }
  template <typename T> inline IOManager operator < (T x) { write(x); return *this; }
} io;

struct Solver {
private:
  static const int N = 80;
  static const int M = 8;
  int n, m, f[2][1 << M][1 << M], mp[N];
  void input() {
    io > n > m;
    for (int i = 1; i <= n; ++ i)
      for (int j = 1; j <= m; ++ j) {
        char c; io > c, (mp[i] <<= 1) += c == '*';
      }
    mp[0] = mp[n + 1] = mp[n + 2] = (1 << m) - 1;
  }
  void init() { memset(f, 0x1f, sizeof f), f[0][mp[0]][mp[1]] = 0; }
  bool check(int s) {
    bool flag = false;
    for (int i = 0; i < m; s >>= 1, ++ i) {
      if (s & 1) flag = false;
      else { if (flag) return false; flag = true; }
    }
    return true;
  }
  void update(int x, int a, int b, int s, int val, int p) {
    if (check(b) && ! ((1 << m) - 1 - a & (1 << m) - 1 - b)) minify(f[x][b][s], val);
    for (int i = p; i < m; ++ i)
      if (! (b & 1 << i)) {
        if (! (s & 1 << i)) update(x, a, b | 1 << i, s | 1 << i, val + 1, i + 1);
        if (i && ! (b & 1 << i - 1)) update(x, a, b | 1 << i | 1 << i - 1, s, val + 1, i + 1);
      }
  }
  void process() {
    int cur = 0;
    for (int i = 1; i <= n + 1; cur ^= 1, ++ i) {
      memset(f[cur ^ 1], 0x1f, sizeof f[cur ^ 1]);
      for (int j = 0; j < 1 << m; ++ j)
        for (int k = 0; k < 1 << m; ++ k)
          if (f[cur][j][k] < 5e8) update(cur ^ 1, j, k, mp[i + 1], f[cur][j][k], 0);
    }
    io < f[cur][(1 << m) - 1][(1 << m) - 1] < '\n';
  }

public:
  void solve() { input(), init(), process(); }
} solver;

int main() {
  solver.solve();
  return 0;
}