topcoder srm 625 div1

problem1 link

假设第$i$种出现的次数为$n_{i}$，总个数为$m$，那么排列数为$T=frac{m!}{prod_{i=1}^{26}(n_{i}!)}$

然后计算回文的个数，只需要考虑前一半，得到个数为$R$，那么答案为$frac{R}{T}$.

为了防止数字太大导致越界，可以分解为质因子的表示方法。

problem2 link

假设终点所在的位置为$(tx,ty)$，那么所有底面是$1x1$的格子$(x,y)$一定满足$(x-tx)mod(3)=0,(y-ty)mod(3)=0$

把每个这样的点拆成两个点然后建立最小割的图。

源点与所有的$b$相连，终点与汇点相连，流量为无穷，割边不会在这里产生。

如果不是洞，那么这个格子拆成的两个点流量为1,表示将这个格子设为洞。

每个格子向周围连边，流量为将中间的两个格子设为洞的代价。

最后最小割就是答案。

problem3 link

首先考虑集合之间的关系。设$f[i][j]$表示前$i$个人分成$j$个集合的方案数。初始化$f[1][1]=n$。那么有：

(1)$f[i+1][j+1]=f[i][j]*j$表示新加一个集合，可以在任意两个集合之间

(2)$f[i+1][j]=f[i][j]*j*2$表示新加的元素与之前的某一个集合在一起，可以放在那个集合的前后，所以有$j*2$种方法

(３)$f[i+1][j-1]=f[i][j]*j$表示合并两个集合，可以在任意两个集合之间插入从而进行合并

最后就是对于$f[x][y]$来说，有多少种方式可以在$n$个位置上放置$x$个使得有$y$个集合并且任意两个集合不相邻。令$m=n-(x-y)$，那么相当于在$m$个位置中放置$y$个，使得任意两个不相邻。由于$f[1][1]=n$那么这$y$个集合的排列已经计算了，所以现在可以假设这$y$个元素的第一个放在$ｍ$个位置的第一个位置，那么第二个位置也不能放置了。所以还剩$m-2$个位置,$y-1$个元素。由于每放置一个元素其后面的位置就不能放置了，所以可以把剩下$y-1$个元素的位置与其后面相邻的位置绑定成一个位置，这样的话，就是$m-2-(y-1)$个位置，$y-1$个元素，即$C_{m-2-(y-1)}^{y-1}=C_{n-(x-y)-2-(y-1)}^{y-1}=C_{n-x-1}^{y-1}$

code for problem1

#include <cmath>
#include <string>
#include <vector>

class PalindromePermutations {
 public:
  double palindromeProbability(const std::string &word) {
    std::vector<int> h(26, 0);
    for (auto e : word) {
      ++h[e - 'a'];
    }
    int old_idx = -1;
    for (int i = 0; i < 26; ++i) {
      if (h[i] % 2 == 1) {
        if (old_idx != -1) {
          return 0.0;
        }
        old_idx = i;
      }
    }
    auto total = Compute(h);
    if (old_idx != -1) {
      --h[old_idx];
    }
    for (auto &e : h) {
      e /= 2;
    }
    auto target = Compute(h);
    double result = 1.0;
    for (int i = 2; i < 50; ++i) {
      result *= std::pow(i, target[i] - total[i]);
    }
    return result;
  }

 private:
  std::vector<int> Compute(const std::vector<int> &h) {
    std::vector<int> result(50, 0);
    auto Add = [&](int x, int sgn) {
      for (int i = 2; i <= x; ++i) {
        int k = i;
        for (int j = 2; j * j <= k; ++j) {
          while (k % j == 0) {
            result[j] += sgn;
            k /= j;
          }
        }
        if (k != 1) {
          result[k] += sgn;
        }
      }
    };
    int n = 0;
    for (auto e : h) {
      Add(e, -1);
      n += e;
    }
    Add(n, 1);
    return result;
  }
};

code for problem2

#include <limits>
#include <unordered_map>
#include <vector>

template <typename FlowType>
class MaxFlowSolver {
  static constexpr FlowType kMaxFlow = std::numeric_limits<FlowType>::max();
  static constexpr FlowType kZeroFlow = static_cast<FlowType>(0);
  struct node {
    int v;
    int next;
    FlowType cap;
  };

 public:
  int VertexNumber() const { return used_index_; }

  FlowType MaxFlow(int source, int sink) {
    source = GetIndex(source);
    sink = GetIndex(sink);

    int n = VertexNumber();
    std::vector<int> pre(n);
    std::vector<int> cur(n);
    std::vector<int> num(n);
    std::vector<int> h(n);
    for (int i = 0; i < n; ++i) {
      cur[i] = head_[i];
      num[i] = 0;
      h[i] = 0;
    }
    int u = source;
    FlowType result = 0;
    while (h[u] < n) {
      if (u == sink) {
        FlowType min_cap = kMaxFlow;
        int v = -1;
        for (int i = source; i != sink; i = edges_[cur[i]].v) {
          int k = cur[i];
          if (edges_[k].cap < min_cap) {
            min_cap = edges_[k].cap;
            v = i;
          }
        }
        result += min_cap;
        u = v;
        for (int i = source; i != sink; i = edges_[cur[i]].v) {
          int k = cur[i];
          edges_[k].cap -= min_cap;
          edges_[k ^ 1].cap += min_cap;
        }
      }
      int index = -1;
      for (int i = cur[u]; i != -1; i = edges_[i].next) {
        if (edges_[i].cap > 0 && h[u] == h[edges_[i].v] + 1) {
          index = i;
          break;
        }
      }
      if (index != -1) {
        cur[u] = index;
        pre[edges_[index].v] = u;
        u = edges_[index].v;
      } else {
        if (--num[h[u]] == 0) {
          break;
        }
        int k = n;
        cur[u] = head_[u];
        for (int i = head_[u]; i != -1; i = edges_[i].next) {
          if (edges_[i].cap > 0 && h[edges_[i].v] < k) {
            k = h[edges_[i].v];
          }
        }
        if (k + 1 < n) {
          num[k + 1] += 1;
        }
        h[u] = k + 1;
        if (u != source) {
          u = pre[u];
        }
      }
    }
    return result;
  }

  MaxFlowSolver() = default;

  void Clear() {
    edges_.clear();
    head_.clear();
    vertex_indexer_.clear();
    used_index_ = 0;
  }

  void InsertEdge(int from, int to, FlowType cap) {
    from = GetIndex(from);
    to = GetIndex(to);
    AddEdge(from, to, cap);
    AddEdge(to, from, kZeroFlow);
  }

 private:
  int GetIndex(int idx) {
    auto iter = vertex_indexer_.find(idx);
    if (iter != vertex_indexer_.end()) {
      return iter->second;
    }
    int map_idx = used_index_++;
    head_.push_back(-1);
    return vertex_indexer_[idx] = map_idx;
  }

  void AddEdge(int from, int to, FlowType cap) {
    node p;
    p.v = to;
    p.cap = cap;
    p.next = head_[from];
    head_[from] = static_cast<int>(edges_.size());
    edges_.emplace_back(p);
  }

  std::vector<node> edges_;
  std::vector<int> head_;

  std::unordered_map<int, int> vertex_indexer_;
  int used_index_ = 0;
};

class BlockTheBlockPuzzle {
  static constexpr int kInfinite = 1000000;

 public:
  int minimumHoles(const std::vector<std::string> &S) {
    MaxFlowSolver<int> solver;
    int n = static_cast<int>(S.size());
    int source = -1;
    int sink = -2;
    int tx = -1, ty = -1;
    for (int i = 0; i < n; ++i) {
      for (int j = 0; j < n; ++j) {
        if (S[i][j] == '$') {
          tx = i;
          ty = j;
        }
      }
    }

    auto P0 = [&](int i, int j) { return i * n + j; };
    auto P1 = [&](int i, int j) { return i * n + j + n * n; };

    for (int i = 0; i < n; ++i) {
      for (int j = 0; j < n; ++j) {
        if (i % 3 == tx % 3 && j % 3 == ty % 3) {
          if (S[i][j] == '$') {
            solver.InsertEdge(P1(i, j), sink, kInfinite);
          }
          if (S[i][j] == 'b') {
            solver.InsertEdge(source, P0(i, j), kInfinite);
          }
          if (S[i][j] != 'H') {
            solver.InsertEdge(P0(i, j), P1(i, j),
                              S[i][j] == '.' ? 1 : kInfinite);
          }
          if (i + 3 < n) {
            auto cost = GetCost(S, i + 1, j, i + 2, j);
            solver.InsertEdge(P1(i, j), P0(i + 3, j), cost);
            solver.InsertEdge(P1(i + 3, j), P0(i, j), cost);
          }
          if (j + 3 < n) {
            auto cost = GetCost(S, i, j + 1, i, j + 2);
            solver.InsertEdge(P1(i, j), P0(i, j + 3), cost);
            solver.InsertEdge(P1(i, j + 3), P0(i, j), cost);
          }
        }
      }
    }
    auto result = solver.MaxFlow(source, sink);
    if (result >= kInfinite) {
      return -1;
    }
    return result;
  }

 private:
  int GetCost(const std::vector<std::string> &s, int x1, int y1, int x2,
              int y2) {
    if (s[x1][y1] == 'b' || s[x2][y2] == 'b') {
      return kInfinite;
    }
    int ans = 0;
    if (s[x1][y1] == '.') {
      ++ans;
    }
    if (s[x2][y2] == '.') {
      ++ans;
    }
    return ans;
  }
};

code for problem3

constexpr int kMod = 1000000007;
constexpr int kMax = 2000;

int f[kMax + 1][kMax + 1];
int C[kMax + 1][kMax + 1];

class Seatfriends {
 public:
  int countseatnumb(int N, int K, int G) {
    f[1][1] = N;
    for (int i = 1; i < K; ++i) {
      for (int j = 1; j <= G; ++j) {
        long long p = f[i][j];
        if (p == 0) {
          continue;
        }
        if (j < G) {
          (f[i + 1][j + 1] += static_cast<int>(p * j % kMod)) %= kMod;
        }
        (f[i + 1][j - 1] += static_cast<int>(p * j % kMod)) %= kMod;
        (f[i + 1][j] += static_cast<int>(p * 2 * j % kMod)) %= kMod;
      }
    }
    if (K == N) {
      return f[K][0];
    }

    C[0][0] = 1;
    for (int i = 1; i <= N; ++i) {
      C[i][0] = C[i][i] = 1;
      for (int j = 1; j < i; ++j)
        C[i][j] = (C[i - 1][j - 1] + C[i - 1][j]) % kMod;
    }

    long long ans = 0;
    for (int j = 1; j <= G; ++j) {
      ans += static_cast<long long>(f[K][j]) * C[N - K - 1][j - 1] % kMod;
    }
    return static_cast<int>(ans % kMod);
  }
};