省选日记 Day11 - Day15

省选日记 Day \(11\) - Day \(15\)

Day \(11\) Apr 14, 2022, Thursday

SDOI2019 染色

一开始一眼看出 \(O(n^3)\) 的做法, 设 \(f_{i, j, k}\) 表示计算到第 \(i\) 列, \((i, 1)\) 为颜色 \(j\), \((i, 2)\) 为颜色 \(k\) 的方案数. 统计 \(U_{i, j}\) 作为所有 \(f_{i, j, x}\) 的总和, 统计 \(D_{i, j}\) 作为所有 \(f_{i, x, j}\) 的总和, \(Sum_i\) 作为所有 \(f\), 转移是:

\[f_{i, j, k} = Sum_{i - 1} - U_{i - 1, j} - D_{i - 1, k} + f_{i - 1, j, k} \]

然后把和已经确定的位置的颜色相悖的 \(f\) 值赋为 \(0\). 我一直认为这个题需要什么奇妙科技以至于想了一个小时没有进展, 看了两眼题解发现貌似用不到什么奇技淫巧, 也是从这个暴力 DP 上发展而来的, 于是继续之前的 DP 进行优化.

如果滚动数组, 可以写成:

\[f_{i, j} += Sum - U_i - D_j \]

发现这个东西可以转化为全局加法, 行减, 列减, 全局求和, 行求和, 列求和, 只保留某行或某列的归零, 只保留单点的归零, 对角线删除. 行减列减容易维护, 只需要每行每列记录增加数量总和 \(PR_i\), \(PC_i\), 全局操作也是一样, 只需要记录全局每个元素增加的数量总和 \(Pl\). 对于单点修改, 我们记录一个矩阵 \(f_{i, j}\) 表示每个点增加的数量总和, 维护 \(Sum\) 表示矩阵 \(f\) 的总和, \(U_i\) 和 \(D_i\) 仍然表示矩阵第 \(i\) 行和第 \(i\) 列的总和.

全局总和是 \(Sum + c(c - 1)Pl + (c - 1)\sum_{i = 1}^n (PR_i + PC_i)\), 第 \(i\) 行的总和是 \(U_i + (c - 1)(Pl + PR_i) + \sum_{j = 1, j \neq i}^n PC_j\), 第 \(i\) 列的总和是 \(D_i + (c - 1)(Pl + PC_i) + \sum_{j = 1, j \neq i}^n PR_j\).

最后是归零, 我们求一个单点的值的时间是 \(O(1)\) 的, 所以可以在 \(O(c)\) 之内求出所有保留位置的值, 直接赋到 \(f\) 里面, 清空所有的 \(PR\), \(PC\), \(Pl\) 值. 我们发现, \(f\) 最多同时只有一行或一列或一个点有值, 所以不如直接存行的总和和列的总和来代表 \(f\).

直接做可以搞到 \(96'\) 的好成绩.

const unsigned long long Mod(1000000009);
inline void Mn(unsigned long long& x) {x -= ((x >= Mod) ? Mod : 0);}
inline void Mn(unsigned& x) {x -= ((x >= Mod) ? Mod : 0);}
unsigned a[100005][2];
unsigned long long mm;
unsigned m, n;
unsigned A, B, C, D, t;
unsigned Cnt(0), Ans(0);
struct Sta {
  unsigned long long Pl, SumPR, SumPC, SumExist;
  unsigned PR[10005], PC[10005], Exist[10005], Pos;
  char Type;
  inline unsigned long long Sum () {
    return (SumExist + mm * Pl + (SumPR + SumPC) * (m - 1)) % Mod;
  }
  inline unsigned long long Get(unsigned x, unsigned y) {
    unsigned Ex(0);
    if(Type == 1) Ex = ((x == Pos) ? Exist[y] : 0);
    if(Type == 2) Ex = ((y == Pos) ? Exist[x] : 0);
    return (Pl + PR[x] + PC[y] + Ex) % Mod;
  }
  inline unsigned long long GetRow(unsigned x) {
    unsigned Ex(0);
    if(Type == 1) Ex = ((x == Pos) ? (Mod + SumExist - Exist[x]) : 0);
    if(Type == 2) Ex = ((x == Pos) ? 0 : Exist[x]);
    return ((Pl + PR[x]) * (m - 1) + Mod - PC[x] + SumPC + Ex) % Mod;
  }
  inline unsigned long long GetColumn(unsigned x) {
    unsigned Ex(0);
    if(Type == 1) Ex = ((x == Pos) ? 0 : Exist[x]);
    if(Type == 2) Ex = ((x == Pos) ? (Mod + SumExist - Exist[x]) : 0);
    return ((Pl + PC[x]) * (m - 1) + Mod - PR[x] + SumPR + Ex) % Mod;
  }
  inline void Clr() {
    Pl = 0;
    memset(PR + 1, 0, m << 2);
    memset(PC + 1, 0, m << 2);
  }
  inline void Udt() {
    SumExist = SumPR = SumPC = 0;
    for (unsigned i(1); i <= m; ++i) SumExist += Exist[i];
    for (unsigned i(1); i <= m; ++i) SumPC += PC[i];
    for (unsigned i(1); i <= m; ++i) SumPR += PR[i];
    SumExist %= Mod;
    SumPC %= Mod;
    SumPR %= Mod;
  }
}cl[2], *Cur(cl + 0), *Lst(cl + 1);
signed main() {
  n = RD(), mm = (m = RD()) - 1, mm = mm * m % Mod;
  for (unsigned i(1); i <= n; ++i) a[i][0] = RD();
  for (unsigned i(1); i <= n; ++i) a[i][1] = RD();
  if(a[1][0]) {
    Cur->Type = 1, Cur->Pos = a[1][0];
    if(a[1][1]) Cur->Exist[a[1][1]] = 1, Cur->SumExist = 1;
    else Cur->PR[Cur->Pos] = 1, Cur->SumPR = 1;
  }
  else {
    if(!a[1][1]) Cur->Type = 0, Cur->Pl = 1; 
    else Cur->Type = 2, Cur->PC[Cur->Pos = a[1][1]] = 1, Cur->SumPC = 1;
  }
  for (unsigned i(2); i <= n; ++i) {
    swap(Lst, Cur), Cur->Pl = Lst->Pl + Lst->Sum(), Mn(Cur->Pl); 
    for (unsigned j(1); j <= m; ++j) Cur->PR[j] = Lst->PR[j] + Mod - Lst->GetRow(j), Mn(Cur->PR[j]);
    for (unsigned j(1); j <= m; ++j) Cur->PC[j] = Lst->PC[j] + Mod - Lst->GetColumn(j), Mn(Cur->PC[j]);
    memcpy(Cur->Exist + 1, Lst->Exist + 1, m << 2), Cur->Type = Lst->Type, Cur->Pos = Lst->Pos;
    if(a[i][0]) {
      if (a[i][1]) {
        unsigned TmpC(Cur->Get(a[i][0], a[i][1]));
        memset(Cur->Exist + 1, 0, m << 2), Cur->Exist[a[i][1]] = TmpC;
      } else {
        for (unsigned j(1); j <= m; ++j) Lst->Exist[j] = Cur->Get(a[i][0], j); Lst->Exist[a[i][0]] = 0;
        memcpy(Cur->Exist + 1, Lst->Exist + 1, m << 2);
      }
      Cur->Clr(), Cur->Type = 1, Cur->Pos = a[i][0];
    } else {
      if (a[i][1]) {
        for (unsigned j(1); j <= m; ++j) Lst->Exist[j] = Cur->Get(j, a[i][1]); Lst->Exist[a[i][1]] = 0; 
        memcpy(Cur->Exist + 1, Lst->Exist + 1, m << 2);
        Cur->Clr(), Cur->Type = 2, Cur->Pos = a[i][1];
      }
    }
    Cur->Udt();
  }
  printf("%llu\n", Cur->Sum());
  return Wild_Donkey;
}

发现有两个 Type, 也就是说 \(Exist\) 可能是行, 也可能是列, 我们使 \(Exist\) 只表示某一行的信息. 这样相当于不存在某一列第一行是 \(0\), 而第二行非零的情况, 如果出现, 转置所维护的矩阵并且上下翻转输入序列. 仍然是 \(96'\).

const unsigned long long Mod(1000000009);
inline void Mn(unsigned long long& x) {x -= ((x >= Mod) ? Mod : 0);}
inline void Mn(unsigned& x) {x -= ((x >= Mod) ? Mod : 0);}
unsigned a[100005][2];
unsigned long long mm;
unsigned m, n;
unsigned A, B, C, D, t;
unsigned Cnt(0), Ans(0);
char Type, EType;
struct Sta {
  unsigned long long SumPR, SumPC, SumExist;
  unsigned PR[10005], PC[10005], Exist[10005], Pos;
  inline unsigned long long Sum () {
    return (SumExist + (SumPR + SumPC) * (m - 1)) % Mod;
  }
  inline unsigned long long GetRow(unsigned x) {
    unsigned Ex(0);
    if(EType ^ Type) Ex = Exist[x];
    else Ex = ((x == Pos) ? SumExist : 0);
    return ((unsigned long long)PR[x] * (m - 1) + Mod - PC[x] + SumPC + Ex) % Mod;
  }
  inline unsigned long long GetColumn(unsigned x) {
    unsigned Ex(0);
    if(EType ^ Type) Ex = ((x == Pos) ? SumExist : 0);
    else Ex = Exist[x];
    return ((unsigned long long)PC[x] * (m - 1) + Mod - PR[x] + SumPR + Ex) % Mod;
  }
  inline unsigned long long Get(unsigned x, unsigned y) {
    if(x == y) return 0;
    unsigned Ex;
    if(EType ^ Type) Ex = ((y == Pos) ? Exist[x] : 0);
    else Ex = ((x == Pos) ? Exist[y] : 0);
    return (PR[x] + PC[y] + Ex) % Mod;
  }
  inline void Clr() {
    memset(PR + 1, 0, m << 2), memset(PC + 1, 0, m << 2), SumPR = SumPC = 0; 
  }
}cl[2], *Cur(cl + 0), *Lst(cl + 1);
inline void Flip() {
  Type ^= 1;
  swap(Cur->SumPR, Cur->SumPC);
  for (unsigned i(1); i <= m; ++i) swap(Cur->PC[i], Cur->PR[i]);
}
signed main() {
  n = RD(), mm = (m = RD()) - 1, mm = mm * m % Mod;
  for (unsigned i(1); i <= n; ++i) a[i][0] = RD();
  for (unsigned i(1); i <= n; ++i) a[i][1] = RD();
  if(a[1][1]) EType = Type = 1, swap(a[1][1], a[1][0]);
  if(a[1][0]) {
    Cur->Pos = a[1][0];
    if(a[1][1]) Cur->Exist[a[1][1]] = 1, Cur->SumExist = 1;
    else Cur->PR[Cur->Pos] = 1, Cur->SumPR = 1;
  } else {for (unsigned i(1); i <= m; ++i) Cur->PR[i] = 1; Cur->SumPR = m;}
  for (unsigned i(2); i <= n; ++i) {
    swap(Lst, Cur), Cur->SumPR = Cur->SumPC = 0;
    unsigned long long Pl(Lst->Sum());
    for (unsigned j(1); j <= m; ++j) 
      Cur->PR[j] = Lst->PR[j] + Mod - Lst->GetRow(j), Mn(Cur->PR[j]), Mn(Cur->PR[j] += Pl), Mn(Cur->SumPR += Cur->PR[j]);
    for (unsigned j(1); j <= m; ++j) 
      Cur->PC[j] = Lst->PC[j] + Mod - Lst->GetColumn(j), Mn(Cur->PC[j]), Mn(Cur->SumPC += Cur->PC[j]);
    memcpy(Cur->Exist + 1, Lst->Exist + 1, m << 2), Cur->Pos = Lst->Pos, Cur->SumExist = Lst->SumExist; 
    if(a[i][1]) {swap(a[i][1], a[i][0]); if(!Type) Flip(); }
    else if(Type) Flip();
    if(a[i][0]) {
      if (a[i][1]) {
        unsigned TmpC(Cur->Get(a[i][0], a[i][1]));
        memset(Cur->Exist + 1, 0, m << 2), Cur->SumExist = Cur->Exist[a[i][1]] = TmpC;
      } else {
        Cur->SumExist = 0;
        for (unsigned j(1); j <= m; ++j) Mn(Cur->SumExist += (Lst->Exist[j] = Cur->Get(a[i][0], j)));
        memcpy(Cur->Exist + 1, Lst->Exist + 1, m << 2);
      }
      Cur->Clr(), Cur->Pos = a[i][0], EType = Type;
    }
  }
  printf("%llu\n", Cur->Sum());
  return Wild_Donkey;
}

Day \(12\) Apr 15, 2022, Friday

还是那道题

看了题解发现我超脱了题解的路数, 以至于题解的方法无法优化这个做法, 而我也完全没有办法优化这个做法. 重新审视我们的转移, 发现每个阶段的转移只和当前列的状态有关, 而当前列为空白时, 所有状态从上一个阶段到下一个阶段的转移是相同的. 而当前列非空白时, 这个阶段有效状态数最多是 \(O(c)\) 的.

仍然是遇到第二行有确定颜色就交换两行, 我们用 \(f_{i, j}\) 表示第 \(i\) 个非空列, 第二行是颜色 \(j\) 的方案数. 处理 \(g_{i, 0/1/2/3/4}\) 用来转移, 表示相邻两个非空列中间隔了 \(i\) 列, 两端状态为 \(0/1/2/3/4\) 五种情况的转移系数. 两端共四个格子, 这五种情况分别是:

• \(0\) 表示四个颜色各不相同
• \(1\) 表示同行的颜色相同而同列的不同
• \(2\) 表示对角线上颜色相同而同行或同列颜色不同
• \(3\) 表示只有一行颜色相同, 其余两个格子为不同颜色
• \(4\) 表示只有一个对角线颜色相同, 其余两个格子为不同颜色.

发现有情况无法转移, 就是两端列存在空白列的情况, 这时需要预处理第一个阶段的 DP 值, 并且通过最后一个阶段的 DP 值算出答案.

\(g\) 数组线性递推即可求出.

const unsigned long long Mod(1000000009);
inline void Mn(unsigned long long& x) {x -= ((x >= Mod) ? Mod : 0);}
inline void Mn(unsigned& x) {x -= ((x >= Mod) ? Mod : 0);}
bitset<100005> Flp;
unsigned a[100005][2], Pos[100005], f[100005], g[100005][5];
unsigned long long mm, M1, M2, M3, M4, Sum;
unsigned m, n;
unsigned Cnt(0), Ans(0);
inline unsigned ColorTwo(unsigned x) {return a[x][Flp[x] ^ 1];}
inline unsigned ColorOne(unsigned x) {return a[x][Flp[x]];}
inline unsigned long long Onesided(unsigned x) {
  if(!x) return 1; --x;
  return ((g[x][0] * M2 % Mod) * M3 + g[x][1] + g[x][2] + ((g[x][3] + g[x][4]) * M2 << 1)) % Mod;
}
inline void Mul(unsigned long long x) {
  Sum = Sum * x % Mod;
  for (unsigned i(1); i <= m; ++i) f[i] = f[i] * x % Mod;
}
inline void Add(unsigned x) {
  Sum = (Sum + (unsigned long long)m * x) % Mod;
  for (unsigned i(1); i <= m; ++i) Mn(f[i] += x);
}
inline void Def(unsigned x) {
  Sum = (m * x) % Mod;
  for (unsigned i(1); i <= m; ++i) f[i] = x;
}
inline void Set(unsigned x, unsigned y) {
  Mn(Sum += y + Mod - f[x]), Mn(Sum), f[x] = y;
}
inline unsigned long long Find(unsigned x) {
  return f[x];
}
signed main() {
  n = RD(), mm = (m = RD()) - 1, mm = mm * m % Mod;
  M1 = m - 1, M2 = m - 2, M3 = m - 3, M4 = m - 4; 
  for (unsigned i(1); i <= n; ++i) a[i][0] = RD();
  for (unsigned i(1); i <= n; ++i) a[i][1] = RD();
  g[0][0] = g[0][2] = g[0][4] = 1, g[0][1] = g[0][3] = 0;
  for (unsigned i(1); i <= n; ++i) {
    g[i][0] = (g[i - 1][0] * ((M3 * M4 + 1) % Mod) + g[i - 1][1] + g[i - 1][2] + ((g[i - 1][3] + g[i - 1][4]) * M3 << 1) % Mod) % Mod;
    g[i][1] = ((g[i - 1][0] * M2 % Mod) * M3 + g[i - 1][2] + (g[i - 1][4] * M2 << 1)) % Mod;
    g[i][2] = ((g[i - 1][0] * M2 % Mod) * M3 + g[i - 1][1] + (g[i - 1][3] * M2 << 1)) % Mod;
    g[i][3] = ((g[i - 1][0] * M3 % Mod) * M3 + g[i - 1][2] + g[i - 1][3] * M2 + g[i - 1][4] * (M3 + M2)) % Mod;
    g[i][4] = ((g[i - 1][0] * M3 % Mod) * M3 + g[i - 1][1] + g[i - 1][4] * M2 + g[i - 1][3] * (M3 + M2)) % Mod;
  }
  for (unsigned i(1); i <= n; ++i) if(a[i][0] | a[i][1]) {Pos[++Cnt] = i; if(a[i][1]) Flp[Cnt] = 1;}
  for (unsigned i(1); i <= Cnt; ++i) a[i][0] = a[Pos[i]][0], a[i][1] = a[Pos[i]][1];
  if(!Cnt) {
    printf("%llu\n", (Onesided(n - 1) * m % Mod) * M1 % Mod);
    return 0;
  }
  unsigned CT(ColorTwo(1)), Val(Onesided(Pos[1] - 1));
  if(CT) Sum = f[CT] = Val;
  else {for (unsigned i(1); i <= m; ++i) f[i] = Val; f[ColorOne(1)] = 0, Sum = Val * M1 % Mod;}
  for (unsigned i(2); i <= Cnt; ++i) {
    if(0) {
      CT = ColorTwo(Cnt), Ans = 0;
      for (unsigned i(1); i <= m; ++i) if(CT ^ i) Mn(Ans += f[i]);
      printf("%llu\n", Ans);
    }
    unsigned A(a[i - 1][0]), B(a[i - 1][1]), C(a[i][0]), D(a[i][1]), *Len(g[Pos[i] - Pos[i - 1] - 1]);
    if(Flp[i]) swap(A, B), swap(C, D), Flp[i - 1] = (Flp[i - 1] ^ 1);
    if(Flp[i - 1]) {
      if(B == C) {
        unsigned long long PTmp(Sum * Len[4] % Mod);
        Mul(Mod + Len[2] - Len[4]), Add(PTmp), Set(ColorOne(i), 0);
      } else {
        unsigned long long FC(Find(C)), PTmp(((Mod + Sum - FC) * Len[0] + FC * Len[3]) % Mod);
        unsigned long long AB((FC * Len[1] + (Mod + Sum - FC) * Len[3]) % Mod);
        Mul(Mod + Len[4] - Len[0]), Add(PTmp), Set(C, 0), Set(B, AB);
      }
    } else {
      if(A == C) {
        unsigned long long PTmp(Sum * Len[3] % Mod);
        Mul(Mod + Len[1] - Len[3]), Add(PTmp), Set(ColorOne(i), 0);
      } else {
        unsigned long long FC(Find(C)), PTmp(((Mod + Sum - FC) * Len[0] + FC * Len[4]) % Mod);
        unsigned long long AB((FC * Len[2] + (Mod + Sum - FC) * Len[4]) % Mod);
        Mul(Mod + Len[3] - Len[0]), Add(PTmp), Set(C, 0), Set(A, AB);
      }
    }
    if(D) {
      unsigned TmpF(Find(D));
      Def(0), Set(D, TmpF);
    }
  }
  CT = ColorTwo(Cnt), Ans = 0;
  for (unsigned i(1); i <= m; ++i) if(CT ^ i) Mn(Ans += f[i]);
  printf("%llu\n", Ans * Onesided(n - Pos[Cnt]) % Mod);
  return Wild_Donkey;
}

发现瓶颈是对于每个阶段的 \(f\) 数组进行全局修改查询和单点修改和查询, 由于模数较大, 所以无法线性, 但是可以通过线段树做到 \(O(n\log c + c)\). 把快速查询的部分分代码复制过来, 删除全局赋值即可拿到 \(100'\) 的好成绩.

const unsigned long long Mod(1000000009);
inline void Mn(unsigned long long& x) {x -= ((x >= Mod) ? Mod : 0);}
inline void Mn(unsigned& x) {x -= ((x >= Mod) ? Mod : 0);}
bitset<100005> Flp;
unsigned a[100005][2], Pos[100005], g[100005][5];
unsigned long long mm, M1, M2, M3, M4;
unsigned OpV, OpP;
unsigned m, n;
unsigned Cnt(0), Ans(0);
inline unsigned ColorTwo(unsigned x) {return a[x][Flp[x] ^ 1];}
inline unsigned ColorOne(unsigned x) {return a[x][Flp[x]];}
struct Node {
  Node *LS, *RS;
  unsigned Sum, Mul, Pls;
  inline void Build(unsigned L, unsigned R);
  inline void PsDw(unsigned long long Llen, unsigned long long Rlen) {
    if(Mul ^ 1) {
      LS->Mul = (unsigned long long)Mul * LS->Mul % Mod;
      LS->Pls = (unsigned long long)Mul * LS->Pls % Mod;
      LS->Sum = (unsigned long long)Mul * LS->Sum % Mod;
      RS->Mul = (unsigned long long)Mul * RS->Mul % Mod;
      RS->Pls = (unsigned long long)Mul * RS->Pls % Mod;
      RS->Sum = (unsigned long long)Mul * RS->Sum % Mod;
      Mul = 1;
    }
    if(Pls) {
      Mn(LS->Pls += Pls), Mn(RS->Pls += Pls);
      LS->Sum = (LS->Sum + Pls * Llen) % Mod;
      RS->Sum = (RS->Sum + Pls * Rlen) % Mod;
      Pls = 0;
    }
  }
  inline void Define(unsigned L, unsigned R) {
    if(L == R) {Mul = 1, Pls = 0, Sum = OpV; return;}
    unsigned Mid((L + R) >> 1);
    this->PsDw(Mid - L + 1, R - Mid);
    if(OpP <= Mid) LS->Define(L, Mid);
    else RS->Define(Mid + 1, R);
    Sum = LS->Sum + RS->Sum, Mn(Sum);
  }
  inline void Qry(unsigned L, unsigned R) {
    if(L == R) {OpV = Sum; return;}
    unsigned Mid((L + R) >> 1);
    this->PsDw(Mid - L + 1, R - Mid);
    if(OpP <= Mid) LS->Qry(L, Mid);
    else RS->Qry(Mid + 1, R);
  }
}N[200005], *CntN(N);
inline void Node::Build(unsigned L, unsigned R) {
  Mul = 1, Pls = 0, Sum = 0;
  if(L == R) return;
  unsigned Mid((L + R) >> 1);
  (LS = ++CntN)->Build(L, Mid);
  (RS = ++CntN)->Build(Mid + 1, R);
}
inline unsigned long long Onesided(unsigned x) {
  if(!x) return 1; --x;
  return ((g[x][0] * M2 % Mod) * M3 + g[x][1] + g[x][2] + ((g[x][3] + g[x][4]) * M2 << 1)) % Mod;
}
inline void Multiple(unsigned long long x) {
  N->Mul = N->Mul * x % Mod;
  N->Pls = N->Pls * x % Mod;
  N->Sum = N->Sum * x % Mod;
}
inline void Add(unsigned long long x) {
  Mn(N->Pls += x);
  N->Sum = (N->Sum + m * x) % Mod;
}
inline void Set(unsigned x, unsigned y) {
  if(!x) return;
  OpP = x, OpV = y, N->Define(1, m);
}
inline unsigned long long Find(unsigned x) {
  if(!x) return 0;
  OpP = x, N->Qry(1, m);
  return OpV;
}
signed main() {
  n = RD(), mm = (m = RD()) - 1, mm = mm * m % Mod;
  M1 = m - 1, M2 = m - 2, M3 = m - 3, M4 = m - 4, N->Build(1, m);
  for (unsigned i(1); i <= n; ++i) a[i][0] = RD();
  for (unsigned i(1); i <= n; ++i) a[i][1] = RD();
  g[0][0] = g[0][2] = g[0][4] = 1, g[0][1] = g[0][3] = 0;
  for (unsigned i(1); i <= n; ++i) {
    g[i][0] = (g[i - 1][0] * ((M3 * M4 + 1) % Mod) + g[i - 1][1] + g[i - 1][2] + ((g[i - 1][3] + g[i - 1][4]) * M3 << 1) % Mod) % Mod;
    g[i][1] = ((g[i - 1][0] * M2 % Mod) * M3 + g[i - 1][2] + (g[i - 1][4] * M2 << 1)) % Mod;
    g[i][2] = ((g[i - 1][0] * M2 % Mod) * M3 + g[i - 1][1] + (g[i - 1][3] * M2 << 1)) % Mod;
    g[i][3] = ((g[i - 1][0] * M3 % Mod) * M3 + g[i - 1][2] + g[i - 1][3] * M2 + g[i - 1][4] * (M3 + M2)) % Mod;
    g[i][4] = ((g[i - 1][0] * M3 % Mod) * M3 + g[i - 1][1] + g[i - 1][4] * M2 + g[i - 1][3] * (M3 + M2)) % Mod;
  }
  for (unsigned i(1); i <= n; ++i) if(a[i][0] | a[i][1]) {Pos[++Cnt] = i; if(a[i][1]) Flp[Cnt] = 1;}
  for (unsigned i(1); i <= Cnt; ++i) a[i][0] = a[Pos[i]][0], a[i][1] = a[Pos[i]][1];
  if(!Cnt) {
    printf("%llu\n", (Onesided(n - 1) * m % Mod) * M1 % Mod);
    return 0;
  }
  unsigned CT(ColorTwo(1)), Val(Onesided(Pos[1] - 1));
  if(CT) Set(CT, Val);
  else Add(Val), Set(ColorOne(1), 0);
  for (unsigned i(2); i <= Cnt; ++i) {
    unsigned A(a[i - 1][0]), B(a[i - 1][1]), C(a[i][0]), D(a[i][1]), *Len(g[Pos[i] - Pos[i - 1] - 1]);
    if(Flp[i]) swap(A, B), swap(C, D), Flp[i - 1] = (Flp[i - 1] ^ 1);
    if(Flp[i - 1]) {
      if(B == C) {
        unsigned long long PTmp((unsigned long long)N->Sum * Len[4] % Mod);
        Multiple(Mod + Len[2] - Len[4]), Add(PTmp), Set(ColorOne(i), 0);
      } else {
        unsigned long long FC(Find(C)), PTmp(((Mod + N->Sum - FC) * Len[0] + FC * Len[3]) % Mod);
        unsigned long long AB((FC * Len[1] + (Mod + N->Sum - FC) * Len[3]) % Mod);
        Multiple(Mod + Len[4] - Len[0]), Add(PTmp), Set(C, 0), Set(B, AB);
      }
    } else {
      if(A == C) {
        unsigned long long PTmp((unsigned long long)N->Sum * Len[3] % Mod);
        Multiple(Mod + Len[1] - Len[3]), Add(PTmp), Set(ColorOne(i), 0);
      } else {
        unsigned long long FC(Find(C)), PTmp(((Mod + N->Sum - FC) * Len[0] + FC * Len[4]) % Mod);
        unsigned long long AB((FC * Len[2] + (Mod + N->Sum - FC) * Len[4]) % Mod);
        Multiple(Mod + Len[3] - Len[0]), Add(PTmp), Set(C, 0), Set(A, AB);
      }
    }
    if(D) {
      unsigned TmpF(Find(D));
      Multiple(0), Set(D, TmpF);
    }
  }
  CT = ColorOne(Cnt), Ans = N->Sum + Mod - Find(CT);
  printf("%llu\n", Ans * Onesided(n - Pos[Cnt]) % Mod);
  return Wild_Donkey;
}

Day \(13\) Apr 16, 2022, Saturday

SDOI2019 热闹的聚会和尴尬的聚会

因为一个方案中, 第一天和第二天是互相独立的, 因此我们要分别使得 \(p\) 和 \(q\) 最大. 如果 \(p\) 和 \(q\) 分别最大的时候也不能满足条件, 则无解.

对于 \(p\), 我们用单调队列从小到大维护每个点的度数, 每次取度最小的点删掉, 这个点的度数作为此时此刻还没删掉的点的集合的 \(p\), 维护一个 \(p\) 的最大值, 和达到这个最大值时删掉的点集, 输出时可以将没删掉的点输出.

\(q\) 的最大值显然是求最大独立集, 这个问题是经典 NPC, 显然不能去求 \(q\) 的最大值, 只能求一个近似解.

一开始想的是, 随便找出一个极大的独立集, 据说这样能过. 方法是: 随机一个排列, 依次选择, 如果一个点是已经被选的邻居, 则跳过, 否则选择.

但是发现每次选一个点, 都会使它当前度数个点不能被选择, 所以感性地, 每次选择度数最小的点. 这样可以证明是一定有解的.

证明是这样的: 如果第 \(i\) 次选择的点使 \(Cg_i\) 个点从可以被选择变成了不能被选择, 一共选则了 \(q\) 个. 那么一定有 \(\sum_{i = 1}^q (Cg_i + 1) = n\).

我们知道, 如果第 \(i\) 个被选择的点使 \(Cg_i\) 个点从可以被选择变成了不能被选择, 那么这个点当时的度数一定大于等于 \(Cg_i\), 而每个点被扩展的时候, 一定会更新 \(p\), 也就是说 \(p \geq Cg_i\).

我们需要满足的条件是 \(\lfloor \frac{n}{p + 1}\rfloor \leq q\), \(\lfloor \frac{n}{q + 1}\rfloor \leq p\), 转化为 \(\lfloor \frac{n}{p + 1}\rfloor < q + 1\), \(\lfloor \frac{n}{q + 1}\rfloor < p + 1\), 最后是 \(n < (q + 1)(p + 1) + n \% (p + 1)\), \(n < (q + 1)(p + 1) + n \% (q + 1)\). 一个更严格的界是 \(n < (q + 1)(p + 1)\), 拆括号可得 \(n < pq + p + q + 1\).

因为 \(p \geq Cg_i\), 所以 \(pq + q \geq \sum_{i = 1}^q (Cg_i + 1) = n\), 因此 \(pq + q + p \geq n\), \(pq + q + p + 1 > n\).

证毕.

因为求独立集的过程也是每次找一个度数最小的点, 因此可以从求 \(p\) 的过程中同时处理.

struct Node {
  vector<Node*> E;
  Node**Pos; 
  unsigned Deg, Ava;
}N[10005], *Stack[10005], **Head[10005];
vector <Node*> Buck[10005];
unsigned Arr[10005], Cnt(0);
unsigned m, n;
unsigned A, B, C, D, t;
unsigned Ans(0), Tmp(0);
inline void Clr() {
  for (unsigned i(1); i <= n; ++i) N[i].E.clear(), N[i].Ava = 0;
  for (unsigned i(1); i <= n; ++i) Buck[i].clear();
  n = RD(), m = RD(), Head[0] = Stack;
}
signed main() {
  t = RD();
  for (unsigned T(1); T <= t; ++T){
    Clr();
    for (unsigned i(1); i <= m; ++i) 
      A = RD(), B = RD(), N[A].E.push_back(N + B), N[B].E.push_back(N + A);
    for (unsigned i(1); i <= n; ++i) Buck[N[i].Deg = N[i].E.size()].push_back(N + i);
    for (unsigned i(1); i <= n; ++i) {
      Head[i] = Head[i - 1]; 
      for (auto j:Buck[i]) *(++Head[i]) = j, j->Pos = Head[i];
    }
    Ans = 0, A = 1, Cnt = 0;
    for (unsigned i(1); i <= n; ++i) {
      Node* Cur(Stack[i]);
      if(Ans < Cur->Deg) A = i, Ans = Cur->Deg;
      if(!(Cur->Ava)) Arr[++Cnt] = Cur - N;
      for (auto j:Cur->E) if(j->Deg) {
        j->Ava |= (Cur->Ava ^ 1);
        swap(*(++Head[--(Cur->Deg)]), *(Cur->Pos));
        swap(Cur->Pos, (*(Cur->Pos))->Pos);
        swap(*(++Head[--(j->Deg)]), *(j->Pos));
        swap(j->Pos, (*(j->Pos))->Pos);
      }
    }
    printf("%u", n - A + 1);
    for (unsigned i(A); i <= n; ++i) printf(" %u", Stack[i] - N); putchar(0x0A);
    printf("%u", Cnt);
    for (unsigned i(1); i <= Cnt; ++i) printf(" %u", Arr[i]); putchar(0x0A);
  }
  return Wild_Donkey;
}

Day \(14\) Apr 17, 2022, Sunday

今天模拟省选联考 2022, 起晚了, 少考了半个多小时. 先看了题, T1 不愧是个模拟, 不过没有那么恐怖的样子, T2 还算正常, T3 一眼看出超出能力范围.

所以先做 T2. 如果枚举一个值域区间, 使得所有非零的点值都在区间内, 我们可以进行树形 DP 就可以 \(O(n)\) 求出我们想要的东西. 一开始把题读错了, 读成了只要能走就不能停, 这个比原问题还要强, 方案数和总和各需要多记一个状态, 用了一段时间竟然干出来了. 发现读错题了, 删掉两个状态搞出来了.

一开始是离散化容斥, 结果被第二个样例卡掉. 改成值域枚举发现还是错, 因为我是按值域长度和 \(K\) 的关系容斥的, 仔细思考发现不用枚举每一个长度的区间, 只要对长度为 \(K + 1\) 和 \(K\) 的区间进行容斥, 求 \(O(V)\) 个区间的答案即可求出. 这就是 \(O(nV)\) 的做法.

PrSl2022 预处理器

T2 过大样例之和开始做 T1.

听了这道题的臭名声, 本来以为是个折磨大模拟, 没想到一会就写完了, 小调过大样例. 果然人越菜越会叫唤.

拒绝 string, 哈希天下第一, 远离超时.

inline char Judge(char x) {
  if(x >= 'a' && x <= 'z') return 1;
  if(x >= 'A' && x <= 'Z') return 1;
  if(x >= '0' && x <= '9') return 1;
  return x == '_'; 
}
struct Modle {
  unsigned Leng;
  char Cond, Conte[105];
}List[105];
unordered_map<unsigned long long, unsigned> M;
unsigned n, Cnt(0);
char Cur[105];
inline void Open(char* f, unsigned Len) {
  unsigned long long Hash(0);
  unsigned j(0);
  for(unsigned i(0); i < Len; ++i) {
    if(Judge(f[i])) Hash *= Base, Hash += f[i];
    else {
      unsigned Me(M[Hash]);
      if((!Me) || (List[Me].Cond)) while (j <= i) putchar(f[j++]);
      else {
        List[Me].Cond = 1, Open(List[Me].Conte, List[Me].Leng);
        List[Me].Cond = 0, j = i + 1, putchar(f[i]);
      }
      Hash = 0;
    }
  }
  unsigned Me(M[Hash]);
  if((!Me) || (List[Me].Cond)) while (j < Len) putchar(f[j++]);
  else {
    List[Me].Cond = 1, Open(List[Me].Conte, List[Me].Leng);
    List[Me].Cond = 0;
  }
}
int main() {
  n = RD();
  for (unsigned i(1); i <= n; ++i) {
    memset(Cur, 0, sizeof(Cur));
    cin.getline(Cur, 102);
    unsigned Len(strlen(Cur));
    if(Cur[0] == '#') {
      unsigned long long Hash(0);
      unsigned Pos;
      if(Cur[1] == 'd') {
        Pos = 8;
        while (Cur[Pos] != ' ') Hash *= Base, Hash += Cur[Pos++];
        M[Hash] = ++Cnt, ++Pos;
        while (Pos < Len) List[Cnt].Conte[(List[Cnt].Leng)++] = Cur[Pos++];
      } else {
        Pos = 7;
        while (Pos < Len) Hash *= Base, Hash += Cur[Pos++];
        List[M[Hash]].Cond = 1;
      }
    } else Open(Cur, Len);
    putchar(0x0A);
  }
}

Day \(15\) Apr 18, 2022, Monday

模拟省选 2022 Day 2, 今天上来就发现之前快读写错了:

inline unsigned RD() {
  unsigned RTmp(0);
  char ch(getchar()); 
  while (ch < '0' && ch > '9') ch = getchar();
  while (ch >= '0' && ch <= '9') RTmp = RTmp * 10 + ch - '0', ch = getchar(); 
  return RTmp;
}

这个地方

while (ch < '0' && ch > '9') ch = getchar();

Day 1 得分玄学了.

今天感冒了, 半个小时读完题, 感觉可以先搞一搞 T2, 于是睡了两个小时.

起来以后发现 T2 的所有权值相等和 \(x = 0\) 可以做. 对于 \(x = y = 1\) 的情况, 我本来写了一个自认为是正确的算法, 但是和大样例差了一点点.

T1 我写了个爆搜容斥, 可能有个 \(10\) 分 \(20\) 分的吧.

T3 没看.