Problem:
Score : 600 points

Problem Statement

There is a grid with N rows and N columns of squares. Let (i, j) be the square at the i-th row from the top and the j-th column from the left.

Each of the central (N-2) \times (N-2) squares in the grid has a black stone on it.
Each of the 2N - 1 squares on the bottom side and the right side has a white stone on it.

Q queries are given. We ask you to process them in order.
There are two kinds of queries. Their input format and description are as follows:

1 x: Place a white stone on (1, x). After that, for each black stone between (1, x) and the first white stone you hit if you go down from (1, x), replace it with a white stone.

2 x: Place a white stone on (x, 1). After that, for each black stone between (x, 1) and the first white stone you hit if you go right from (x, 1), replace it with a white stone.

How many black stones are there on the grid after processing all Q queries?

Constraints

3 \leq N \leq 2\times 10^5

0 \leq Q \leq \min(2N-4,2\times 10^5)

2 \leq x \leq N-1

Queries are pairwise distinct.

Input

Input is given from Standard Input in the following format:

N Q
Query_1
\vdots
Query_Q

Output

Print how many black stones there are on the grid after processing all Q queries.

Sample Input 1

5 5
1 3
2 3
1 4
2 2
1 2

Sample Output 1

1

After each query, the grid changes in the following way:

Sample Input 2

200000 0

Sample Output 2

39999200004

Sample Input 3

176527 15
1 81279
2 22308
2 133061
1 80744
2 44603
1 170938
2 139754
2 15220
1 172794
1 159290
2 156968
1 56426
2 77429
1 97459
2 71282

Sample Output 3

31159505795

Solution in C++ (GCC 9.2.1) (judge verdict: Accepted)

#include <bits/stdc++.h>

#ifdef DEBUG
#include <Mylib/Debug/debug.cpp>
#else
#define dump(...) ((void)0)
#endif

template <typename T, typename U>
bool chmin(T &a, const U &b){
  return (a > b ? a = b, true : false);
}

template <typename T, typename U>
bool chmax(T &a, const U &b){
  return (a < b ? a = b, true : false);
}

template <typename T, size_t N, typename U>
void fill_array(T (&a)[N], const U &v){
  std::fill((U*)a, (U*)(a + N), v);
}

template <typename T, size_t N, size_t I = N>
auto make_vector(const std::array<int, N> &a, T value = T()){
  static_assert(I >= 1);
  static_assert(N >= 1);
  if constexpr (I == 1){
    return std::vector<T>(a[N - I], value);
  }else{
    return std::vector(a[N - I], make_vector<T, N, I - 1>(a, value));
  }
}

template <typename T>
std::ostream& operator<<(std::ostream &s, const std::vector<T> &a){
  for(auto it = a.begin(); it != a.end(); ++it){
    if(it != a.begin()) s << " ";
    s << *it;
  }
  return s;
}

template <typename T>
std::istream& operator>>(std::istream &s, std::vector<T> &a){
  for(auto &x : a) s >> x;
  return s;
}


namespace haar_lib {
  template <typename Monoid>
  class dual_segment_tree {
    using value_type = typename Monoid::value_type;
    const static Monoid M;

    const int depth, size, hsize;
    std::vector<value_type> data;

    void propagate(int i){
      if(i < hsize){
        data[i << 1 | 0] = M(data[i], data[i << 1 | 0]);
        data[i << 1 | 1] = M(data[i], data[i << 1 | 1]);
        data[i] = M();
      }
    }

    void propagate_top_down(int i){
      std::vector<int> temp;
      while(i > 1){
        i >>= 1;
        temp.push_back(i);
      }

      for(auto it = temp.rbegin(); it != temp.rend(); ++it) propagate(*it);
    }

  public:
    dual_segment_tree(int n):
      depth(n > 1 ? 32 - __builtin_clz(n - 1) + 1 : 1),
      size(1 << depth), hsize(size / 2),
      data(size, M())
    {}

    void update(int l, int r, const value_type &x){
      propagate_top_down(l + hsize);
      propagate_top_down(r + hsize);

      int L = l + hsize;
      int R = r + hsize;

      while(L < R){
        if(R & 1) --R, data[R] = M(x, data[R]);
        if(L & 1) data[L] = M(x, data[L]), ++L;
        L >>= 1, R >>= 1;
      }
    }

    value_type operator[](int i){
      propagate_top_down(i + hsize);
      return data[i + hsize];
    }

    template <typename T>
    void init_with_vector(const std::vector<T> &a){
      data.assign(size, M());
      for(int i = 0; i < (int)a.size(); ++i) data[hsize + i] = a[i];
    }

    template <typename T>
    void init(const T &val){
      init_with_vector(std::vector<value_type>(hsize, val));
    }
  };
}

namespace haar_lib {
  template <typename T>
  struct min_monoid {
    using value_type = std::optional<T>;

    value_type operator()() const {return {};}
    value_type operator()(const value_type &a, const value_type &b) const {
      if(not a) return b;
      if(not b) return a;
      return {std::min(*a, *b)};
    }
  };
}


namespace haar_lib {}

namespace solver {
  using namespace haar_lib;

  constexpr int m1000000007 = 1000000007;
  constexpr int m998244353 = 998244353;

  void init(){
    std::cin.tie(0);
    std::ios::sync_with_stdio(false);
    std::cout << std::fixed << std::setprecision(12);
    std::cerr << std::fixed << std::setprecision(12);
    std::cin.exceptions(std::ios_base::failbit);
  }

  void solve(){
    int64_t N, Q; std::cin >> N >> Q;

    auto seg_row = dual_segment_tree<min_monoid<int>>(N);
    auto seg_col = dual_segment_tree<min_monoid<int>>(N);

    seg_row.init(N - 1);
    seg_col.init(N - 1);

    int64_t ans = (N - 2) * (N - 2);

    while(Q--){
      int type; std::cin >> type;
      int x; std::cin >> x;
      --x;

      if(type == 1){
        int y = seg_col[x].value();
        ans -= y - 1;
        seg_row.update(0, y, x);
      }else{
        int y = seg_row[x].value();
        ans -= y - 1;
        seg_col.update(0, y, x);
      }

      //dump(ans);
    }

    std::cout << ans << "\n";
  }
}

int main(){
  solver::init();
  while(true){
    try{
      solver::solve();
    }catch(const std::istream::failure &e){
      break;
    }
  }
  return 0;
}