yukicoder No.879 Range Mod 2 Query

Struct generator is available on Github now.
github.com

The lazy information on LZ segtree usually has linearity. So, it can be expressed by a matrix. This time, we prepare a vector having (value, odd or not, even or not, 1). The first operation can be written as

\begin{align}
\begin{pmatrix}
0 & 1 & 0 & 0 \\
0 & 1 & 0 & 0 \\
0 & 0 & 1 & 0 \\
0 & 0 & 0 & 1
\end{pmatrix}
\begin{pmatrix}
x_{\textrm{sum}} \\
x_{\textrm{odd}} \\
x_{\textrm{even}} \\
x_{\textrm{size}}
\end{pmatrix}
=
\begin{pmatrix}
x_{\textrm{odd}} \\
x_{\textrm{odd}} \\
x_{\textrm{even}} \\
x_{\textrm{size}}
\end{pmatrix}
\end{align}

the second operation can be written as

\begin{align}
\begin{pmatrix}
1 & 0 & 0 & x \\
0 & 0 & 1 & 0 \\
0 & 1 & 0 & 0 \\
0 & 0 & 0 & 1
\end{pmatrix}
\begin{pmatrix}
x_{\textrm{sum}} \\
x_{\textrm{odd}} \\
x_{\textrm{even}} \\
x_{\textrm{size}}
\end{pmatrix}
=
\begin{pmatrix}
x_{\textrm{sum}}+x * x_{\textrm{size}} \\
x_{\textrm{even}} \\
x_{\textrm{odd}} \\
x_{\textrm{size}}
\end{pmatrix}
\end{align}

if x is odd, and

\begin{align}
\begin{pmatrix}
1 & 0 & 0 & x \\
0 & 1 & 0 & 0 \\
0 & 0 & 1 & 0 \\
0 & 0 & 0 & 1
\end{pmatrix}
\begin{pmatrix}
x_{\textrm{sum}} \\
x_{\textrm{odd}} \\
x_{\textrm{even}} \\
x_{\textrm{size}}
\end{pmatrix}
=
\begin{pmatrix}
x_{\textrm{sum}}+x*x_{\textrm{size}} \\
x_{\textrm{odd}} \\
x_{\textrm{even}} \\
x_{\textrm{size}}
\end{pmatrix}
\end{align}

if x is even. We can solve it by using this fact, but it may be inefficient regarding as time and space. For this reason, I made a generator which takes matrices and automatically produce structs having the same meaning of given matricies. More details is written in the github.

The following code is a solution for this problem. All of the definitions of mat and vec are automatically generated.

4 IntPlusTimes

typeSwap
0 1 0 0
0 1 0 0
0 0 1 0
0 0 0 1

typePlusOdd
1 0 0 *
0 0 1 0
0 1 0 0
0 0 0 1

typePlusEven
1 0 0 *
0 1 0 0
0 0 1 0
0 0 0 1

#include <bits/stdc++.h>
 
#define rep(i, n) for (int i = 0; i < (n); i++)
#define repr(i, n) for (int i = (n) - 1; i >= 0; i--)
#define range(a) a.begin(), a.end()
 
using namespace std;
using ll = long long;

struct semiring {
  long long x;
  semiring(long long x_ = 0) : x(x_) {}
  static semiring zero() { return semiring(0); }
  static semiring one() { return semiring(1); }
  friend semiring operator+(semiring a, semiring b) { return semiring(a.x + b.x); }
  friend semiring operator*(semiring a, semiring b) { return semiring(a.x * b.x); }
};
struct vec {
  semiring x0;
  semiring x1;
  semiring x2;
  semiring x3;
};
vec operator+(vec a, vec b) {
  a.x0 = a.x0 + b.x0;
  a.x1 = a.x1 + b.x1;
  a.x2 = a.x2 + b.x2;
  a.x3 = a.x3 + b.x3;
  return a;
}
struct mat {
  int k;
  semiring x0;
};
mat operator*(mat a, mat b) {
  mat c;
  if (a.k == 0 && b.k == 0) { c.k = 2; c.x0 = a.x0; }
  if (a.k == 0 && b.k == 1) { c.k = 0; c.x0 = a.x0; }
  if (a.k == 0 && b.k == 2) { c.k = 0; c.x0 = a.x0; }
  if (a.k == 0 && b.k == 3) { c.k = 2; c.x0 = a.x0; }
  if (a.k == 0 && b.k == 4) { c.k = 0; c.x0 = a.x0; }
  if (a.k == 0 && b.k == 5) { c.k = 2; c.x0 = a.x0; }
  if (a.k == 1 && b.k == 0) { c.k = 3; c.x0 = a.x0; }
  if (a.k == 1 && b.k == 1) { c.k = 1; c.x0 = a.x0; }
  if (a.k == 1 && b.k == 2) { c.k = 1; c.x0 = a.x0; }
  if (a.k == 1 && b.k == 3) { c.k = 3; c.x0 = a.x0; }
  if (a.k == 1 && b.k == 4) { c.k = 1; c.x0 = a.x0; }
  if (a.k == 1 && b.k == 5) { c.k = 3; c.x0 = a.x0; }
  if (a.k == 2 && b.k == 0) { c.k = 0; c.x0 = a.x0; }
  if (a.k == 2 && b.k == 1) { c.k = 2; c.x0 = a.x0; }
  if (a.k == 2 && b.k == 2) { c.k = 2; c.x0 = a.x0; }
  if (a.k == 2 && b.k == 3) { c.k = 0; c.x0 = a.x0; }
  if (a.k == 2 && b.k == 4) { c.k = 2; c.x0 = a.x0; }
  if (a.k == 2 && b.k == 5) { c.k = 0; c.x0 = a.x0; }
  if (a.k == 3 && b.k == 0) { c.k = 1; c.x0 = a.x0; }
  if (a.k == 3 && b.k == 1) { c.k = 3; c.x0 = a.x0; }
  if (a.k == 3 && b.k == 2) { c.k = 3; c.x0 = a.x0; }
  if (a.k == 3 && b.k == 3) { c.k = 1; c.x0 = a.x0; }
  if (a.k == 3 && b.k == 4) { c.k = 3; c.x0 = a.x0; }
  if (a.k == 3 && b.k == 5) { c.k = 1; c.x0 = a.x0; }
  if (a.k == 4 && b.k == 0) { c.k = 0; c.x0 = b.x0 + a.x0; }
  if (a.k == 4 && b.k == 1) { c.k = 1; c.x0 = b.x0 + a.x0; }
  if (a.k == 4 && b.k == 2) { c.k = 2; c.x0 = b.x0 + a.x0; }
  if (a.k == 4 && b.k == 3) { c.k = 3; c.x0 = b.x0 + a.x0; }
  if (a.k == 4 && b.k == 4) { c.k = 4; c.x0 = b.x0 + a.x0; }
  if (a.k == 4 && b.k == 5) { c.k = 5; c.x0 = b.x0 + a.x0; }
  if (a.k == 5 && b.k == 0) { c.k = 1; c.x0 = b.x0 + a.x0; }
  if (a.k == 5 && b.k == 1) { c.k = 0; c.x0 = b.x0 + a.x0; }
  if (a.k == 5 && b.k == 2) { c.k = 3; c.x0 = b.x0 + a.x0; }
  if (a.k == 5 && b.k == 3) { c.k = 2; c.x0 = b.x0 + a.x0; }
  if (a.k == 5 && b.k == 4) { c.k = 5; c.x0 = b.x0 + a.x0; }
  if (a.k == 5 && b.k == 5) { c.k = 4; c.x0 = b.x0 + a.x0; }
  return c;
}
vec operator*(mat a, vec b) {
  vec c;
  if (a.k == 0) { c.x0 = b.x2 + a.x0*b.x3; c.x1 = b.x2; c.x2 = b.x1; c.x3 = b.x3; }
  if (a.k == 1) { c.x0 = b.x2 + a.x0*b.x3; c.x1 = b.x1; c.x2 = b.x2; c.x3 = b.x3; }
  if (a.k == 2) { c.x0 = b.x1 + a.x0*b.x3; c.x1 = b.x1; c.x2 = b.x2; c.x3 = b.x3; }
  if (a.k == 3) { c.x0 = b.x1 + a.x0*b.x3; c.x1 = b.x2; c.x2 = b.x1; c.x3 = b.x3; }
  if (a.k == 4) { c.x0 = b.x0 + a.x0*b.x3; c.x1 = b.x1; c.x2 = b.x2; c.x3 = b.x3; }
  if (a.k == 5) { c.x0 = b.x0 + a.x0*b.x3; c.x1 = b.x2; c.x2 = b.x1; c.x3 = b.x3; }
  return c;
}
mat identity() {
  mat a;
  a.k = 4;
  a.x0 = semiring::zero();
  return a;
}
mat typeA() {
  mat a;
  a.k = 2;
  a.x0 = semiring::zero();
  return a;
}
mat typeB(semiring x0) {
  mat a;
  a.k = 5;
  a.x0 = x0;
  return a;
}
mat typeC(semiring x0) {
  mat a;
  a.k = 4;
  a.x0 = x0;
  return a;
}


constexpr int U = 1 << 17;
vec dat[U * 2];
mat lazy[U * 2];

void apply(int k, mat v) {
  lazy[k] = v * lazy[k];
  dat[k] = v * dat[k];
}

void push(int k) {
  apply(k * 2 + 0, lazy[k]);
  apply(k * 2 + 1, lazy[k]);
  lazy[k] = identity();
}

void pull(int k) {
  dat[k] = dat[k * 2] + dat[k * 2 + 1];
}

void update(int a, int b, mat v, int k = 1, int l = 0, int r = U) {
  if (r <= a || b <= l) return;
  if (a <= l && r <= b) {
    apply(k, v);
    return;
  }
  push(k);
  int m = (l + r) / 2;
  update(a, b, v, k * 2 + 0, l, m);
  update(a, b, v, k * 2 + 1, m, r);
  pull(k);
}

vec query(int a, int b, int k = 1, int l = 0, int r = U) {
  if (r <= a || b <= l) return vec{};
  if (a <= l && r <= b) return dat[k];
  push(k);
  int m = (l + r) / 2;
  vec vl = query(a, b, k * 2 + 0, l, m);
  vec vr = query(a, b, k * 2 + 1, m, r);
  return vl + vr;
}

int main() {
  cin.tie(nullptr);
  ios::sync_with_stdio(false);
  int N, Q; cin >> N >> Q;
  vector<int> A(N);
  rep(i, U * 2) lazy[i] = identity();
  rep(i, N) {
    cin >> A[i];
    dat[i + U].x0 = A[i];
    dat[i + U].x1 = A[i] % 2;
    dat[i + U].x2 = (A[i] + 1) % 2;
    dat[i + U].x3 = 1;
  }
  for (int i = U - 1; i >= 1; i--) pull(i);
  while (Q--) {
    int type; cin >> type;
    if (type == 1) {
      int l, r; cin >> l >> r; l--;
      update(l, r, typeA());
    } else if (type == 2) {
      int l, r, x; cin >> l >> r >> x; l--;
      if (x % 2 == 1) {
        update(l, r, typeB(x));
      } else {
        update(l, r, typeC(x));
      }
    } else {
      int l, r; cin >> l >> r; l--;
      vec ans = query(l, r);
      cout << ans.x0.x << '\n';
    }
  }
}