CF719E. Sasha and Array [线段树维护矩阵]

CF719E. Sasha and Array

题意：

对长度为 n 的数列进行 m 次操作, 操作为:

1. a[l..r] 每一项都加一个常数 C, 其中 0 ≤ C ≤ 10^9
2. 求 F[a[l]]+F[a[l+1]]+...F[a[r]] mod 1e9+7 的余数

---

矩阵快速幂求斐波那契

矩阵满足乘法分配律和结合律！

所以可以每个节点维护矩阵/矩阵和，区间加相当于区间乘矩阵

注意：不要把快速幂写在里面，复杂度平添一个log。把\(B^C\)算出来之后传进去就好了

#include 
#include 
#include 
#include 
using namespace std;
typedef long long ll;
#define lc x<<1
#define rc x<<1|1
#define mid ((l+r)>>1)
#define lson lc, l, mid
#define rson rc, mid+1, r
const int N = 1e5+5, P = 1e9+7;

int n, m;
struct Matrix {
    ll a[2][2];
    ll* operator [](int x) {return a[x];}
    Matrix(int p=0) {
        if(!p) a[0][0] = a[0][1] = a[1][0] = a[1][1] = 0;
        else a[0][0] = a[1][1] = 1, a[0][1] = a[1][0] = 0;
    }
} B;

Matrix operator + (Matrix a, Matrix b) {
    Matrix c;
    for(int i=0; i<2; i++)
        for(int j=0; j<2; j++)
            c[i][j] = (a[i][j] + b[i][j]) %P;
    return c;
}

Matrix operator * (Matrix a, Matrix b) {
    Matrix c;
    for(int i=0; i<2; i++)
        for(int j=0; j<2; j++) {
            ll &x = c[i][j];
            for(int k=0; k<2; k++) 
                x = (x + a[i][k] * b[k][j] %P) %P;
        }
    return c;
}

Matrix operator ^ (Matrix a, int b) {
    Matrix ans(1); 
    ans[0][0] = ans[1][1] = 1;
    for(; b; b>>=1, a=a*a)
        if(b & 1) ans = ans*a;
    return ans;
}

struct meow {
    Matrix f;
    Matrix v;
    int c;
    meow() {f[0][0] = 1; v[0][0] = v[1][1] = 1;}
} t[N<<2];

void paint(int x, int l, int r, int d, Matrix &v) {
    t[x].c += d;
    t[x].v = v * t[x].v;
    t[x].f = v * t[x].f;
}
void push_down(int x, int l, int r) {
    if(t[x].c) {
        paint(lson, t[x].c, t[x].v);
        paint(rson, t[x].c, t[x].v);
        t[x].c = 0;
        t[x].v = Matrix(1);
    }
}
void merge(int x) {
    t[x].f = t[lc].f + t[rc].f;
}

void build(int x, int l, int r) {
    if(l == r) {
        cin >> t[x].c;
        if(t[x].c > 1) t[x].f = (B ^ (t[x].c - 1)) * t[x].f;
    } else {
        build(lson);
        build(rson);
        merge(x);
    }
}

void Add(int x, int l, int r, int ql, int qr, int d, Matrix &v) {
    if(ql <= l && r <= qr) paint(x, l, r, d, v);
    else {
        push_down(x, l, r);
        if(ql <= mid) Add(lson, ql, qr, d, v);
        if(mid < qr)  Add(rson, ql, qr, d,v );
        merge(x);
    }
}

ll Que(int x, int l, int r, int ql, int qr) {
    if(ql <= l && r <= qr) return t[x].f[0][0];
    else {
        push_down(x, l, r);
        ll ans = 0;
        if(ql <= mid) ans = (ans + Que(lson, ql, qr)) %P;
        if(mid < qr)  ans = (ans + Que(rson, ql, qr)) %P;
        return ans;
    }
}

int main() {
    //freopen("in", "r", stdin);
    ios::sync_with_stdio(false); cin.tie(); cout.tie();

    B[0][0] = B[0][1] = B[1][0] = 1;
    cin >> n >> m;
    build(1, 1, n);


    for(int i=1; i<=m; i++) {
        int tp, l, r, x;
        cin >> tp >> l >> r;
        if(tp == 1) {
            cin >> x;
            Matrix t = B^x;
            Add(1, 1, n, l, r, x, t);
        } else cout << Que(1, 1, n, l, r) << '\n';
    }