这题我做的好麻烦啊。。。
一开始想分块来着，后来发现可以直接线段树
首先考虑一个性质，我们如果有数列的相邻两项f[i]和 f[i+1]那么用这两项向后推k项其线性表示系数一定（表示为f[i+k]=a∗f[i]+b∗f[i+1]+c的形式)，那么这样我们预处理这些系数，注意到维护的是一个乘积的形式，那么我们要维护这个必须得维护8个量，将其写成3 * 3矩阵的形式转移会比较科学,注意a=0的特判。
说实话网上有些做法感觉很不科学啊。。。
比如很多人初始化线段树的时候都暴力求的f函数，感觉不太科学啊。。。我的做法是BSGS预处理矩阵，这样查询单点就从33∗log变为33，不过我跑的好慢啊。。。还有就是标记下传的问题，网上竟然有下传是O(标记数）的做法，感觉和暴力没啥区别。
复杂度：O(Qlogn∗34+Maxval−−−−−−−√∗33)
上code:
#include<iostream>
#include<cstring>
#include<string>
#include<cstdio>
#include<cmath>
#define N 300002
#define M 50002
using namespace std;
typedef long long LL;
struct Mat
{
int g[3][3];
};
const int P = 1e9 + 7;
int n,Q,a,b,A[N][4],inva,tag[N << 2],B;
Mat Seg[N << 2];
inline void in(int &x)
{
char c;
while (!isdigit(c = getchar()));
x = (c ^ 48);
while (isdigit(c = getchar())) x = 10 * x + (c ^ 48);
}
inline void inc(int &x,int y)
{
x += y;
if (x >= P) x -= P;
}
inline void init()
{
in(n); in(Q);
in(a); in(b);
a %= P; b %= P;
for (int i = 1;i <= n; ++i)
in(A[i][2]);
}
int coe[M][3],recoe[M][3];
inline int quick(int x,int y)
{
int res = 1,base = x;
for (;y;y >>= 1)
{
if (y & 1) res = 1LL * base * res % P;
base = 1LL * base * base % P;
}
return res;
}
inline void Special()
{
for (int i = 3;i < M - 1; ++i)
{
recoe[i][0] = recoe[i - 1][0];
recoe[i][1] = recoe[i - 1][1];
recoe[i][2] = (recoe[i - 1][2] + P - b) % P;
}
}
inline void Get_Coe()
{
inva = quick(a,P - 2);
coe[0][0] = 0; coe[0][1] = 0; coe[0][2] = 1;
coe[1][0] = 0; coe[1][1] = 1; coe[1][2] = 0;
coe[2][0] = 1; coe[2][1] = 0; coe[2][2] = 0;
for (int i = 3;i < M - 1; ++i)
{
coe[i][0] = (coe[i - 1][0] + 1LL * coe[i - 2][0] * a % P) % P;
coe[i][1] = (coe[i - 1][1] + 1LL * coe[i - 2][1] * a % P) % P;
coe[i][2] = ((coe[i - 1][2] + 1LL * coe[i - 2][2] * a % P) % P + b) % P;
}
recoe[0][0] = 0; recoe[0][1] = 0; recoe[0][2] = 1;
recoe[1][0] = 1; recoe[1][1] = 0; recoe[1][2] = 0;
recoe[2][0] = 0; recoe[2][1] = 1; recoe[2][2] = 0;
if (!a)
{
Special();
return;
}
for (int i = 3;i < M - 1; ++i)
{
recoe[i][0] = 1LL * inva * (( -recoe[i - 1][0] + recoe[i - 2][0] + P) % P) % P;
recoe[i][1] = 1LL * inva * (( -recoe[i - 1][1] + recoe[i - 2][1] + P) % P) % P;
recoe[i][2] = 1LL * inva * ((( -recoe[i - 1][2] + recoe[i - 2][2] + P) % P + P - b) % P) % P;
}
}
inline Mat mul(Mat x,Mat y)
{
Mat c;
memset(c.g,0,sizeof(c.g));
for (int k = 0;k < 3; ++k)
for (int i = 0;i < 3; ++i)
for (int j = 0;j < 3; ++j)
if (x.g[i][k]&&y.g[k][j])
inc(c.g[i][j],1LL * x.g[i][k] * y.g[k][j] % P);
return c;
}
Mat Small[M],Big[M];
inline void Mat_init(Mat &x)
{
for (int i = 0;i < 3; ++i)
for (int j = 0;j < 3; ++j)
x.g[i][j] = (i == j);
}
inline void BSGS()
{
B = (int)(sqrt(2000000000)) + 5;
Mat now;
now.g[0][0] = 1; now.g[0][1] = 1; now.g[0][2] = 0;
now.g[1][0] = a; now.g[1][1] = 0; now.g[1][2] = 0;
now.g[2][0] = 1; now.g[2][1] = 0; now.g[2][2] = 1;
Mat_init(Small[0]);
for (int i = 1;i <= B; ++i)
Small[i] = mul(Small[i - 1],now);
now = Small[B];
Mat_init(Big[0]);
for (int i = 1;i <= B; ++i)
Big[i] = mul(Big[i - 1],now);
}
inline void update(int rt)
{
for (int i = 0;i < 3; ++i)
for (int j = 0;j < 3; ++j)
Seg[rt].g[i][j] = (Seg[rt << 1].g[i][j] + Seg[rt << 1|1].g[i][j]) % P;
}
Mat fu;
inline int add_Mul_it(int p,int q)
{
int S = 0;
for (int i = 0;i < 3; ++i)
for (int j = 0;j < 3; ++j)
inc(S,1LL * coe[p][i] * coe[q][j] % P * fu.g[i][j] % P);
return S;
}
inline int dec_Mul_it(int p,int q)
{
int S = 0;
for (int i = 0;i < 3; ++i)
for (int j = 0;j < 3; ++j)
inc(S,1LL * recoe[p][i] * recoe[q][j] % P * fu.g[i][j] % P);
return S;
}
inline void Deal(int l,int r,int rt,bool kind,int left,int right)
{
fu = Seg[rt];
memset(Seg[rt].g,0,sizeof(Seg[rt].g));
if (!kind)
{
for (int i = 0;i < 3; ++i)
for (int j = 0;j < 3; ++j)
{
int p = (i == 2) ? 0 : (left + 2 - i),q = (j == 2) ? 0 : (right + 2 - j);
Seg[rt].g[i][j] = add_Mul_it(p,q);
}
return;
}
for (int i = 0;i < 3; ++i)
for (int j = 0;j < 3; ++j)
{
int p = (i == 2) ? 0 : (left + i + 1),q = (j == 2) ? 0 : (right + j + 1);
Seg[rt].g[i][j] = dec_Mul_it(p,q);
}
}
inline void pushdown(int l,int r,int rt)
{
if (tag[rt])
{
int mid = (r + l) >> 1;
bool pd = (tag[rt] < 0);
int x = (tag[rt] > 0) ? tag[rt] : -tag[rt];
Deal(l,mid,rt << 1,pd,x,x);
Deal(mid + 1,r,rt << 1|1,pd,x,x);
tag[rt << 1] += tag[rt]; tag[rt << 1|1] += tag[rt];
}
tag[rt] = 0;
}
inline void build(int l,int r,int rt)
{
int mid = (r + l) >> 1;
tag[rt] = 0;
if (l == r)
{
if (l == 1 || l == n)
{
memset(Seg[rt].g,0,sizeof(Seg[rt].g));
return;
}
for (int i = 0;i < 3; ++i)
for (int j = 0;j < 3; ++j)
{
int p = (i == 2) ? 1 : A[l - 1][3 - i],q = (j == 2) ? 1 : A[l + 1][j ^ 1];
Seg[rt].g[i][j] = 1LL * p * q % P;
}
return;
}
build(l,mid,rt << 1);
build(mid + 1,r,rt << 1|1);
update(rt);
}
inline void change(int l,int r,int rt,int ll,int rr,bool kind,int left,int right)
{
int mid = (r + l) >> 1;
if (ll <= l&&rr >= r)
{
Deal(l,r,rt,kind,left,right);
(kind) ? --tag[rt] : ++tag[rt];
return;
}
pushdown(l,r,rt);
if (ll <= mid) change(l,mid,rt << 1,ll,rr,kind,left,right);
if (rr > mid) change(mid + 1,r,rt << 1|1,ll,rr,kind,left,right);
update(rt);
}
inline void change1(int l,int r,int rt,int pos,bool kind,int left,int right)
{
if (pos == 1||pos == n) return;
int mid = (r + l) >> 1;
if (l == r)
{
Deal(l,r,rt,kind,left,right);
return;
}
pushdown(l,r,rt);
if (pos <= mid) change1(l,mid,rt << 1,pos,kind,left,right);
if (pos > mid) change1(mid + 1,r,rt << 1|1,pos,kind,left,right);
update(rt);
}
inline int query(int l,int r,int rt,int ll,int rr)
{
if (ll > rr) return 0;
int mid = (r + l) >> 1;
if (ll <= l&&rr >= r)
return Seg[rt].g[0][0];
pushdown(l,r,rt);
int SS = 0;
if (ll <= mid) inc(SS,query(l,mid,rt << 1,ll,rr));
if (rr > mid) inc(SS,query(mid + 1,r,rt << 1|1,ll,rr));
update(rt);
return SS;
}
inline int Calc(int x)
{
if (x <= 2) return x;
int p = (x - 2) / B,q = (x - 2) % B;
Mat c = mul(Big[p],Small[q]);
int Sum = 0;
inc(Sum,2LL * c.g[0][0] % P);
inc(Sum,c.g[1][0]);
inc(Sum,1LL * b * c.g[2][0] % P);
return Sum;
}
inline void PRE()
{
Get_Coe();
BSGS();
for (int i = 1;i <= n; ++i)
A[i][0] = Calc(A[i][2] - 2),
A[i][1] = Calc(A[i][2] - 1),
A[i][3] = Calc(A[i][2] + 1),
A[i][2] = Calc(A[i][2]);
build(1,n,1);
}
int L,R;
inline void Plus(bool pd)
{
if (R == L)
{
if (L > 1) change1(1,n,1,L - 1,pd,0,1);
if (R < n) change1(1,n,1,R + 1,pd,1,0);
return;
}
if (R - L > 1)
change(1,n,1,L + 1,R - 1,pd,1,1);
change1(1,n,1,L,pd,0,1);
if (L > 1) change1(1,n,1,L - 1,pd,0,1);
change1(1,n,1,R,pd,1,0);
if (R < n) change1(1,n,1,R + 1,pd,1,0);
}
inline void QUERY()
{
char ch[10];
for (int i = 1;i <= Q; ++i)
{
scanf("%s",ch);
in(L); in(R);
if (ch[0] == 'p') Plus(0);
if (ch[0] == 'm') Plus(1);
if (ch[0] == 'q')
printf("%d\n",query(1,n,1,L + 1,R - 1));
}
}
inline void DO_IT()
{
PRE();
QUERY();
}
int main()
{
init();
DO_IT();
return 0;
}