洛谷 P3328 【[SDOI2015]音质检测】

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;
}