智乃酱的静态数组维护问题多项式（拉格朗日插值求高阶差分）

题目链接

智乃酱的静态数组维护问题多项式

题目大意

有 $n$ 个数 $a[1],a[2],\cdots ,a[n]$.

$m$ 次操作，每次操作给出一个多项式函数 $f(x)=\begin{array}{r}\sum _{i=0}^k{c}_i{x}^i\end{array}$ 和区间 $[l,r]$ 表示进行操作 $a[i]+=f(i-l+1)i\in [l,r]$.

$m$ 次操作之后进行 $q$ 次询问，每次查询区间 $[l,r]$ 的和.

$1\le n,m,q\le 10^5,0\le a[i],c_i\le 10^9,0\le k\le 5,1\le l\le r\le n$.

分析

可以把多项式函数 $f(x)$ 按照 $x^i$ 拆开分别考虑.

对每个 $x^i$ 做 $6$ 次差分，记差分数组为 $d$，不难发现 $d[l],d[l+1],\cdots ,d[l+5],d[r+1],d[r+2],\cdots ,d[r+6]$ 是关于 $r-l+1$ 的 $5$ 次多项式函数，该函数可通过拉格朗日插值求得.

详见代码.

代码实现

#include <bits/stdc++.h>
using namespace std;

template <typename T>
void read(T& n)
{
    n = 0; int f = 1; char ch = getchar();
    while(!isdigit(ch)) {if(ch == '-') f = -1; ch = getchar();}
    while( isdigit(ch)) {n = n * 10 + ch - '0', ch = getchar();}
    n *= f;
}

template <typename T>
void print(T n)
{
    if(n < 0) putchar('-'), n = -n;
    if(n > 9) print(n / 10);
    putchar(n % 10 + '0');
}

template <typename T>
void print(T n, char ch)
{
    print(n), putchar(ch);
}

typedef long long ll;

const int M = (int)1e5;
const int N = (int)5;
const double eps = 1e-6;
const int mod = (int)1e9 + 7;
const int inf = 0x3f3f3f3f;

int n, m, q;
int a[M + 5];
int b[N + 2][3 * N + 3];
int y[N + 1][2 * N + 3][N + 2];
int d[M + 5];
int pre[N + 3], suf[N + 3];
int fac[N + 1], inv[N + 1], facinv[N + 1];

ll quick(ll a, ll b, ll p = mod)
{
    ll s = 1;
    while(b)
    {
        if(b & 1) s = s * a % p;
        a = a * a % p;
        b >>= 1;
    }
    return s % p;
}

void init()
{
    fac[0] = facinv[0] = fac[1] = inv[1] = facinv[1] = 1;
    for(int i = 2; i <= N; ++i)
    {
        fac[i] = 1ll * fac[i - 1] * i % mod;
        inv[i] = 1ll * (mod - mod / i) * inv[mod % i] % mod;
        facinv[i] = 1ll * facinv[i - 1] * inv[i] % mod;
    }
    for(int i = 0; i <= N; ++i)
    {
        for(int j = 2 * N + 2; j <= 3 * N + 2; ++j)
        {
            for(int k = 1; k <= j - N - 1; ++k) b[0][k] = quick(k, i);
            for(int k = j - N; k <= j; ++k) b[0][k] = 0;
            for(int k = 1; k <= N + 1; ++k)
            {
                for(int l = 1; l <= j; ++l) b[k][l] = b[k - 1][l] - b[k - 1][l - 1];
            }
            for(int k = 1; k <= N + 1; ++k) y[i][k][j - 2 * N - 1] = b[N + 1][k];
            for(int k = j - N; k <= j; ++k) y[i][k - j + 2 * N + 2][j - 2 * N - 1] = b[N + 1][k];
        }
    }
}

int L(int y[], int x)
{
    if(1 <= x && x <= N + 1) return y[x];
    pre[0] = 1; for(int i = 1; i <= N; ++i) pre[i] = 1ll * pre[i - 1] * (x - i) % mod;
    suf[N + 2] = 1; for(int i = N + 1; i >= 0; --i) suf[i] = 1ll * suf[i + 1] * (x - i) % mod;
    int s = 0, cur;
    for(int i = 1; i <= N + 1; ++i)
    {
        cur = 1ll * pre[i - 1] * suf[i + 1] % mod;
        cur = 1ll * cur * facinv[i - 1] % mod * facinv[N + 1 - i] % mod;
        if((N + 1 - i) & 1) cur = -cur;
        cur = 1ll * cur * y[i] % mod;
        (s += cur) %= mod;
    }
    return s;
}

void dif(int k, int c, int l, int r)
{
    for(int i = 1; i <= N + 1; ++i)  (d[i + l - 1] += 1ll * L(y[k][i], r - l + 1 - N) * c % mod) %= mod;
    for(int i = N + 2; i <= 2 * N + 2; ++i) (d[i - N + r - 1] += 1ll * L(y[k][i], r - l + 1 - N) * c % mod) %= mod;
}

void work()
{
    read(n), read(m), read(q);
    for(int i = 1; i <= n; ++i) read(a[i]), (a[i] += a[i - 1]) %= mod;
    for(int i = 1, l, r, k, c; i <= m; ++i)
    {
        read(l), read(r), read(k);
        for(int j = k; j >= 0; --j)
        {
            read(c);
            dif(j, c, l, r);
        }
    }
    for(int i = N + 1; i >= 0; --i)
    {
        for(int j = 1; j <= n; ++j) (d[j] += d[j - 1]) %= mod;
    }
    for(int i = 1, l, r, s; i <= q; ++i)
    {
        read(l), read(r);
        (s = a[r] - a[l - 1]) %= mod;
        (s += (d[r] - d[l - 1]) % mod) %= mod;
        print((s % mod + mod) % mod, '\n');
    }
}

int main()
{
//    freopen("input.txt", "r", stdin);
//    freopen("output.txt", "w", stdout);
//    int T; read(T);
//    for(int ca = 1; ca <= T; ++ca)
//    {
////        printf("Case #%d:", ca);
//        work();
//    }
    init();
    work();
//    cerr << 1.0 * clock() / CLOCKS_PER_SEC << "\n";
    return 0;
}