维护一个序列,初始全为\(1\)

支持两种操作:

1.对于所有的位置\(i\),将它的值乘上\(i + a\)

2.询问\(a\)处的值

\(q=120000\) 20s 512M

——————

如果把第一个操作看成乘上一个\(x + a_i\),第二个操作看成询问\(x = a_i\)处多项式的值,那么这是一个裸的动态多点求值

首先暴力是可以AC的……恩,开个O2就行……

直接上CDQ+静态多点求值的话是\(O(n \log ^3 n)\)的,只能跑过60分

#pragma GCC optimize(2)
#include <bits/stdc++.h>
using namespace std; const int N = ;
const int MOD = ;
const int G = ;
int powi(int a, int b)
{
int c = ;
for (; b; b >>= , a = 1ll * a * a % MOD)
if (b & ) c = 1ll * c * a % MOD;
return c;
}
void NTT(int A[], int n, int f)
{
for (int i = ; i < n; ++ i)
{
int j = ;
for (int p = , q = n >> ; p < n; p <<= , q >>= )
if (i & p) j |= q;
if (i < j) swap(A[i], A[j]);
}
for (int i = ; i <= n; i <<= )
{
int w1 = powi(G, (MOD - ) / i);
if (f < ) w1 = powi(w1, MOD - );
for (int j = ; j < n; j += i)
{
int w = ;
for (int k = j; k < j + (i >> ); ++ k)
{
int u = A[k], v = 1ll * A[k + (i >> )] * w % MOD;
A[k] = (u + v) % MOD;
A[k + (i >> )] = (u - v + MOD) % MOD;
w = 1ll * w * w1 % MOD;
}
}
} int p = powi(n, MOD - );
if (f < ) for (int i = ; i < n; ++ i) A[i] = 1ll * A[i] * p % MOD;
} int Rev_tmp1[N], Rev_tmp2[N];
void Rev(int A[], int G[], int n)
{
if (n == )
{
G[] = powi(A[], MOD - );
}
else
{
Rev(A, G, n / );
for (int i = ; i < n; ++ i) Rev_tmp1[i] = A[i];
for (int i = n; i < (n << ); ++ i) Rev_tmp1[i] = ;
for (int i = ; i < (n >> ); ++ i) Rev_tmp2[i] = G[i];
for (int i = (n >> ); i < (n << ); ++ i) Rev_tmp2[i] = ;
NTT(Rev_tmp1, n << , );
NTT(Rev_tmp2, n << , );
for (int i = ; i < (n << ); ++ i) Rev_tmp1[i] = (( - 1ll * Rev_tmp1[i] * Rev_tmp2[i]) % MOD + MOD) * Rev_tmp2[i] % MOD;
NTT(Rev_tmp1, n << , -);
for (int i = ; i < n; ++ i) G[i] = Rev_tmp1[i];
}
} int Mod_tmp1[N], Mod_tmp2[N], Mod_tmp3[N], Mod_tmp4[N];
void Mod(int A[], int B[], int D[], int n, int m)
{
if (n < m)
{
for (int i = ; i < n; ++ i) D[i] = A[i];
return;
}
int nn = ;
while (nn < (n - m + )) nn <<= ;
for (int i = ; i < n; ++ i) Mod_tmp3[i] = A[i];
for (int i = ; i < m; ++ i) Mod_tmp4[i] = B[i];
for (int i = , j = n - ; i < j; ++ i, -- j) swap(Mod_tmp3[i], Mod_tmp3[j]);
for (int i = , j = m - ; i < j; ++ i, -- j) swap(Mod_tmp4[i], Mod_tmp4[j]);
for (int i = m; i < nn; ++ i) Mod_tmp4[i] = ;
Rev(Mod_tmp4, Mod_tmp2, nn); for (int i = ; i < n - m + ; ++ i) Mod_tmp1[i] = Mod_tmp3[i];
for (int i = n - m + ; i < (nn << ); ++ i) Mod_tmp1[i] = ;
for (int i = n - m + ; i < (nn << ); ++ i) Mod_tmp2[i] = ;
NTT(Mod_tmp1, nn << , );
NTT(Mod_tmp2, nn << , );
for (int i = ; i < (nn << ); ++ i) Mod_tmp1[i] = 1ll * Mod_tmp1[i] * Mod_tmp2[i] % MOD;
NTT(Mod_tmp1, nn << , -); while (nn < n) nn <<= ;
for (int i = , j = n - m; i < j; ++ i, -- j) swap(Mod_tmp1[i], Mod_tmp1[j]);
for (int i = n - m + ; i < (nn << ); ++ i) Mod_tmp1[i] = ;
for (int i = ; i < m; ++ i) Mod_tmp2[i] = B[i];
for (int i = m; i < (nn << ); ++ i) Mod_tmp2[i] = ;
NTT(Mod_tmp1, nn << , );
NTT(Mod_tmp2, nn << , );
for (int i = ; i < (nn << ); ++ i) Mod_tmp1[i] = 1ll * Mod_tmp1[i] * Mod_tmp2[i] % MOD;
NTT(Mod_tmp1, nn << , -); for (int i = ; i < m - ; ++ i) D[i] = (A[i] - Mod_tmp1[i] + MOD) % MOD;
} int AAA[N];
int Mul_tmp[][N], Mul_tmp1[N], Mul_tmp2[N];
void Mul(int x, int l, int r)
{
if (r - l == )
{
Mul_tmp[x][l * + ] = ;
Mul_tmp[x][l * ] = AAA[l];
return;
}
int m = (l + r) / ;
Mul(x + , l, m);
Mul(x + , m, r);
for (int i = ; i < ((m - l) << ); ++ i) Mul_tmp1[i] = Mul_tmp[x + ][i + l * ];
for (int i = ; i < ((r - m) << ); ++ i) Mul_tmp2[i] = Mul_tmp[x + ][i + m * ];
int nn = ;
while (nn < (r - l) * ) nn <<= ;
for (int i = ((m - l) << ); i < nn; ++ i) Mul_tmp1[i] = ;
for (int i = ((r - m) << ); i < nn; ++ i) Mul_tmp2[i] = ;
NTT(Mul_tmp1, nn, );
NTT(Mul_tmp2, nn, );
for (int i = ; i < nn; ++ i) Mul_tmp1[i] = 1ll * Mul_tmp1[i] * Mul_tmp2[i] % MOD;
NTT(Mul_tmp1, nn, -);
for (int i = ; i < ((r - l) << ); ++ i) Mul_tmp[x][i + l * ] = Mul_tmp1[i];
} int get_val_tmp[][N];
void get_val(int fin[], int x, int l, int r)
{
if (r - l == )
{
fin[l] = get_val_tmp[x][l * ];
return;
}
int m = (l + r) / ;
Mod(get_val_tmp[x] + l * , Mul_tmp[x + ] + l * , get_val_tmp[x + ] + l * , (r - l) * , (m - l) + );
Mod(get_val_tmp[x] + l * , Mul_tmp[x + ] + m * , get_val_tmp[x + ] + m * , (r - l) * , (r - m) + ); for (int i = m + l; i < m * ; ++ i) get_val_tmp[x + ][i] = ;
for (int i = r + m; i < r * ; ++ i) get_val_tmp[x + ][i] = ; get_val(fin, x + , l, m);
get_val(fin, x + , m, r);
} void GGG(int A[], int B[], int D[], int n, int m)
{
for (int i = ; i < m * ; ++ i) get_val_tmp[][i] = ;
for (int i = ; i < m; ++ i) AAA[i] = MOD - B[i];
Mul(, , m);
Mod(A, Mul_tmp[], get_val_tmp[], n, m + );
get_val(D, , , m);
} int typ[N], val[N], ans[N]; int solve_tmp1[N], solve_tmp2[N], solve_tmp3[N];
void solve(int l, int r)
{
if (r - l == ) return;
int m = (l + r) / ;
solve(l, m);
solve(m, r);
int tot1 = , tot2 = , tot3 = ;
for (int i = l; i < m; ++ i) if (typ[i] == ) AAA[tot1 ++] = val[i];
for (int i = m; i < r; ++ i) if (typ[i] == ) solve_tmp2[tot2 ++] = val[i];
if (!tot1 || !tot2) return;
Mul(, , tot1);
for (int i = ; i < tot1 + ; ++ i) solve_tmp1[i] = Mul_tmp[][i];
for (int i = tot1 + ; i < tot1 * ; ++ i) solve_tmp1[i] = ;
GGG(solve_tmp1, solve_tmp2, solve_tmp3, tot1 + , tot2);
for (int i = m; i < r; ++ i) if (typ[i] == ) ans[i] = 1ll * ans[i] * solve_tmp3[tot3 ++] % MOD;
}
int main()
{
int a, b, c, n;
scanf("%d%d%d", &a, &b, &c); n = b + c;
for (int i = ; i < n; ++ i)
{
scanf("%d%d", &typ[i], &val[i]); ans[i] = ; val[i] = ((val[i]) % MOD + MOD) % MOD;
}
solve(, n);
for (int i = ; i < n; ++ i)
if (typ[i] == ) printf("%d\n", ans[i]);
}

令第\(i-1\)次询问到第\(i\)次询问之间的多项式之积为\(A_i(x)\)

那么显然有 \(ans_i = (\prod_{p=1}^{i} A_p(x)) \mod (x - a_i)\)

令\(f_{l,r}=(\prod_{p=1}^{l} A_p(x)) \mod (\prod_{p = l}^{r-1} (x - a_p)) \)

于是\(ans_i = f_{i,i+1}\)

注意到

\(f_{l,mid} = f_{l,r} \mod (\prod_{p = l}^{mid-1} (x - a_p)) \)

\(f_{mid,r} =( f_{l,r} \prod_{p=l+1}^{mid} A_p(x))  \mod (\prod_{p = mid}^{r-1} (x - a_p)) \)

\(f_{1,q}=A_1(x) \)

预处理 需要用的 \(x-a_i\)的积 和 需要用的 \(A_i(x) \) 的积(就是一个多项式分治乘法啦

一次递归只需要跑一次多项式乘法和两次多项式取膜,多项式的度数是\(O(len)\)的,因此一次递归的复杂度是\(O(len \log len)\)的

\(T(n) = 2 T(\frac{n}{2})  + O(n \log n))\)

最终复杂度\(O(n \log ^2 n)\)

代码写的和上面的分析有些不同,所以有点萎(捂脸

#pragma GCC optimize(2)
#include <bits/stdc++.h>
using namespace std; const int N = ;
const int MOD = ;
const int G = ;
int powi(int a, int b)
{
int c = ;
for (; b; b >>= , a = 1ll * a * a % MOD)
if (b & ) c = 1ll * c * a % MOD;
return c;
}
void NTT(int A[], int n, int f)
{
for (int i = , j = ; i < n; ++ i)
{
if (i > j) swap(A[i], A[j]);
for (int l = n >> ; (j ^= l) < l; l >>= );
}
for (int i = ; i <= n; i <<= )
{
int w1 = powi(G, (MOD - ) / i);
if (f < ) w1 = powi(w1, MOD - );
for (int j = ; j < n; j += i)
{
int w = ;
for (int k = j; k < j + (i >> ); ++ k)
{
int u = A[k], v = 1ll * A[k + (i >> )] * w % MOD;
A[k] = (u + v >= MOD? u + v - MOD: u + v);
A[k + (i >> )] = (u - v < ? u - v + MOD: u - v);
w = 1ll * w * w1 % MOD;
}
}
} int p = powi(n, MOD - );
if (f < ) for (int i = ; i < n; ++ i) A[i] = 1ll * A[i] * p % MOD;
} int Rev_tmp1[N], Rev_tmp2[N];
void Rev(int A[], int G[], int n)
{
if (n == )
{
G[] = powi(A[], MOD - );
}
else
{
Rev(A, G, n / );
for (int i = ; i < n; ++ i) Rev_tmp1[i] = A[i];
for (int i = n; i < (n << ); ++ i) Rev_tmp1[i] = ;
for (int i = ; i < (n >> ); ++ i) Rev_tmp2[i] = G[i];
for (int i = (n >> ); i < (n << ); ++ i) Rev_tmp2[i] = ;
NTT(Rev_tmp1, n << , );
NTT(Rev_tmp2, n << , );
for (int i = ; i < (n << ); ++ i) Rev_tmp1[i] = (( - 1ll * Rev_tmp1[i] * Rev_tmp2[i]) % MOD + MOD) * Rev_tmp2[i] % MOD;
NTT(Rev_tmp1, n << , -);
for (int i = ; i < n; ++ i) G[i] = Rev_tmp1[i];
}
} int Mod_tmp1[N], Mod_tmp2[N], Mod_tmp3[N], Mod_tmp4[N];
void Mod(int A[], int B[], int D[], int n, int m)
{
while (B[m - ] == ) m --;
if (n < m)
{
for (int i = ; i < n; ++ i) D[i] = A[i];
return;
}
int nn = ;
while (nn < (n - m + )) nn <<= ;
for (int i = ; i < n; ++ i) Mod_tmp3[i] = A[i];
for (int i = ; i < m; ++ i) Mod_tmp4[i] = B[i];
for (int i = , j = n - ; i < j; ++ i, -- j) swap(Mod_tmp3[i], Mod_tmp3[j]);
for (int i = , j = m - ; i < j; ++ i, -- j) swap(Mod_tmp4[i], Mod_tmp4[j]);
for (int i = m; i < nn; ++ i) Mod_tmp4[i] = ;
Rev(Mod_tmp4, Mod_tmp2, nn); for (int i = ; i < n - m + ; ++ i) Mod_tmp1[i] = Mod_tmp3[i];
for (int i = n - m + ; i < (nn << ); ++ i) Mod_tmp1[i] = ;
for (int i = n - m + ; i < (nn << ); ++ i) Mod_tmp2[i] = ;
NTT(Mod_tmp1, nn << , );
NTT(Mod_tmp2, nn << , );
for (int i = ; i < (nn << ); ++ i) Mod_tmp1[i] = 1ll * Mod_tmp1[i] * Mod_tmp2[i] % MOD;
NTT(Mod_tmp1, nn << , -); while (nn < n) nn <<= ;
for (int i = , j = n - m; i < j; ++ i, -- j) swap(Mod_tmp1[i], Mod_tmp1[j]);
for (int i = n - m + ; i < (nn << ); ++ i) Mod_tmp1[i] = ;
for (int i = ; i < m; ++ i) Mod_tmp2[i] = B[i];
for (int i = m; i < (nn << ); ++ i) Mod_tmp2[i] = ;
NTT(Mod_tmp1, nn << , );
NTT(Mod_tmp2, nn << , );
for (int i = ; i < (nn << ); ++ i) Mod_tmp1[i] = 1ll * Mod_tmp1[i] * Mod_tmp2[i] % MOD;
NTT(Mod_tmp1, nn << , -); for (int i = ; i < m - ; ++ i) D[i] = (A[i] - Mod_tmp1[i] + MOD) % MOD;
} int AAA[N], BBB[N];
int Mul1[][N], Mul2[][N], Mul_tmp1[N], Mul_tmp2[N];
void Mul(int Mul_tmp[][N], int x, int l, int r)
{
if (r - l == )
{
if (!BBB[l])
{
Mul_tmp[x][l * + ] = ;
Mul_tmp[x][l * ] = AAA[l];
}
else
{
Mul_tmp[x][l * + ] = ;
Mul_tmp[x][l * ] = ;
}
return;
}
int m = (l + r) / ;
Mul(Mul_tmp, x + , l, m);
Mul(Mul_tmp, x + , m, r);
for (int i = ; i < ((m - l) << ); ++ i) Mul_tmp1[i] = Mul_tmp[x + ][i + l * ];
for (int i = ; i < ((r - m) << ); ++ i) Mul_tmp2[i] = Mul_tmp[x + ][i + m * ];
int nn = ;
while (nn < (r - l) * ) nn <<= ;
for (int i = ((m - l) << ); i < nn; ++ i) Mul_tmp1[i] = ;
for (int i = ((r - m) << ); i < nn; ++ i) Mul_tmp2[i] = ;
NTT(Mul_tmp1, nn, );
NTT(Mul_tmp2, nn, );
for (int i = ; i < nn; ++ i) Mul_tmp1[i] = 1ll * Mul_tmp1[i] * Mul_tmp2[i] % MOD;
NTT(Mul_tmp1, nn, -);
for (int i = ; i < ((r - l) << ); ++ i) Mul_tmp[x][i + l * ] = Mul_tmp1[i];
} int cnt1[N], cnt2[N];
int ans[N], typ[N], val[N];
int solve_tmp[][N], solve_tmp1[N], solve_tmp2[N];
void solve(int x, int l, int r)
{
if (r - l == )
{
ans[l] = solve_tmp[x][l * ];
return;
}
if (cnt1[r] - cnt1[l] == ) return;
int m = (l + r) / ;
Mod(solve_tmp[x] + l * , Mul1[x + ] + l * , solve_tmp[x + ] + l * , (r - l) * , (m - l) * ); int nn = ;
while (nn < (r - l) * ) nn *= ; for (int i = ; i < (r - l) * ; ++ i) solve_tmp1[i] = solve_tmp[x][i + l * ];
for (int i = (r - l) * ; i < nn; ++ i) solve_tmp1[i] = ;
for (int i = ; i < (m - l) * ; ++ i) solve_tmp2[i] = Mul2[x + ][i + l * ];
for (int i = (m - l) * ; i < nn; ++ i) solve_tmp2[i] = ;
NTT(solve_tmp1, nn, );
NTT(solve_tmp2, nn, );
for (int i = ; i < nn; ++ i) solve_tmp1[i] = 1ll * solve_tmp1[i] * solve_tmp2[i] % MOD;
NTT(solve_tmp1, nn, -);
Mod(solve_tmp1, Mul1[x + ] + m * , solve_tmp[x + ] + m * , (r - l) * , (r - m) * ); solve(x + , l, m);
solve(x + , m, r); } namespace IO {
const int SIZE=(int)(1e6);
char buf[SIZE]; FILE *in,*out;
int cur; inline void init() {
cur=SIZE;in=stdin;out=stdout;
} inline char nextChar() {
if(cur==SIZE) {
fread(buf,SIZE,,in);cur=;
} return buf[cur++];
} inline int nextInt() {
bool st=,neg=;
char c;int num=;
while((c=nextChar())!=EOF) {
if(c=='-') st=neg=;
else if(c>='' && c<='') {
st=;num=num*+c-'';
} else if(st) break;
} if(neg) num=-num;
return num;
} inline void printChar(char c) {
buf[cur++]=c;
if(cur==SIZE) {
fwrite(buf,SIZE,,out);cur=;
}
} inline void printInt(int x) {
if(x<) {
printChar('-');printInt(-x);
return;
}
if(x>=) printInt(x/);printChar(''+x%);
} inline void close() {
if(cur>) fwrite(buf,cur,,out);cur=;
}
} int main()
{
using namespace IO;
int a, b, c, n;
init();
a = nextInt();
b = nextInt();
c = nextInt();
n = b + c;
for (int i = ; i < n; ++ i)
{
typ[i] = nextInt();
val[i] = nextInt();
ans[i] = ;
cnt1[i + ] = cnt1[i] + (typ[i] == );
cnt2[i + ] = cnt2[i] + (typ[i] == );
val[i] = ((val[i]) % MOD + MOD) % MOD;
}
for (int i = ; i < n; ++ i) if (typ[i] == ) BBB[i] = , AAA[i] = val[i]; else BBB[i] = ;
Mul(Mul2, , , n); for (int i = ; i < n; ++ i) if (typ[i] == ) BBB[i] = , AAA[i] = MOD - val[i]; else BBB[i] = ;
Mul(Mul1, , , n); solve_tmp[][] = ;
solve(, , n); cur = ;
for (int i = ; i < n; ++ i)
if (typ[i] == ) printInt(ans[i]), printChar('\n');
close();
}

换了一个NTT板子,终于跑的比暴力快了QAQ

学会了多项式技巧(划去

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