$P5240 Derivation$

Posted 2021-02-14 qf-breeze

神仙题。

第一场月赛的题目我到第二场月赛完了才写【由此可见我是真的菜

题目就是个大模拟加乘显然，幂的话需要将原函数、导函数的函数值用扩展欧拉定理展开 (log) 层。时间复杂度 (O(T |S| log^2p))

因为求导时要对指数减一，你可能会用加 (模数-1) 来实现，并且如果你的扩展欧拉定理写法是小于模数时是正常值，超过模数时用真实值 + 模数代替，就会导致错误，因为正常的快速幂中 (0^0=1)

但 (0^P=0) ((P≠0))

以下不是本人写的程序 是验题人写的

#include <bits/stdc++.h>
#define fi first
#define se second
#define pii pair<int,int>
#define mp make_pair
#define pb push_back
#define space putchar(' ')
#define enter putchar('
')
#define MAXN 10005
#define eps 1e-12
//#define ivorysi
using namespace std;
typedef long long int64;
typedef unsigned int u32;
typedef double db;
template<class T>
void read(T &res) {
    res = 0;T f = 1;char c = getchar();
    while(c < '0' || c > '9') {
        if(c == '-') f = -1;
        c = getchar();
    }
    while(c >= '0' && c <= '9') {
        res = res * 10 + c - '0';
        c = getchar();
    }
    res *= f;
}
template<class T>
void out(T x) {
    if(x < 0) {x = -x;putchar('-');}
    if(x >= 10) {
        out(x / 10);
    }
    putchar('0' + x % 10);
}
const int MOD = 998244353;
char s[MAXN],que[MAXN];
int val[2],tot,phi[MAXN];
vector<pii > sta[2];
bool f[2][MAXN][2];
int eu(int x) {
    int t = x;
    for(int i = 2 ; i <= x / i ; ++i) {
        if(x % i == 0) {
            t = t / i * (i - 1);
            while(x % i == 0) x /= i;
        }
    }
    if(x > 1) t = t / x * (x - 1);
    return t;
}
int fpow(int x,int c,int mod,bool &on) {
    int64 res = 1,t = x;
    if(c == 0) {on = 0;return 1;}
    if(x == 1 || x == 0) return x;

    while(c && !on) {
        res = res * x;--c;
        if(res >= MOD) {res %= mod;on = 1;break;}
    }
    while(c) {
        if(c & 1) res = res * t % mod;
        t = t * t % mod;
        c >>= 1;
    }
    return res;
}
void Solve() {
    scanf("%s",s + 2);
    read(val[0]);read(val[1]);
    s[1] = '(';
    int L = strlen(s + 1);
    s[L + 1] = ')';++L;
    sta[0].clear();sta[1].clear();
    tot = 0;
    int lev = 0;
    for(int i = 1 ; i <= L ; ++i) {
        if(s[i] >= '0' && s[i] <= '9') {
            if(s[i - 1] < '0' || s[i - 1] > '9') {
                for(int k = 0 ; k <= 1 ; ++k) {
                    int v = s[i] - '0';
                    sta[k].pb(mp(v,(lev == 0 ? 0 : v)));
                    f[k][sta[k].size() - 1][0] = 0;
                    f[k][sta[k].size() - 1][1] = 0;
                }
            }
            else {
                for(int k = 0 ; k <= 1 ; ++k) {
                    auto t = sta[k].back();
                    int64 a = 1LL * t.fi * 10 + s[i] - '0';
                    int64 b = 1LL * t.se * 10 + s[i] - '0';
                    if(a >= MOD) {
                        f[k][sta[k].size() - 1][0] = 1;
                        a %= phi[lev];
                    }
                    if(b >= MOD) {
                        f[k][sta[k].size() - 1][1] = 1;
                        b %= phi[max(lev - 1,0)];
                    }
                    if(lev == 0) b = 0;
                    t = mp(a,b);
                    sta[k].pop_back();sta[k].push_back(t);
                }
            }
        }
        else if(s[i] == ')') {
            if(!tot) break;
            if(que[tot] == '^') --lev;
            for(int k = 0 ; k <= 1 ; ++k) {
                int si = sta[k].size() - 1;
                auto t0 = sta[k][si - 1],t1 = sta[k][si];
                sta[k].pop_back();sta[k].pop_back();
                if(que[tot] == '^') {
                    if(f[k][si][0]) t1.fi += phi[lev + 1];
                    if(f[k][si][1]) t1.se += phi[lev];
                    if(lev) {
                        int a = fpow(t0.fi,t1.fi,phi[lev],f[k][si - 1][0]);
                        int b = fpow(t0.se,t1.se,phi[lev - 1],f[k][si - 1][1]);
                        sta[k].pb(mp(a,b));
                    }
                    else {
                        int a = fpow(t0.fi,t1.fi,MOD,f[k][si - 1][0]);
                        int b = 1LL * t1.se * t0.se % MOD;
                        if(t0.fi != 0)
                            b = 1LL * b * fpow(t0.fi,(1LL * t1.fi + (MOD - 2)) % (MOD - 1),MOD,f[k][si - 1][1]) % MOD;
                        else {
                            if(f[k][si][0]) b = 0;
                            else if(t1.fi - 1) b = 0;
                        }
                        sta[k].pb(mp(a,b));
                    }
                }
                else if(que[tot] == '+') {
                    f[k][si - 1][0] |= f[k][si][0];
                    f[k][si - 1][1] |= f[k][si][1];
                    t0.fi = t0.fi + t1.fi;
                    t0.se = t0.se + t1.se;
                    if(t0.fi >= MOD) {t0.fi %= phi[lev];f[k][si - 1][0] |= 1;}
                    if(t0.se >= MOD) {t0.se %= phi[max(0,lev - 1)];f[k][si - 1][1] |= 1;}
                    sta[k].pb(t0);
                }
                else if(que[tot] == '*') {
                    if(lev == 0) {
                        int64 a = 1LL * t0.fi * t1.fi % MOD;
                        int64 b = (1LL * t0.fi * t1.se % MOD + 1LL * t1.fi * t0.se % MOD) % MOD;
                        sta[k].pb(mp((int)a,(int)b));
                    }
                    else {
                        if((!f[k][si][0] && t1.fi == 0) || (!f[k][si - 1][0] && t0.fi == 0)) f[k][si - 1][0] = 0;
                        else f[k][si - 1][0] |= f[k][si][0];
                        if((!f[k][si][1] && t1.fi == 0) || (!f[k][si - 1][1] && t0.fi == 0)) f[k][si - 1][1] = 0;
                        else f[k][si - 1][1] |= f[k][si][1];
                        int64 a = 1LL * t0.fi * t1.fi;
                        int64 b = 1LL * t0.se * t1.se;
                        if(a >= MOD) {a %= phi[lev];f[k][si - 1][0] |= 1;}
                        if(b >= MOD) {b %= phi[lev - 1];f[k][si - 1][1] |= 1;}
                        sta[k].pb(mp((int)a,(int)b));
                    }
                }
            }
            --tot;
        }
        else if(s[i] == 'x') {
            sta[0].pb(mp(val[0],1));
            sta[1].pb(mp(val[1],1));
        }
        else {
            if(s[i] != '(') que[++tot] = s[i];
            if(s[i] == '^') ++lev;
        }
    }
    out(sta[0][0].se);space;out(sta[1][0].se);enter;
}
int main() {
#ifdef ivorysi
    freopen("f2.in","r",stdin);
#endif
    int T;
    read(T);
    phi[0] = MOD;
    for(int i = 1 ; i <= 10000 ; ++i) {
        phi[i] = eu(phi[i - 1]);
    }
    for(int i = 1 ; i <= T ; ++i) {
        Solve();
    }
}