NTT经典例题

CCPC-Winter-Camp-day6-A——NTT经典例题

对于上面格式，如果想求出每个i的值可以使用卷积求出，因为阶乘j和阶乘i-j相乘的值为(i+(i-j))=i

补充一个二次剩余定理

P5491 【模板】二次剩余 - 洛谷 | 计算机科学教育新生态 (luogu.com.cn)

//#include<bits/stdc++.h>
#include<iostream>
#include<algorithm>
#include<numeric>
#include<cstring>//rfind("string"),s.find(string,begin)!=s.npos,find_first _of(),find_last_of()
#include<string>//to_string(value),s.substr(int begin, int length);
#include<cstdio>
#include<cmath>
#include<vector>//res.erase(unique(res.begin(), res.end()), res.end()),resize(n)//size of vector,vector<int>().swap(at[mx])
#include<queue>//priority_queue(big)  /priority_queue<int, vector<int>, greater<int>> q(small)
#include<stack>
#include<map>
#include<set>
#include<unordered_map>
#include<unordered_set>
#include<bitset>
#include<random>
#include<chrono>
//#include<ext/pb_ds/assoc_container.hpp>//gp_hash_table
//#include<ext/pb_ds/hash_policy.hpp>
//using namespace __gnu_pbds;
std::mt19937_64 rnd(std::chrono::steady_clock::now().time_since_epoch().count());
using namespace std;
#define int long long//__int128 2^127-1(GCC)
#define PII pair<int,int>
struct num {
	int x;// 实部
	int y;// 虚部(即虚数单位√w的系数)
};

int t, w, n, p;

num mul(num a, num b, int p) {// 复数乘法 
	num res;
	res.x = ((a.x * b.x % p + a.y * b.y % p * w % p) % p + p) % p;// x = a.x*b.x + a.y*b.y*w
	res.y = ((a.x * b.y % p + a.y * b.x % p) % p + p) % p;// y = a.x*b.y + a.y*b.x
	return res;
}
int qpow_r(int a, int b, int p) {// 实数快速幂 
	int res = 1;
	while (b) {
		if (b & 1) res = res * a % p;
		a = a * a % p;
		b >>= 1;
	}
	return res;
}
int qpow_i(num a, int b, int p) {// 复数快速幂  
	num res = { 1,0 };
	while (b) {
		if (b & 1) res = mul(res, a, p);
		a = mul(a, a, p);
		b >>= 1;
	}
	return res.x % p;// 只用返回实数部分，因为虚数部分没了 
}
int cipolla(int n, int p) {
	n %= p;
	if (qpow_r(n, (p - 1) / 2, p) == -1 + p) return -1;// 据欧拉准则判定是否有解 

	int a;
	while (1) {// 找出一个符合条件的a
		a = rand() % p;
		w = (((a * a) % p - n) % p + p) % p;// w = a^2 - n，虚数单位的平方
		if (qpow_r(w, (p - 1) / 2, p) == -1 + p) break;
	}

	num x = { a,1 };
	return qpow_i(x, (p + 1) / 2, p);
}
signed main() {
	srand(time(0));
	cin >> t;
	while (t--) {
		cin >> n >> p;
		if (!n) {
			printf("0\n");
			continue;
		}

		int ans1 = cipolla(n, p), ans2 = -ans1 + p;// 另一个解就是其相反数,ans1正数解 
		if (ans1 == -1) printf("Hola!\n");//无解
		else {
			if (ans1 > ans2) swap(ans1, ans2);
			if (ans1 == ans2) printf("%lld\n", ans1);
			else printf("%lld %lld\n", ans1, ans2);
		}
	}

	return 0;
}

NTT背包合并

NN​​​​​​​Fly (nowcoder.com)

PowerPoint 演示文稿 (nowcoder.com)

有点像数位dp，其中用到背包合并可以使用多项式解决，如果n个背包合并可以使用线段树和启发式合并类似的思想

//#include<bits/stdc++.h>
#include<iostream>
#include<algorithm>
#include<numeric>
#include<cstring>//rfind("string"),s.find(string,begin)!=s.npos,find_first _of(),find_last_of()
#include<string>//to_string(value),s.substr(int begin, int length);
#include<cstdio>
#include<cmath>
#include<vector>//res.erase(unique(res.begin(), res.end()), res.end()),resize(n)//size of vector,vector<int>().swap(at[mx])
#include<queue>//priority_queue(big)  /priority_queue<int, vector<int>, greater<int>> q(small)
#include<stack>
#include<map>
#include<set>
#include<unordered_map>
#include<unordered_set>
#include<bitset>
#include<random>
#include<chrono>
//#include<ext/pb_ds/assoc_container.hpp>//gp_hash_table
//#include<ext/pb_ds/hash_policy.hpp>
//using namespace __gnu_pbds;
std::mt19937_64 rnd(std::chrono::steady_clock::now().time_since_epoch().count());
using namespace std;
#define int long long//__int128 2^127-1(GCC)
#define PII pair<int,int>
const int N = 3e6 + 5, mod = 998244353;
namespace ntt {
    const int g = 3;
    int a[N], b[N];
    int r[N], tot, bit;
    int invg;
    int qpow(int a, int b) {
        int res = 1;
        while (b) {
            if (b & 1) res = 1ll * res * a % mod;
            a = 1ll * a * a % mod;
            b >>= 1;
        }
        return res;
    }
    void add(int& a, int b) {
        a += b;
        if (a >= mod) a -= mod;
    }
    void NTT(int a[], int inv) {
        for (int i = 0; i < tot; i++)
            if (i < r[i])
                swap(a[i], a[r[i]]);
        for (int mid = 1; mid < tot; mid <<= 1) {
            int g1 = qpow(inv == 1 ? g : invg, (mod - 1) / (mid << 1));
            for (int i = 0; i < tot; i += mid << 1) {
                for (int j = 0, gk = 1; j < mid; j++, gk = 1ll * gk * g1 % mod) {
                    int x = a[i + j], y = 1ll * gk * a[i + j + mid] % mod;
                    a[i + j] = (x + y) % mod, a[i + j + mid] = (x - y + mod) % mod;
                }
            }
        }
        if (inv == -1) {
            int invtot = qpow(tot, mod - 2);
            for (int i = 0; i < tot; i++) {
                a[i] = 1ll * a[i] * invtot % mod;
            }
        }
    }
    struct Poly {
        vector<int> coef;
        int deg;

        int& operator[](int x) {
            return coef[x];
        }

        Poly(int deg = -1) : deg(deg) {
            coef = vector<int>(deg + 1, 0);
        }

        void norm(int deg) {
            this->deg = deg;
            coef.resize(deg + 1);
        }
    };
    void init(int len) {
        bit = tot = 0;
        while ((1ll << bit) <= len) bit++;
        tot = 1ll << bit;
        for (int i = 0; i < tot; i++) a[i] = b[i] = 0;
        for (int i = 1; i < tot; i++) r[i] = (r[i >> 1] >> 1) | ((i & 1) << (bit - 1));
    }
    Poly operator*(const Poly& f, const Poly& g) {
        Poly res(f.deg + g.deg);
        if (f.deg <= 8 || g.deg <= 8) {
            for (int i = 0; i <= f.deg; i++)
                for (int j = 0; j <= g.deg; j++)
                    add(res[i + j], 1ll * f.coef[i] * g.coef[j] % mod);
            return res;
        }
        init(res.deg);
        copy(f.coef.begin(), f.coef.end(), a);
        copy(g.coef.begin(), g.coef.end(), b);
        NTT(a, 1), NTT(b, 1);
        for (int i = 0; i < tot; i++) a[i] = 1ll * a[i] * b[i] % mod;
        NTT(a, -1);
        copy(a, a + res.deg + 1, res.coef.begin());
        return res;
    }
    int __ = []
    {
        invg = qpow(g, mod - 2);
        return 0;
    }();
}
using namespace ntt;
signed main()
{
    ios_base::sync_with_stdio(0); cin.tie(0), cout.tie(0);
    int n, k;
    int m;
    cin >> n >> m >> k;
    vector<int>a(n + 1);
    vector<Poly>v;
    int sum = 0;
    for (int i = 1; i <= n; i++) {
        cin >> a[i];
        sum += a[i];
        Poly f(a[i]);
        f[0] = f[a[i]] = 1;
        v.emplace_back(f);
    }
    auto solve = [&](auto self, int l, int r)->Poly {
        if (l == r) return v[l];
        int mid = l + r >> 1;
        return self(self, l, mid) * self(self, mid + 1, r);
    };

    Poly f = solve(solve, 0, v.size() - 1);
    //assert(f.deg == sum);
    vector<vector<int>>ban(60, vector<int>());
    //array<vector<int>, 60>ban;
    while (k--)
    {
        int b, c;
        cin >> b >> c;
        ban[c].push_back(b);
    }
    vector<Poly>dp(2);
    dp[0].norm(0);
    dp[0][0] = 1;
    for (int i = 0; i < 60; i++) {
        Poly g = f;
        sort(ban[i].begin(), ban[i].end());
        ban[i].erase(unique(ban[i].begin(), ban[i].end()), ban[i].end());
        for (auto x : ban[i]) {
            for (int j = a[x]; j <= sum; j++) {
                g[j] -= g[j - a[x]];
                if (g[j] < 0)g[j] += mod;
            }
        }
        vector<Poly>f(2), ndp(2);
        f[0] = dp[0] * g;
        f[1] = dp[1] * g;
        for (auto t : { 0,1 }) ndp[t].norm(f[t].deg / 2);
        for (auto t : { 0,1 }) {
            for (int j = 0; j <= f[t].deg; j++) {
                if (j % 2 == (m >> i & 1)) {
                    add(ndp[t][j / 2], f[t][j]);
                }
                else if (j % 2 > (m >> i & 1)) {
                    add(ndp[1][j / 2], f[t][j]);
                }
                else {
                    add(ndp[0][j / 2], f[t][j]);
                }
            }
        }
        dp = ndp;
    }
    cout << dp[0][0] << "\n";
}