LibreOJ 2565 「SDOI2018」旧试题

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拿这个代码交了 JZOJ 4991（此题的 \(A,B,C \le 5000\) 弱化版），成功拿到 14ms 最优解（

\[\newcommand\ffrac[2]{\genfrac\lfloor\rfloor{}{}{ #1 }{ #2 }} \begin{align*} & \sum\limits_{i=1}^A\sum\limits_{j=1}^B\sum\limits_{k=1}^C d(ijk) \\ = & \sum\limits_{i=1}^A\sum\limits_{j=1}^B\sum\limits_{k=1}^C\sum\limits_{x|i}\sum\limits_{y|j}\sum\limits_{z|k} [\gcd(x,y)=1][\gcd(x,z)=1][\gcd(y,z)=1] \\ = & \sum\limits_{x=1}^A\sum\limits_{y=1}^B\sum\limits_{z=1}^C\ffrac Ax\ffrac By\ffrac Cz [\gcd(x,y)=1][\gcd(x,z)=1][\gcd(y,z)=1] \\ = & \sum\limits_{x=1}^A\sum\limits_{y=1}^B\sum\limits_{z=1}^C\ffrac Ax\ffrac By\ffrac Cz \sum\limits_{i|\gcd(x,y)}\mu(i) \sum\limits_{j|\gcd(x,z)}\mu(j) \sum\limits_{k|\gcd(y,z)}\mu(k) \\ = & \sum\limits_{i=1}^{\min(A,B)}\mu(i) \sum\limits_{j=1}^{\min(A,C)}\mu(j) \sum\limits_{k=1}^{\min(B,C)}\mu(k) \sum\limits_{\text{lcm}(i,j)|x}\ffrac Ax \sum\limits_{\text{lcm}(i,k)|y}\ffrac By \sum\limits_{\text{lcm}(j,k)|z}\ffrac Cz \end{align*}\]

记 \(f_0(n) = \sum\limits_{n|x}\ffrac Ax\)，类似地有 \(f_1,f_2\)，上式变为 \[\sum\limits_{i=1}^{\min(A,B)}\mu(i) \sum\limits_{j=1}^{\min(A,C)}\mu(j) \sum\limits_{k=1}^{\min(B,C)}\mu(k) f_0(\text{lcm}(i,j)) f_1(\text{lcm}(i,k)) f_2(\text{lcm}(j,k))\]

注意到可以把枚举的三元组 \((i,j,k)\) 看做一堆三元环然后做三元环计数，同时统计答案。
注意到有很多无贡献的边可以直接删掉，包括 \(\mu(u) = 0,\mu(v) = 0,\text{lcm}(u,v) > \min(A,B,C)\) 的 \(u,v\) 的边。
于是我们剪枝剪掉一堆边之后边就变得很少了，直接暴力上三元环计数。

同时注意三元环计数无法考虑自环，特判一下有两个数相等或是三个数均相等的三元组即可。
据说 Cache Miss 使得 vector 存图会快很多？
参考了 GKxx 大佬的代码。

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#include <cstdio>
#include <cstring>
#include <vector>
#include <algorithm>
using namespace std;
const int N = 1e5;
const long long mod = 1e9 + 7;
int T,A,B,C,mx,mn;
int vis[N + 5],cnt,prime[N + 5],mu[N + 5];
long long f[3][N + 5],ans;
int deg[N + 5];
vector<int> e[N + 5];
vector<long long> v[N + 5];
struct Edge
{
    int u,v,w;
} g[(N << 4) + 5];
int tot;
int main()
{
    mu[1] = 1;
    for(register int i = 2;i <= N;++i)
    {
        if(!vis[i])
            mu[prime[++cnt] = i] = -1;
        for(register int j = 1;j <= cnt && i * prime[j] <= N;++j)
        {
            vis[i * prime[j]] = 1;
            if(!(i % prime[j]))
                break;
            else
                mu[i * prime[j]] = -mu[i];
        }
    }
    memset(vis,0,sizeof vis);
    scanf("%d",&T);
    for(;T;--T)
    {
        scanf("%d%d%d",&A,&B,&C),mx = max(A,max(B,C)),mn = min(A,min(B,C)),ans = 0,memset(f,0,sizeof f);
        for(register int i = 1;i <= A;++i)
            for(register int j = i;j <= A;j += i)
                f[0][i] += A / j;
        for(register int i = 1;i <= B;++i)
            for(register int j = i;j <= B;j += i)
                f[1][i] += B / j;
        for(register int i = 1;i <= C;++i)
            for(register int j = i;j <= C;j += i)
                f[2][i] += C / j;
        for(register int i = 1;i <= mn;++i)
            mu[i] && (ans += mu[i] * mu[i] * mu[i] * f[0][i] * f[1][i] * f[2][i]);
        memset(deg,0,sizeof deg),tot = 0;
        for(register int d = 1;d <= mx;++d)
            for(register int i = 1;i * d <= mx;++i)
                if(mu[i * d])
                    for(register int j = i + 1,x,y,lcm;(long long)i * j * d <= mx;++j)
                        if(mu[j * d] && __gcd(i,j) == 1)
                            x = i * d,y = j * d,lcm = i * j * d,
                            ans += mu[x] * mu[x] * mu[y] * (
                                f[0][x] * f[1][lcm] * f[2][lcm] +
                                f[0][lcm] * f[1][x] * f[2][lcm] +
                                f[0][lcm] * f[1][lcm] * f[2][x]),
                            ans += mu[x] * mu[y] * mu[y] * (
                                f[0][y] * f[1][lcm] * f[2][lcm] +
                                f[0][lcm] * f[1][y] * f[2][lcm] +
                                f[0][lcm] * f[1][lcm] * f[2][y]),
                            ++deg[x],++deg[y],g[++tot] = (Edge){x,y,lcm};
        for(register int i = 1;i <= mx;++i)
            e[i].clear(),v[i].clear();
        for(register int i = 1;i <= tot;++i)
            if(deg[g[i].u] > deg[g[i].v] || (deg[g[i].u] == deg[g[i].v] && g[i].u < g[i].v))
                e[g[i].u].push_back(g[i].v),v[g[i].u].push_back(g[i].w);
            else
                e[g[i].v].push_back(g[i].u),v[g[i].v].push_back(g[i].w);
        for(register int x = 1;x <= mx;++x)
            if(mu[x])
            {
                for(register int i = 0;i < e[x].size();++i)
                    vis[e[x][i]] = v[x][i];
                for(register int i = 0,y,xy;i < e[x].size() && (y = e[x][i],xy = v[x][i],1);++i)
                    if(mu[y])
                        for(register int j = 0,z,xz,yz;j < e[y].size() && (z = e[y][j],xz = vis[z],yz = v[y][j],1);++j)
                            if(xz && mu[z])
                                ans += mu[x] * mu[y] * mu[z] * (
                                    f[0][xy] * f[1][xz] * f[2][yz] +
                                    f[0][xy] * f[1][yz] * f[2][xz] +
                                    f[0][xz] * f[1][xy] * f[2][yz] +
                                    f[0][xz] * f[1][yz] * f[2][xy] +
                                    f[0][yz] * f[1][xy] * f[2][xz] +
                                    f[0][yz] * f[1][xz] * f[2][xy]
                                );
                for(register int i = 0;i < e[x].size();++i)
                    vis[e[x][i]] = 0;
            }
        printf("%lld\n",ans % mod);
    }
}