[POI2007] ZAP-Queries (莫比乌斯反演)

[POI2007] ZAP-Queries

题目描述

Byteasar the Cryptographer works on breaking the code of BSA (Byteotian Security Agency). He has alreadyfound out that whilst deciphering a message he will have to answer multiple queries of the form"for givenintegers aa, bb and dd, find the number of integer pairs (x,y)(x,y) satisfying the following conditions:

1le xle a1≤x≤a,1le yle b1≤y≤b,gcd(x,y)=dgcd(x,y)=d, where gcd(x,y)gcd(x,y) is the greatest common divisor of xx and yy".

Byteasar would like to automate his work, so he has asked for your help.

TaskWrite a programme which:

reads from the standard input a list of queries, which the Byteasar has to give answer to, calculates answers to the queries, writes the outcome to the standard output.

FGD正在破解一段密码，他需要回答很多类似的问题：对于给定的整数a,b和d，有多少正整数对x,y，满足x<=a，y<=b，并且gcd(x,y)=d。作为FGD的同学，FGD希望得到你的帮助。

输入输出格式

输入格式：

The first line of the standard input contains one integer nn (1le nle 50 0001≤n≤50 000),denoting the number of queries.

The following nn lines contain three integers each: aa, bb and dd(1le dle a,ble 50 0001≤d≤a,b≤50 000), separated by single spaces.

Each triplet denotes a single query.

输出格式：

Your programme should write nn lines to the standard output. The ii'th line should contain a single integer: theanswer to the ii'th query from the standard input.

输入输出样例

输入样例#1：

2
4 5 2
6 4 3

输出样例#1：

3
2

Solution

预备知识:莫比乌斯反演,整除分块

不会的看这位dalao的博客莫比乌斯反演

本蒟蒻的整除分块

根据题意

[ans=sum_{i=1}^a sum_{j=1}^b [{gcd(i,j)=d}] ]

[ans=sum_{i=1}^{a/d} sum_{j=1}^{b/d}[gcd(i,j)=1] ]

下面就是反演

[ans=sum_{i=1}^{a/d} sum_{j=1}^{b/d} sum_{p|gcd(i,j)}mu(p) ]

但是这样枚举还是(O(n^2)),所以我们换一个变量枚举,把最后一个求和提到前面,因为p既是i的因子又是j的因子,所以枚举范围就是(min(a/d,b/d)),那么继续推公式

[ans=sum_{p=1}^{min(a/d,b/d)}{mu(p)} sum_{i=1}^{a/d} sum_{j=1}^{b/d} lfloorfrac{a}{p imes d} floor lfloorfrac{b}{p imes d} floor ]

如果对于后面的式子不理解,可以这么看,令(x=a/d,y=b/d)
(p)是(x,y)的一个因子,在(x)的范围内有(lfloorfrac{x}{p} floor)个(p)的倍数,对于(y)同理,所以每个因子(p)都有(lfloorfrac{x}{p} floorlfloorfrac{y}{p} floor)的贡献

而对于后面的两个求和我们是可以用前缀和预处理出来的,这个时候是可以做到(O(n))了,但是由于多组数据,所以我们发现,对于一段连续的p,因为a和b的值是确定的,所以(lfloorfrac{a}{p imes d} floorlfloorfrac{b}{p imes d} floor)的值也是确定的,这中间有许多重复的值,那么我们就可以使用整除分块优化到(O(sqrt n))

(有错误欢迎指出)

Code

#include<bits/stdc++.h>
#define lol long long
#define il inline
#define rg register
#define Min(a,b) (a)<(b)?(a):(b)
#define Max(a,b) (a)>(b)?(a):(b)

using namespace std;

const int N=5e4+10;

void in(int &ans)
{
    ans=0; int f=1; char i=getchar();
    while(i<'0' || i>'9') {if(i=='-') f=-1; i=getchar();}
    while(i>='0' && i<='9') ans=(ans<<1)+(ans<<3)+i-'0', i=getchar();
    ans*=f;
}

int n,m,d,tot,ans,T;
int mu[N],sum[N],prime[N];
bool vis[N];

il void get_mu() {
    mu[1]=1;
    for(int i=2;i<=N-10;i++) {
        if(!vis[i]) prime[++tot]=i,mu[i]=-1;
        for(int j=1;j<=tot && prime[j]*i<=N-10;j++) {
            vis[i*prime[j]]=1;
            if(i%prime[j]==0) break;
            else mu[i*prime[j]]=-mu[i];
        }
    }
    for(int i=1;i<=N-10;i++) sum[i]=sum[i-1]+mu[i];
}

int main()
{
    in(T); get_mu();
    while(T--) {
        in(n),in(m),in(d); int nn=n/=d,mm=m/=d,ans=0;
        for(rg int i=1,pos,p=Min(n,m);i<=p;i=pos+1) {
            pos=Min(n/(n/i),m/(m/i));
            ans+=(sum[pos]-sum[i-1])*(nn/i)*(mm/i);
        }
        printf("%d
",ans);
    }
    return 0;
}