hdu 5869 Different GCD Subarray Query BIT+GCD 2016ICPC 大连网络赛

Posted 2020-08-09 smilesundream

Different GCD Subarray Query

Time Limit: 6000/3000 MS (Java/Others)    Memory Limit: 65536/65536 K (Java/Others)
Total Submission(s): 828    Accepted Submission(s): 300

Problem Description

This is a simple problem. The teacher gives Bob a list of problems about GCD (Greatest Common Divisor). After studying some of them, Bob thinks that GCD is so interesting. One day, he comes up with a new problem about GCD. Easy as it looks, Bob cannot figure it out himself. Now he turns to you for help, and here is the problem:
  
  Given an array a of N positive integers a1,a2,?aN−1,aN; a subarray of a is defined as a continuous interval between a1 and aN. In other words, ai,ai+1,?,aj−1,aj is a subarray of a, for 1≤i≤j≤N. For a query in the form (L,R), tell the number of different GCDs contributed by all subarrays of the interval [L,R].
  

 

 

Input

There are several tests, process till the end of input.
  
  For each test, the first line consists of two integers N and Q, denoting the length of the array and the number of queries, respectively. N positive integers are listed in the second line, followed by Q lines each containing two integers L,R for a query.

You can assume that 
  
    1≤N,Q≤100000 
    
   1≤ai≤1000000

 

 

Output

For each query, output the answer in one line.

 

 

Sample Input

5 3 1 3 4 6 9 3 5 2 5 1 5

 

 

Sample Output

6 6 6

 

 

Source

2016 ACM/ICPC Asia Regional Dalian Online

 题意：有n个数字依次存放在一个数组中(n<=1e5),每个数字<=1e6,数组中每个子序列可以产生一个整个子序列的最大公约数，有q个询问(q<=1e5)，每次询问包括两个数字，l,r询问下标从l,到r的区间内一共有多少个不同的GCD；

#include <cstdio>
#include <iostream>
#include <algorithm>
#include <cstring>
#include <iostream>
#include <cmath>
#include <map>
#include <stack>
#include <queue>
#include <vector>
#include <bitset>
#include <set>
#define MM(a,b) memset(a,b,sizeof(a));
#define inf 0x3f3f3f3f
using namespace std;
typedef long long ll;
#define CT continue
#define SC scanf
const int N=1e5+10;
int n,qur,a[N],tree[N],ans[N],pos[10*N];

struct node{
   int l,id;
};

int gcd(int a,int b)
{
   if(b==0) return a;
   else return gcd(b,a%b);
}

int lowbit(int i)
{
   return i&(-i);
}

int add(int pos,int val)
{
    while(pos<=n){
        tree[pos]+=val;
        pos+=lowbit(pos);
    }
}

int query(int r)
{
    int s=0;
    while(r>=1){
        s+=tree[r];
        r-=lowbit(r);
    }
    return s;
}

vector<node> q[N],lgcd[N];

void solve()
{
    for(int i=1;i<=n;i++){
        if(pos[a[i]]!=-1) add(pos[a[i]],-1);
        pos[a[i]]=i;
        add(i,1);
        int val=a[i];
        for(int j=i-1;j>=1;j--){
           int k=gcd(val,a[j]);
           if(pos[k]<j){
              if(pos[k]!=-1) add(pos[k],-1);
              pos[k]=j;
              add(j,1);
           }
           if(k==1) break;
           val=k;
        }
        for(int j=0;j<q[i].size();j++){
            int l=q[i][j].l,id=q[i][j].id;
            ans[id]=query(i)-query(l-1);
        }
    }
}

int main()
{
    while(~SC("%d%d",&n,&qur)){
       MM(tree,0);
       MM(pos,-1);
       for(int i=1;i<=n;i++) {
          SC("%d",&a[i]);
          q[i].clear();
       }
       for(int i=1;i<=qur;i++) {
           int l,r;
           SC("%d%d",&l,&r);
           q[r].push_back((node){l,i});
       }
       solve();
       for(int i=1;i<=qur;i++) printf("%d\n",ans[i]);
    }
    return 0;
}

　　分析：

错因分析：比赛的时候想到了每个数字的gcd并不会很多，，但是想到的解决方法是，先统计出从1到i（1<=i<=n）的各个位置所拥有的gcd种类数，，然后对于一个区间[l,r]，用种类数r-种类数l-1，，，，但是这样有个很显然的错误就是[l,l-1]内出现的gcd有可能在[l,r]内再次出现，所以这样肯定就错了

纠正与解答：对于这样的问题，

1.我们可以从1-n依次固定右端点，然后从i向前扫，得到一个gcd，然后用BIT维护其gcd最靠右位置，在BIT中+1(可以想象，固定右端点后，越靠右，则不管怎样的区间，都尽可能包含)

2.最多在loga时间内gcd衰减至1.复杂度nlogn*logn;