hdu 5975---Aninteresting game（树状数组）

Posted 2020-09-21 茶飘香~

题目链接

 

Problem Description

Let’s play a game.We add numbers 1,2...n in increasing order from 1 and put them into some sets.
When we add i,we must create a new set, and put iinto it.And meanwhile we have to bring [i-lowbit(i)+1,i-1] from their original sets, and put them into the new set,too.When we put one integer into a set,it costs us one unit physical strength. But bringing integer from old set does not cost any physical strength.
After we add 1,2...n,we have q queries now.There are two different kinds of query:
1 L R:query the cost of strength after we add all of [L,R](1≤L≤R≤n)
2 x:query the units of strength we cost for putting x(1≤x≤n) into some sets.

 

Input

There are several cases,process till end of the input.
For each case,the first line contains two integers n and q.Then q lines follow.Each line contains one query.The form of query has been shown above.
n≤10^18,q≤10^5

 

Output

For each query, please output one line containing your answer for this query

 

Sample Input

10 2

1 8 9

2 6

 

Sample Output

9

2

 

Hint

lowbit(i) =i&(-i).It means the size of the lowest nonzero bits in binary of i. For example, 610=1102, lowbit(6) =102= 210 When we add 8,we should bring [1,7] and 8 into new set. When we add 9,we should bring [9,8] (empty) and 9 into new set. So the first answer is 8+1=9. When we add 6 and 8,we should put 6 into new sets. So the second answer is 2.

 

题意：每次查询有两种操作

           op1：求加入L~R的数时所消耗的单元

           op2：求将x加入集合或移动到其它集合所消耗的单元（即由x引起消耗的单元）

 

思路：op1：每次加入一个数i 那么会移动[i-lowbit(i)+1 , i-1] ，总的消耗是i-(i-lowbit(i)+1) +1=lowbit(i) 所以每次加入一个数对应的消耗是2的幂次，那么求L~R即可以枚举幂次，即： ans+=(n/(1<<i)-n/(1<<(i+1)))*(1<<i)

                 解释一下，n/(1<<i)-n/(1<<(i+1))表示长为2^i的消耗的数的个数，例如：n=10 ， 包含长为2的数是2，6，10 为什么4，8不是，因为它们虽然是2的倍数，但更是4的倍数，包含更长的区间了，所以这部分要减去。

        op2：由树状数组可知 [i-lowbit(i)+1 , i-1] 是以i为根节点对应的区间，如果假如的数能够移动i ，那么这个数对应的孩子区间一定包含i ，所以从x向上一直找父节点即可。

 

代码如下：

#include <iostream>
#include <algorithm>
#include <cstring>
#include <cstdio>
using namespace std;
typedef long long LL;
LL lowbit(LL x)
{
    return x&(-x);
}
LL query(LL x,LL n)
{
    LL ans=0;
    while(x<=n)
    {
        ans++;
        x+=lowbit(x);
    }
    return ans;
}
LL cal(LL x)
{
    LL ans=0;
    LL tmp=1;
    for(LL i=0; tmp<=x; i++)
        ans+=(x/(tmp)-x/(tmp<<1))*tmp,tmp<<=1;
    return ans;
}
int main()
{
    LL n,q;
    while(scanf("%lld%lld",&n,&q)!=EOF)
    {
        while(q--)
        {
            int op;
            scanf("%d",&op);
            if(op==1)
            {
                LL x,y;
                scanf("%lld%lld",&x,&y);
                LL ans=cal(y)-cal(x-1);
                printf("%lld\n",ans);
            }
            else
            {
                LL x;
                scanf("%lld",&x);
                LL ans=query(x,n);
                printf("%lld\n",ans);
            }
        }
    }
    return 0;
}