2017级寒假讲座二分三分训练题 F - Yukari's Birthday （二分法的用法之一）

题目：F - Yukari's Birthday
Today is Yukari's n-th birthday. Ran and Chen hold a celebration party for her. Now comes the most important part, birthday cake! But it's a big challenge for them to place n candles on the top of the cake. As Yukari has lived for such a long long time, though she herself insists that she is a 17-year-old girl.
To make the birthday cake look more beautiful, Ran and Chen decide to place them like r ≥ 1 concentric circles. They place k i candles equidistantly on the i-th circle, where k ≥ 2, 1 ≤ i ≤ r. And it's optional to place at most one candle at the center of the cake. In case that there are a lot of different pairs of r and k satisfying these restrictions, they want to minimize r × k. If there is still a tie, minimize r.
Input
There are about 10,000 test cases. Process to the end of file.
Each test consists of only an integer 18 ≤ n ≤ 10 12.
Output
For each test case, output r and k.
Sample Input
18
111
1111
Sample Output
1 17
2 10
3 10

大意：找到r圈，i为1到r圈，依次为k的i次方的蜡烛个数，找到这个r和k。

思路：通过提前的计算得出，r最大只能为40，所以可以通过枚举进行，然后里面的k的选择出来的方法是通过二分法进行的。这里要注意的是为了避免数据溢出的操作！！；

小技巧：关于相乘可能导致的数据溢出，提前判断的方法是通过相处来判断，以后要成一个习惯即通过k的n次方这种，一开始先设立的为1，然后之后进行先乘在求和的操作，而不是设立一开始的k，然后先求和再乘，后者不能进行判断是否在一开始求和前有溢出的可能（尽量在求和前判断，而非求和后判断）；

代码：

#include<stdio.h>

int main()
{
    long long r,k,i,j,n;
    while(scanf("%lld",&n)!=EOF){
            long long left,right,mid,mids,mins=n-1,min_r=1,min_k=n-1;
            long long  sum;
            for(r=1;r<=40;r++){
                left=1;right=n;
                while(right>=left){
                    sum=0;
                    mid=(right+left)/2;
                    mids=1;//对于这个有很大的可能出现溢出的问题的时，尽量一开始先是1，毕竟有可能一上来就未必满足相加的条件
                    for(i=1;i<=r;i++)
                    {
                        if(n/mids<mid)
                        {
                            sum=n+1;
                            break;}
                        mids*=mid;
                        sum+=mids;
                        if(sum>n)
                            break;

                    }
                   // printf("%lld\n",sum);
                    if(sum==n-1||sum==n){
                        if(mins>r*mid)
                        {mins=r*mid;min_r=r;min_k=mid;}
                    }
                    if(sum>n)
                        right=mid-1;
                    else
                        left=mid+1;
              //  printf("%lld\n",min_k);
              //  printf("%lld %lld\n",right,left);
                }
            }
            printf("%lld %lld\n",min_r,min_k);
    }
    return 0;
}