Aizu - 2305 Beautiful Currency (二分 + DFS遍历)

Beautiful Currency

KM country has N kinds of coins and each coin has its value a_i.

The king of the country, Kita_masa, thought that the current currency system is poor, and he decided to make it beautiful by changing the values of some (possibly no) coins.

A currency system is called beautiful if each coin has an integer value and the (i+1)-th smallest value is divisible by the i-th smallest value for all i (1 ¥leq i ¥leq N-1).

For example, the set {1, 5, 10, 50, 100, 500} is considered as a beautiful system, while the set {1, 5, 10, 25, 50, 100} is NOT, because 25 is not divisible by 10.

Since changing the currency system may confuse citizens, the king, Kita_masa, wants to minimize the maximum value of the confusion ratios. Here, the confusion ratio for the change in the i-th coin is defined as |a_i - b_i| / a_i, where a_i and b_i is the value of i-th coin before and after the structure changes, respectively.

Note that Kita_masa can change the value of each existing coin, but he cannot introduce new coins nor eliminate existing coins. After the modification, the values of two or more coins may coincide.

Input

Each dataset contains two lines. The first line contains a single integer, N, and the second line contains N integers, {a_i}.

You may assume the following constraints:

1 ¥leq N ¥leq 20

1 ¥leq a_1 ¥lt a_2 ¥lt... ¥lt a_N ¥lt 10^5

Output

Output one number that represents the minimum of the maximum value of the confusion ratios. The value may be printed with an arbitrary number of decimal digits, but may not contain an absolute error greater than or equal to 10^{-8}.

Sample Input 1

3
6 11 12

Output for the Sample Input 1

0.090909090909

Sample Input 2

3
6 11 24

Output for the Sample Input 2

0.090909090909

Sample Input 3

3
6 11 30

Output for the Sample Input 3

0.166666666667

二分P可以得到每一个数的变化范围[n - n * P, n + n * P]中，接着就是检查是否存在一个序列满足后一个是前一个的倍数。

#include <map>
#include <set>
#include <cstdio>
#include <cmath>
#include <cstring>
#include <vector>
#include <queue>
#include <iostream>
#include <string>
#include <sstream>
#include <cstdlib>
#include <ctime>
#include <cctype>
#include <algorithm>
using namespace std;


#define pb push_back
#define mp make_pair
#define fillchar(a, x) memset(a, x, sizeof(a))
#define copy(a, b) memcpy(a, b, sizeof(a))
#define S_queue<P> priority_queue<P, vector<P>,greater<P> >
#define FIN freopen("D://imput.txt", "r", stdin)


typedef long long LL;
typedef pair<int, int > PII;
typedef unsigned long long uLL;
template<typename T>
void print(T* p, T* q, string Gap = " "){int d = p < q ? 1 : -1;while(p != q){cout << *p;p += d;if(p != q) cout << Gap; }cout << endl;}
template<typename T>
void print(const T &a, string bes = "") {int len = bes.length();if(len >= 2)cout << bes[0] << a << bes[1] << endl;else cout << a << endl;}

const int INF = 0x3f3f3f3f;
const int MAXM = 2e1 + 5;
const int MAXN = 1e2 + 5;
const double eps = 1e-8;
int A[MAXM], n;
double M;

bool DFS(int x, int id, double Max, double m){//是否存在一个序列满足条件
    if(id >= n) {
        M = min(Max, M);//更新最大值
        return true;
    }
    bool flag = false;
    int f = (int)(A[id] * m);
    int cnt = 1,Ma = A[id] + f,Mi = A[id] - f > 1 ? A[id] - f: 1;
    while(cnt * x <= Ma){
        if(cnt * x < Mi) {
            cnt ++;
            continue;
        }
        if(cnt * x >= Mi && cnt * x <= Ma){
            double f_t = fabs(cnt * x - A[id]) / A[id] * 1.0;
            if(DFS(cnt * x, id + 1, max(Max, f_t), m)) {
                    flag = true;
            }
        }
        cnt ++;
    }
    return flag;
}

bool C(double m){
    bool flag = false;
    int f = (int)(A[0] * m);
    int Mi = A[0] - f > 1? A[0] - f : 1, Ma = A[0] + f;
    for(int i = Mi;i <= Ma;i ++){
        if(DFS(i, 1, fabs(A[0] - i) / A[0] * 1.0, m)) {
                flag = true;
        }
    }
    return flag;
}

int main(){
    //FIN;
    while(cin >> n){
        for(int i = 0;i < n;i ++){
            cin >> A[i];
        }
        M = INF;
        double lb = -1,ub = 1.0;
        while(ub - lb > eps){
            double mid = (ub + lb) / 2.0;
            if(C(mid)) ub = mid;
            else lb = mid;
        }
        printf("%.12lf\n", M);
    }
    return 0;
}