UVA 1594 Ducci Sequence（两极问题）

    

 

                     Ducci Sequence

Time Limit:3000MS     Memory Limit:0KB     64bit IO Format:%lld & %llu

 

Description

A Ducci sequence is a sequence of n-tuples of integers. Given an n-tuple of integers (a₁, a₂, ^... , a_n), the next n-tuple in the sequence is formed by taking the absolute differences of neighboring integers:

( a₁, a₂, ^... , a_n)  (| a₁ - a₂|,| a₂ - a₃|, ^... ,| a_n - a₁|)

Ducci sequences either reach a tuple of zeros or fall into a periodic loop. For example, the 4-tuple sequence starting with 8,11,2,7 takes 5 steps to reach the zeros tuple:

 (8, 11, 2, 7)  (3, 9, 5, 1)  (6, 4, 4, 2)  (2, 0, 2, 4)  (2, 2, 2, 2)  (0, 0, 0, 0).

The 5-tuple sequence starting with 4,2,0,2,0 enters a loop after 2 steps:

(4, 2, 0, 2, 0)  (2, 2, 2, 2, 4)  ( 0, 0, 0, 2, 2)  (0, 0, 2, 0, 2)  (0, 2, 2, 2, 2)  (2, 0, 0, 0, 2) (2, 0, 0, 2, 0)  (2, 0, 2, 2, 2)  (2, 2, 0, 0, 0)  (0, 2, 0, 0, 2)  (2, 2, 0, 2, 2)  (0, 2, 2, 0, 0) (2, 0, 2, 0, 0)  (2, 2, 2, 0, 2)  (0, 0, 2, 2, 0)  (0, 2, 0, 2, 0)  (2, 2, 2, 2, 0)  ( 0, 0, 0, 2, 2)  ^...

 Given an n-tuple of integers, write a program to decide if the sequence is reaching to a zeros tuple or a periodic loop.

Input 

Your program is to read the input from standard input. The input consists of T test cases. The number of test cases T is given in the first line of the input. Each test case starts with a line containing an integer n(3n15), which represents the size of a tuple in the Ducci sequences. In the following line, n integers are given which represents the n-tuple of integers. The range of integers are from 0 to 1,000. You may assume that the maximum number of steps of a Ducci sequence reaching zeros tuple or making a loop does not exceed 1,000.

Output 

Your program is to write to standard output. Print exactly one line for each test case. Print `LOOP' if the Ducci sequence falls into a periodic loop, print `ZERO' if the Ducci sequence reaches to a zeros tuple.

The following shows sample input and output for four test cases.

Sample Input 

4 
4 
8 11 2 7 
5 
4 2 0 2 0 
7 
0 0 0 0 0 0 0 
6 
1 2 3 1 2 3

Sample Output 

ZERO 
LOOP 
ZERO 
LOOP

题解：给定数组，依次求前一个减后一个的值的绝对值，如果是最后一个则是最后一个减第一个的值的绝对值，

最多循环1000次，如果出现数组的值全部变为0，则为ZERO，否则为LOOP，所以只求全部为0的情况即可。

#include<iostream>
using namespace std;
int main()
{int a[20];
    int i,j,t,n,sum;
    cin>>t;
    while(t--)
    {
        cin>>n;
        for(i=0; i<n; i++)
            cin>>a[i];
        for(i=0; i<1000; i++)
        {  sum=0;
            int s=a[0];
            for(j=0; j<n-1; j++)
            {
                 if(a[j]>a[j+1])
                        a[j]=a[j]-a[j+1];
                    else
                        a[j]=a[j+1]-a[j];
                    sum+=a[j];

            }

                    if(a[n-1]>s)
                        a[n-1]=a[n-1]-s;
                    else a[n-1]=s-a[n-1];
                    sum+=a[n-1];
                if(sum==0)
                    break;


        }

    if(sum==0)
        cout<<"ZERO"<<endl;
    else
        cout<<"LOOP"<<endl;


    }
    return 0;
}