Ducci Sequence解题报告

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

题意：给你一个数组，相邻数相加（最后一个数应该加第一个数），然后又可以得到一个新的数组，反复进行这样的步骤，结果会有两种，数组每个成员都是零，或者在这样的过程中出现了周期，即出现了和以前已经出现过的相同的项。最后要求判断是哪一种情况

思路：如果按照一般的解题思想，每得到一个新的数组都去判断是zero，还是loop。zero还好说，如果判断loop那就要每次遍历一遍数组，程序肯定会超时。、所以~~~反正结果只有两种可能，不是loop，就是zero，只判断zero就好了

代码：

#include"iostream"
#include"cstring"
using namespace std;

const int maxn=1010;
const int maxm=16;
int len;
int loop=0,zero=0;

int a[maxn][maxm];

int swift(int number)
{
	return (number<0)?-number:number;
}

void Init()
{
cin>>len;
for(int i=0;i<len;i++)
cin>>a[0][i];
}

void Work()
{
int flag=0;
for(int j=1;j<maxn;j++)
{
for(int i=0;i<len;i++)
{
	if(i==len-1)
		a[j][i]=swift(a[j-1][i]-a[j-1][0]);
	else
        a[j][i]=swift(a[j-1][i]-a[j-1][i+1]);
}

for(int h=0;h<len;h++)
if(a[j][h]!=0) {flag=1;break;}
if(flag==0) {zero=1;return;}
else flag=0;
}
}

int main()
{
	int n;
	cin>>n;
	while(n--)
	{
		Init();
		Work();
		if(zero)
		{
        cout<<"ZERO"<<endl;
		 zero=0;
		}
		else
		cout<<"LOOP"<<endl;
	}
	return 0;
}