Ducci Sequence

最新推荐文章于 2020-08-23 13:54:00 发布

revstar_

最新推荐文章于 2020-08-23 13:54:00 发布

阅读量1.2k

点赞数

CC 4.0 BY-SA版权

分类专栏：算法竞赛入门经典

本文链接：https://blog.youkuaiyun.com/a1967919189/article/details/46925271

算法竞赛入门经典专栏收录该内容

5 篇文章

订阅专栏

本文介绍了一种名为杜奇序列的数学现象，该序列通过计算整数元组相邻元素的绝对差形成新的元组。文章详细解释了杜奇序列要么达到全零状态要么进入周期循环的特性，并提供了一个用于判断序列走向的程序实现。

题目：

Ducci Sequence

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:

( a ₁, a ₂, ^..., a _n) $\displaystyle \rightarrow$ (| a ₁ - a ₂|,| a ₂ - a ₃|, ^...,| a _n - a ₁|)

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) $\displaystyle \rightarrow$ (3, 9, 5, 1) $\displaystyle \rightarrow$ (6, 4, 4, 2) $\displaystyle \rightarrow$ (2, 0, 2, 4) $\displaystyle \rightarrow$ (2, 2, 2, 2) $\displaystyle \rightarrow$ (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) $\displaystyle \rightarrow$ (2, 2, 2, 2, 4) $\displaystyle \rightarrow$ ( 0, 0, 0, 2, 2) $\displaystyle \rightarrow$ (0, 0, 2, 0, 2) $\displaystyle \rightarrow$ (0, 2, 2, 2, 2) $\displaystyle \rightarrow$ (2, 0, 0, 0, 2) $\displaystyle \rightarrow$

(2, 0, 0, 2, 0) $\displaystyle \rightarrow$ (2, 0, 2, 2, 2) $\displaystyle \rightarrow$ (2, 2, 0, 0, 0) $\displaystyle \rightarrow$ (0, 2, 0, 0, 2) $\displaystyle \rightarrow$ (2, 2, 0, 2, 2) $\displaystyle \rightarrow$ (0, 2, 2, 0, 0) $\displaystyle \rightarrow$

(2, 0, 2, 0, 0) $\displaystyle \rightarrow$ (2, 2, 2, 0, 2) $\displaystyle \rightarrow$ (0, 0, 2, 2, 0) $\displaystyle \rightarrow$ (0, 2, 0, 2, 0) $\displaystyle \rightarrow$ (2, 2, 2, 2, 0) $\displaystyle \rightarrow$ ( 0, 0, 0, 2, 2) $\displaystyle \rightarrow$ ^...

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(3 $\le$ n $\le$ 15), 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

代码如下：

#include<iostream>
#include<cmath>
using namespace std;
int const maxn=1000;
int const maxm=16;
int len;
int zero=0;
int a[maxn][maxm];
int main()
{
    int n;
    cin>>n;
   while(n--)
   {
       cin>>len;
       for(int i=0;i<len;i++)
        cin>>a[0][i];
        int flag=0;
        for(int i=1;i<maxn;i++)
      {
          for(int j=0;j<len;j++)
        {
            if(j==len-1)
                a[i][j]=abs(a[i-1][j]-a[i-1][0]);
                else
                a[i][j]=abs(a[i-1][j]-a[i-1][j+1]);

        }

      for(int h=0;h<len;h++)
if(a[i][h]!=0) {flag=1;break;}
if(flag==0) {zero=1;}
else flag=0;
   }
   if(zero)
   {
       cout<<"ZERO"<<endl;
   zero=0;
   }
   else
    cout<<"LOOP"<<endl;
   }
    return 0;
}