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:

 

 

(  a ₁,  a ₂,  ^... ,  a _n)   (|  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)   (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; }