紫书第五章训练 uva 1594 Ducci Sequence by crq

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

题意分析：对一个数组不断按照以下过程进行变换（每个数变成后面一个元素与自身相减后取绝对值，最后一个数的下一个是第一个数）：

(a1, a2, · · · , an) → (|a1 − a2|, |a2 − a3|, · · · , |an − a1|)

到最后要么变成全部为0（输出ZERO）， 要么形成死循环(LOOP)。

解决方法：因为数组元素个数最大才15，直接模拟，数组用vector记录，并用map记录每次变换后数组元素：

map<vector<int>, int> mp;

每次变换后查找map中是否已经存在，若已经存在则输出LOOP，结束。如果变换后结果是全部为0的数组，输出ZERO，结束。

AC源码：

#include <iostream>
#include <string>
#include <vector>
#include <sstream>
#include <map>
using namespace std;

map<vector<int>, int> mp;
int myabs(int a)
{
    return a>0?a:-a;
}

bool Zero(vector<int> &vec)
{
    for(int i=0;i<vec.size();i++)
    {
        if(vec[i]!=0)
            return false;
    }
    return true;
}

void Change(vector<int>& vec)
{
    int n = vec.size();
    int x = vec[0];
    for(int i=0;i<n-1;i++)
    {
        vec[i] = myabs(vec[i+1]-vec[i]);
    }
    vec[n-1] = myabs(vec[n-1]-x);
}

int main()
{
//    freopen("d:\data1.in","r",stdin);
    int cas;
    cin>>cas;
    while(cas--)
    {
        mp.clear();
        int i, n;
        vector<int> vec;
        cin>>n;
        for(i=0;i<n;i++)
        {
            int x;
            cin>>x;
            vec.push_back(x);
        }
        int flag = 0;
        while(true)
        {
            if(mp[vec])
            {
                flag = 2;//loop
                break;
            }
            mp[vec] = 1;
            if(Zero(vec))
            {
                flag = 1;//zero
                break;
            }
            Change(vec);
        }
        if(flag==1)
            cout<<"ZERO"<<endl;
        else if(flag==2)
            cout<<"LOOP"<<endl;
    } 
    return 0;
}