CodeForces 1344 A Hilberts Hotel 數學

1. 題目描述

1.1. Limit

Time Limit: 1000 ms

Memory Limit: 131,072 kB

1.2. Problem Description

Hilbert’s Hotel is a very unusual hotel since the number of rooms is infinite! In fact, there is exactly one room for every integer, including zero and negative integers. Even stranger, the hotel is currently at full capacity, meaning there is exactly one guest in every room. The hotel’s manager, David Hilbert himself, decides he wants to shuffle the guests around because he thinks this will create a vacancy (a room without a guest).

For any integer $k$ and positive integer $n$ , let $k \mod n$ denote the remainder when $k$ is divided by $n$. More formally, $r=k \mod n$ is the smallest non-negative integer suchthat $k−r$ is divisible by $n$. It always holds that $0 \le k \mod n \le n−1$. For example, $100 \mod 12 = 4$ and $(−1337) \mod 3 = 1$.

Then the shuffling works as follows. There isan array of $n$ integers $a_0,a_1,\dots,a_{n−1}$. Then for each integer $k$, the guest in room $k$ is moved to room number $k+a_k \mod n$.

After this shuffling process, determine if there is still exactly one guest assigned to each room. That is, there are no vacancies or rooms with multiple guests.

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1.3. Input

Each test consists of multiple test cases. The first line contains a single integer $t(1 \le t \le 10^4)$ — the number of test cases.

Next $2t$ lines contain descriptions of test cases.The first line of each test case contains a single integer $(1\le n \le2 \cdot 10^5)$ — the length of the array.

The second line of each test case contains 𝑛n integers $a_0,a_1,\dots,a_{n-1}$ $(−10^9 \le a_i \le 10^9)$.

It is guaranteed that the sum of $n$ over all test cases does not exceed $2 \cdot 10^5$.

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1.4. Output

For each test case, output a single line containing “YES” if there is exactly one guest assigned to each room after the shuffling process, or “NO” otherwise. You can print each letter in any case (upper or lower).

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1.5. Sample Input

6
1
14
2
1 -1
4
5 5 5 1
3
3 2 1
2
0 1
5
-239 -2 -100 -3 -11

1.6. Sample Output

YES
YES
YES
NO
NO
YES

1.7. Notes

In the first test case, every guest is shifted by $14$ rooms, so the assignment is still unique.

In the second test case, even guests move to the right by $1$ room, and odd guests move to the left by $1$ room. We can show that the assignment is still unique.

In the third test case, every fourth guest moves to the right by $1$ room, and the other guests move to the right by $5$ rooms. We can show that the assignment is still unique.

In the fourth test case, guests $0$ and $1$ are both assigned to room $3$ .

In the fifth test case, guests $1$ and $2$ are both assigned to room $2$ .

1.8. Source

CodeForces 1344 A Hilbert’s Hotel

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2. 解讀

計算 $((a_n + n) \%n + n ) \% n$，$n \in [0, n - 1]$，判斷其結果是否有重複，用 set 存儲後，比較 set.size() 和$n$ 的大小即可。

因爲 $a_i$ 的範圍爲$(−10^9 \le a_i \le 10^9)$。

• $a_n + n$爲正數時，結果爲 $(a_n + n)\%n$。

• $a_n + n$爲負數時，結果爲 $n + (n + a_n)\%n$。

將正負數的情況統一起來，結果爲$((a_n + n) \%n + n ) \% n$。

3. 代碼

#include <iostream>
#include <set>
#include <string.h>
using namespace std;

// 集合存儲
set<long long> st;

int main()
{
    // test case
    int t;
    // 個數
    long long n;
    long long buffer;
    scanf("%d", &t);
    // test case
    for (int i = 0; i < t; i++) {
        scanf("%lld", &n);
        // 初始化
        st.clear();
        // 輸入
        for (long long j = 0; j < n; j++) {
            scanf("%lld", &buffer);
            st.insert(((j + buffer) % n + n) % n);
        }

        printf("%s\n", (long long)st.size() == n ? "YES" : "NO");
    }
}

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