C. Did We Get Everything Covered?

C. Did We Get Everything Covered?

You are given two integers $n$ and $k$ along with a string $s$.

Your task is to check whether all possible strings of length $n$ that can be formed using the first $k$ lowercase English alphabets occur as a subsequence of $s$. If the answer is NO, you also need to print a string of length $n$ that can be formed using the first $k$ lowercase English alphabets which does not occur as a subsequence of $s$.

If there are multiple answers, you may print any of them.

Note: A string $a$ is called a subsequence of another string $b$ if $a$ can be obtained by deleting some (possibly zero) characters from $b$ without changing the order of the remaining characters.

Input

The first line of input contains a single integer $t \, (1 \le t \le 10^5)$, the number of test cases.

The first line of each test case contains $3$ integers $n \, (1 \le n \le 26), \: k \, (1 \le k \le 26), \: m \, (1 \le m \le 1000)$, where $n$ and $k$ are the same as described in the input and $m$ is the length of the string $s$.

The second line of each test case contains a single string $s$ of length $m$, comprising only of the first $k$ lowercase English alphabets.

It is guaranteed that the sum of $m$ and the sum of $n$ over all test cases does not exceed $10^6$.

Output

For each test case, print YES if all possible strings of length $n$ that can be formed using the first $k$ lowercase English alphabets occur as a subsequence of $s$, else print NO.

If your answer is NO, print a string of length $n$ that can be formed using the first $k$ lowercase English alphabets which does not occur as a subsequence of $s$ in the next line.

You may print each letter of YES or NO in any case (for example, YES, yES, YeS will all be recognized as a positive answer).

Example

input

3
2 2 4
abba
2 2 3
abb
3 3 10
aabbccabab

output

YES
NO
aa
NO
ccc

Note

For the first test case, all possible strings (aa, ab, ba, bb) of length $2$ that can be formed using the first $2$ English alphabets occur as a subsequence of abba.

For the second test case, the string aa is not a subsequence of abb.

 

解题思路

  受 A 题 We Got Everything Covered! 的启发，如果将前 $k$ 个字符构成一组，并把至少 $n$ 组拼接成字符串，那么字符串所有的子序列一定包含 $k^n$ 种由前 $k$ 个字符构成的长度为 $n$ 的串。不过这只说明了充分性，并不能说明必要性，即如果某字符串所有的子序列包含由前 $k$ 个字符构成的长度为 $n$ 的串，则该字符串一定至少包含 $n$ 个由前 $k$ 个字符构成的组。比赛的时候没想到也没去猜所以没做出来。

  事实上这个必要性是成立的。我们依次找到 $s$ 中由前 $k$ 个字符构成的组，即依次遍历 $s$ 的每个字符，用一个哈希表记录某个字符是否出现过，以及一个变量 $c$ 统计当成组中有多少种字符。当遍历到 $s_i$，哈希表的 $s_i$ 处记为 $\text{true}$，如果在变化之前为 $\text{false}$ 则 $c$ 加 $1$。如果此时的 $c=k$，说明已经找到一组，清空哈希表和 $c$ 继续找下一组。

  如果我们每次都选择每组中最后一个字符来尝试构成一个反例，显然如果字符串中小于 $n$ 组，那么就一定可以构造出一个反例，只需将最后一组中没出现过的字符填到反例后面直到长度变成 $n$。这种构造方案也证明了必要性。

  AC 代码如下，时间复杂度为 $O(m)$：

#include <bits/stdc++.h>
using namespace std;

typedef long long LL;

const int N = 1010;

char s[N];
bool vis[26];

void solve() {
    int n, m, k;
    scanf("%d %d %d %s", &n, &k, &m, s);
    string ans;
    memset(vis, 0, sizeof(vis));
    for (int i = 0, c = 0; i < m; i++) {
        if (!vis[s[i] - 'a']) c++;
        vis[s[i] - 'a'] = true;
        if (c == k) {
            ans.push_back(s[i]);
            memset(vis, 0, sizeof(vis));
            c = 0;
        }
    }
    if (ans.size() >= n) {
        printf("YES\n");
    }
    else {
        printf("NO\n");
        for (int i = 0; i < 26; i++) {
            if (!vis[i]) {
                while (ans.size() < n) {
                    ans.push_back('a' + i);
                }
                printf("%s\n", ans.c_str());
                return;
            }
        }
    }
}

int main() {
    int t;
    scanf("%d", &t);
    while (t--) {
        solve();
    }
    
    return 0;
}

 

参考资料

  Codeforces Round 921 (Div. 1, Div. 2) Editorial：https://codeforces.com/contest/1925/problem/C