CF1703D.Double Strings

You are given $n$ strings $s_1, s_2, \dots, s_n$ of length at most $\mathbf{8}$ .

For each string $s_i$ , determine if there exist two strings $s_j$ and $s_k$ such that $s_i = s_j + s_k$ . That is, $s_i$ is the concatenation of $s_j$ and $s_k$ . Note that $j$ can be equal to $k$ .

Recall that the concatenation of strings $s$ and $t$ is $s + t = s_1 s_2 \dots s_p t_1 t_2 \dots t_q$ , where $p$ and $q$ are the lengths of strings $s$ and $t$ respectively. For example, concatenation of "code" and "forces" is "codeforces".

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

The first line of each test case contains a single integer $n$ ( $1 \leq n \leq 10^5$ ) — the number of strings.

Then $n$ lines follow, the $i$ -th of which contains non-empty string $s_i$ of length at most $\mathbf{8}$ , consisting of lowercase English letters. Among the given $n$ strings, there may be equal (duplicates).

The sum of $n$ over all test cases doesn't exceed $10^5$ .

For each test case, output a binary string of length $n$ . The $i$ -th bit should be $\texttt{1}$ if there exist two strings $s_j$ and $s_k$ where $s_i = s_j + s_k$ , and $\texttt{0}$ otherwise. Note that $j$ can be equal to $k$ .

In the first test case, we have the following:

• $s_1 = s_2 + s_2$ , since $\texttt{abab} = \texttt{ab} + \texttt{ab}$ . Remember that $j$ can be equal to $k$ .
• $s_2$ is not the concatenation of any two strings in the list.
• $s_3 = s_2 + s_5$ , since $\texttt{abc} = \texttt{ab} + \texttt{c}$ .
• $s_4$ is not the concatenation of any two strings in the list.
• $s_5$ is not the concatenation of any two strings in the list.

Since only $s_1$ and $s_3$ satisfy the conditions, only the first and third bits in the answer should be $\texttt{1}$ , so the answer is $\texttt{10100}$ .