1742F · Smaller

Problem: Alperen has two strings, $s$ and $t$ which are both initially equal to "a".

He will perform $q$ operations of two types on the given strings:

• $1 \;\; k \;\; x$ — Append the string $x$ exactly $k$ times at the end of string $s$. In other words, $s := s + \underbrace{x + \dots + x}_{k \text{ times}}$.
• $2 \;\; k \;\; x$ — Append the string $x$ exactly $k$ times at the end of string $t$. In other words, $t := t + \underbrace{x + \dots + x}_{k \text{ times}}$.

After each operation, determine if it is possible to rearrange the characters of $s$ and $t$ such that $s$ is lexicographically smaller$^{\dagger}$ than $t$.

Note that the strings change after performing each operation and don't go back to their initial states.

$^{\dagger}$ Simply speaking, the lexicographical order is the order in which words are listed in a dictionary. A formal definition is as follows: string $p$ is lexicographically smaller than string $q$ if there exists a position $i$ such that $p_i < q_i$, and for all $j < i$, $p_j = q_j$. If no such $i$ exists, then $p$ is lexicographically smaller than $q$ if the length of $p$ is less than the length of $q$. For example, $\texttt{abdc} < \texttt{abe}$ and $\texttt{abc} < \texttt{abcd}$, where we write $p < q$ if $p$ is lexicographically smaller than $q$.

Input Format: The first line of the input contains an integer $t$ ($1 \leq t \leq 10^4$) — the number of test cases.

The first line of each test case contains an integer $q$ $(1 \leq q \leq 10^5)$ — the number of operations Alperen will perform.

Then $q$ lines follow, each containing two positive integers $d$ and $k$ ($1 \leq d \leq 2$; $1 \leq k \leq 10^5$) and a non-empty string $x$ consisting of lowercase English letters — the type of the operation, the number of times we will append string $x$ and the string we need to append respectively.

It is guaranteed that the sum of $q$ over all test cases doesn't exceed $10^5$ and that the sum of lengths of all strings $x$ in the input doesn't exceed $5 \cdot 10^5$.

Output Format: For each operation, output "YES", if it is possible to arrange the elements in both strings in such a way that $s$ is lexicographically smaller than $t$ and "NO" otherwise.

Note: In the first test case, the strings are initially $s =$ "a" and $t =$ "a".

After the first operation the string $t$ becomes "aaa". Since "a" is already lexicographically smaller than "aaa", the answer for this operation should be "YES".

After the second operation string $s$ becomes "aaa", and since $t$ is also equal to "aaa", we can't arrange $s$ in any way such that it is lexicographically smaller than $t$, so the answer is "NO".

After the third operation string $t$ becomes "aaaaaa" and $s$ is already lexicographically smaller than it so the answer is "YES".

After the fourth operation $s$ becomes "aaabb" and there is no way to make it lexicographically smaller than "aaaaaa" so the answer is "NO".

After the fifth operation the string $t$ becomes "aaaaaaabcaabcaabca", and we can rearrange the strings to: "bbaaa" and "caaaaaabcaabcaabaa" so that $s$ is lexicographically smaller than $t$, so we should answer "YES".