Character Blocking
Contest: Codeforces Round 878 (Div. 3)
Difficulty rating: 1600
Tags: data structures, hashing, implementation

You are given two strings of equal length $$$s_1$$$ and $$$s_2$$$, consisting of lowercase Latin letters, and an integer $$$t$$$.

You need to answer $$$q$$$ queries, numbered from $$$1$$$ to $$$q$$$. The $$$i$$$-th query comes in the $$$i$$$-th second of time. Each query is one of three types:

- block the characters at position $$$pos$$$ (indexed from $$$1$$$) in both strings for $$$t$$$ seconds;
- swap two unblocked characters;
- determine if the two strings are equal at the time of the query, ignoring blocked characters.

Note that in queries of the second type, the characters being swapped can be from the same string or from $$$s_1$$$ and $$$s_2$$$.

Input format:
The first line of the input contains a single integer $$$T$$$ ($$$1 \le T \le 10^4$$$) — the number of test cases.

Then follow the descriptions of the test cases.

The first line of each test case contains a string $$$s_1$$$ consisting of lowercase Latin letters (length no more than $$$2 \cdot 10^5$$$).

The second line of each test case contains a string $$$s_2$$$ consisting of lowercase Latin letters (length no more than $$$2 \cdot 10^5$$$).

The strings have equal length.

The third line of each test case contains two integers $$$t$$$ and $$$q$$$ ($$$1 \le t, q \le 2 \cdot 10^5$$$). The number $$$t$$$ indicates the number of seconds for which a character is blocked. The number $$$q$$$ corresponds to the number of queries.

Each of the next $$$q$$$ lines of each test case contains a single query. Each query is one of three types:

- "$$$1\ \ \ pos$$$" — block the characters at position $$$pos$$$ in both strings for $$$t$$$ seconds;
- "$$$2\ \ \ 1/\;\!2\ \ \ pos_1\ \ \ 1/\;\!2\ \ \ pos_2$$$" — swap two unblocked characters. The second number in the query indicates the number of the string from which the first character for the swap is taken. The third number in the query indicates the position in that string of that character. The fourth number in the query indicates the number of the string from which the second character for the swap is taken. The fifth number in the query indicates the position in that string of that character;
- "$$$3$$$" — determine if the two strings are equal at the time of the query, ignoring blocked characters.

For queries of the first type, it is guaranteed that at the time of the query, the characters at position $$$pos$$$ are not blocked.

For queries of the second type, it is guaranteed that the characters being swapped are not blocked.

All values of $$$pos, pos_1, pos_2$$$ are in the range from $$$1$$$ to the length of the strings.

The sum of the values of $$$q$$$ over all test cases, as well as the total length of the strings $$$s_1$$$, does not exceed $$$2 \cdot 10^5$$$.

Output format:
For each query of the third type, output "YES" if the two strings $$$s_1$$$ and $$$s_2$$$ are equal at the time of the query, ignoring blocked characters, and "NO" otherwise.

You can output each letter in any case (lowercase or uppercase). For example, the strings "yEs", "yes", "Yes" and "YES" will be accepted as a positive answer.

Constraints:
Time limit: 2.0 s
Memory limit: 256.0 MB

Note:
Let's look at the strings $$$s_1$$$ and $$$s_2$$$ after each of the $$$q$$$ queries. Blocked characters will be denoted in red.

First example input:

($$$codeforces$$$, $$$codeblocks$$$) $$$\rightarrow$$$ ($$$codeforces$$$, $$$codeblocks$$$) $$$\rightarrow$$$ ($$$code\color{red}{f}orces$$$, $$$code\color{red}{b}locks$$$) $$$\rightarrow$$$ ($$$code\color{red}{fo}rces$$$, $$$code\color{red}{bl}ocks$$$) $$$\rightarrow$$$ ($$$code\color{red}{for}ces$$$, $$$code\color{red}{blo}cks$$$) $$$\rightarrow$$$ ($$$code\color{red}{for}c\color{red}{e}s$$$, $$$code\color{red}{blo}c\color{red}{k}s$$$) $$$\rightarrow$$$ ($$$code\color{red}{for}c\color{red}{e}s$$$, $$$code\color{red}{blo}c\color{red}{k}s$$$) $$$\rightarrow$$$ ($$$codef\color{red}{or}c\color{red}{e}s$$$, $$$codeb\color{red}{lo}c\color{red}{k}s$$$)

Second example input:

($$$cool$$$, $$$club$$$) $$$\rightarrow$$$ ($$$cuol$$$, $$$clob$$$) $$$\rightarrow$$$ ($$$cuol$$$, $$$cbol$$$) $$$\rightarrow$$$ ($$$c\color{red}{u}ol$$$, $$$c\color{red}{b}ol$$$) $$$\rightarrow$$$ ($$$c\color{red}{u}ol$$$, $$$c\color{red}{b}ol$$$) $$$\rightarrow$$$ ($$$cuol$$$, $$$cbol$$$)

Example input:
2
codeforces
codeblocks
5 7
3
1 5
1 6
1 7
1 9
3
3
cool
club
2 5
2 1 2 2 3
2 2 2 2 4
1 2
3
3

Example output:
NO
YES
NO
YES
NO

Editorial:
Two strings are equal if and only if there is no position $$$pos$$$ such that the characters at position $$$pos$$$ are not blocked and $$$s_1[pos] \neq s_2[pos]$$$ (we will call such a position bad). We will use this observation to maintain the current number of bad positions, denoted by $$$cnt$$$.
Let $$$I_{pos}$$$ be an indicator variable. $$$I_{pos} = 1$$$ if position $$$pos$$$ is bad, otherwise $$$I_{pos} = 0$$$.
During an operation (blocking or swapping), we only need to subtract the indicator variables of all positions affected by the operation from $$$cnt$$$. There will be $$$\mathcal{O}(1)$$$ of them. Then, we modify the string according to the operation and add new indicator variables to $$$cnt$$$.
To correctly handle blocking queries, or more precisely, to unblock positions in time, we will use a queue. After each blocking query, we will add a pair of numbers to the queue. The first number of the pair is the position to unblock, and the second number is the time to unblock. Now, before each operation, we will unblock positions by looking at the head of the queue.