5741：Helter Skelter

A non-empty string $s$ is called binary, if it consists only of characters "0" and "1". A substring $s[l \dots r]$ $(1 \le l \le r \le |s|)$ of string $s = s_{1}s_{2} \dots s_{|s|}$ (where $|s|$ is the length of string $s$) is string $s_{l}s_{l + 1}...s_{r}$.

Professor Zhang has got a long binary string $s$ starting with "0", and he wants to know whether there is a substring of the string $s$ that the occurrences of "0" and "1" in the substring is exact $a$ and $b$, respectively, where $a$ and $b$ are two given numbers.

Since the binary string is very long, we will compress it. The compression method is:
1. Split the string to runs of same characters.
2. Any two adjacent runs consist of different characters. Use the length of each run to represent the string.

For example, the runs of a binary string 00101100011110111101001111111 are {00, 1, 0, 11, 000, 1111, 0, 1111, 0, 1, 00, 1111111}, so it will be compressed into {2, 1, 1, 2, 3, 4, 1, 4, 1, 1, 2, 7}.

There are multiple test cases. The first line of input contains an integer $T$, indicating the number of test cases. For each test case:

The first line contains two integers $n$ and $m$ $(1 \le n \le 1000, 1 \le m \le 5 \cdot 10^5)$ - the number of runs and the number of queries. The next line contains $n$ integers: $x_1, x_2, ..., x_n$ $(1 \le x_i \le 10^6)$, indicating the length of the each run.

Each of the following $m$ lines contains two integers $a_i$ and $b_i$ $(0 \le a_i, b_i \le |s|, 1 \le a_i + b_i \le |s|)$, which means Professor Zhang wants to know whether there is a substring of the string $s$ that the occurrences of "0" and "1" in the substring is exact $a_i$ and $b_i$.

For each test case, a binary string of length $n$. The $i$-th number is "1" if the answer for the $i$-th query is yes, or "0" otherwise.