CF2250B.String Construction

You are given two integers $n$ and $k$.

Construct a binary string$^{\text{∗}}$ $s$ of length $n$, such that both of the following conditions hold:

• The absolute difference between the number of characters $\mathtt{0}$ and the number of characters $\mathtt{1}$ in $s$ is at most $1$.
• There are exactly $k$ pairs of adjacent equal characters in $s$. Formally, there are exactly $k$ indices $i$ ($1 \le i \le n-1$) satisfying $s_i = s_{i + 1}$.

Or determine that no such string exists.

$^{\text{∗}}$A binary string is a string where each character is either $\mathtt{0}$ or $\mathtt{1}$.

给你两个整数 $n$ 和 $k$。

构造一个长度为 $n$ 的二进制字符串$^{\text{∗}}$ $s$，使得以下两个条件均成立：

• 字符串 $s$ 中字符 $\mathtt{0}$ 的个数与字符 $\mathtt{1}$ 的个数之差的绝对值至多为 $1$。
• $s$ 中恰好有 $k$ 对相邻且相等的字符。形式化地说，恰好存在 $k$ 个下标 $i$（$1 \le i \le n-1$），满足 $s_i = s_{i + 1}$。

或者判断不存在满足条件的字符串。

$^{\text{∗}}$二进制字符串是指每个字符均为 $\mathtt{0}$ 或 $\mathtt{1}$ 的字符串。

Each test contains multiple test cases. The first line contains the number of test cases $t$ ($1 \le t \le 1000$). The description of the test cases follows.

The only line of each test case contains two integers $n$ and $k$ ($2 \le n \le 2 \cdot 10^5$, $0 \le k \le n-1$).

It is guaranteed that the sum of $n$ over all test cases does not exceed $2 \cdot 10^5$.

每个测试包含多个测试用例。第一行包含测试用例的数量 $t$（$1 \le t \le 1000$）。随后是各测试用例的描述。

每个测试用例仅有一行，包含两个整数 $n$ 和 $k$（$2 \le n \le 2 \cdot 10^5$，$0 \le k \le n-1$）。

保证所有测试用例的 $n$ 值之和不超过 $2 \cdot 10^5$。

For each test case, output a binary string $s$ of length $n$ — the string you constructed. Print $-1$ if such a string does not exist.

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

对于每个测试用例，输出一个长度为 $n$ 的二进制字符串 $s$ —— 即你构造出的字符串。如果不存在这样的字符串，则输出 $-1$。

如果有多个答案，你可以输出其中任意一个。

In the first test case, one possible answer is $s=\mathtt{01110}$. It contains three characters $\mathtt{1}$ and two characters $\mathtt{0}$, and there are exactly $2$ adjacent equal pairs in $s$: $(s_2, s_3)$ and $(s_3, s_4)$.

In the second test case, $k=n-1$. All characters in $s$ should be equal, so the numbers of characters $\mathtt{0}$ and $\mathtt{1}$ could not differ by at most $1$. Thus, the answer is $-1$.

In the third test case, note that $\mathtt{010110}$ is also a possible answer.

在第一个测试用例中，一个可能的答案是 $s=\mathtt{01110}$。它包含三个字符 $\mathtt{1}$ 和两个字符 $\mathtt{0}$，且 $s$ 中恰好有 $2$ 对相邻相等的字符：$(s_2, s_3)$ 和 $(s_3, s_4)$。

在第二个测试用例中，$k=n-1$。$s$ 中所有字符必须相同，因此字符 $\mathtt{0}$ 和 $\mathtt{1}$ 的数量不可能至多相差 $1$。故答案为 $-1$。

在第三个测试用例中，请注意 $\mathtt{010110}$ 也是一个可能的答案。