Score : $600$ points

问题描述

给定一个包含 $N$ 个正整数的序列 $A = (A_1, \dots, A_N)$。

处理 $Q$ 个查询。在第 $i$（$1 \leq i \leq Q$）个查询中，判断是否可以从 $A_{L_i}, A_{L_i + 1}, \dots, A_{R_i}$ 中选择一个或多个元素，使得它们的按位异或值为 $X_i$。

什么是按位异或（$\mathrm{XOR}$）？

对于整数 $A$ 和 $B$ 的按位异或运算 $A\ \mathrm{XOR}\ B$ 定义如下：

• 当 $A\ \mathrm{XOR}\ B$ 以二进制表示时，位于 $2^k$ 位（$k \geq 0$）上的数字是 $1$，当且仅当 $A$ 和 $B$ 中恰好有一个是 $1$；否则为 $0$。

例如，我们有 $3\ \mathrm{XOR}\ 5 = 6$（以二进制表示：$011\ \mathrm{XOR}\ 101 = 110$）。

以上为通义千问 qwen-max 翻译，仅供参考。

Problem Statement

Given is a sequence of $N$ positive integers $A = (A_1, \dots, A_N)$.

Process $Q$ queries. In the $i$-th query $(1 \leq i \leq Q)$, determine whether it is possible to choose one or more elements from $A_{L_i}, A_{L_i + 1}, \dots, A_{R_i}$ so that their $\mathrm{XOR}$ is $X_i$.

What is $\mathrm{XOR}$?

The bitwise $\mathrm{XOR}$ of integers $A$ and $B$, $A\ \mathrm{XOR}\ B$, is defined as follows:

• When $A\ \mathrm{XOR}\ B$ is written in base two, the digit in the $2^k$'s place ($k \geq 0$) is $1$ if exactly one of $A$ and $B$ is $1$, and $0$ otherwise.

For example, we have $3\ \mathrm{XOR}\ 5 = 6$ (in base two: $011\ \mathrm{XOR}\ 101 = 110$).

Constraints

• $1 \leq N \leq 4 \times 10^5$
• $1 \leq Q \leq 2 \times 10^5$
• $1 \leq A_i \lt 2^{60}$
• $1 \leq L_i \leq R_i \leq N$
• $1 \leq X_i \lt 2^{60}$
• All values in input are integers.

Input

Input is given from Standard Input in the following format:

$N$ $Q$

$A_1$ $\ldots$ $A_N$

$L_1$ $R_1$ $X_1$

$\vdots$

$L_Q$ $R_Q$ $X_Q$

Output

Print $Q$ lines. The $i$-th line $(1 \leq i \leq Q)$ should contain Yes if it is possible to choose one or more elements from $A_{L_i}, A_{L_i + 1}, \dots, A_{R_i}$ so that their $\mathrm{XOR}$ is $X_i$, and No otherwise.

Sample Input 1

5 2
3 1 4 1 5
1 3 7
2 5 7

Sample Output 1

Yes
No

In the first query, you can choose $A_1$ and $A_3$, whose $\mathrm{XOR}$ is $7$.

In the second query, there is no way to choose elements so that their $\mathrm{XOR}$ is $7$.

Sample Input 2

10 10
8 45 56 9 38 28 33 5 15 19
10 10 53
3 8 60
1 10 29
5 7 62
3 7 51
8 8 52
1 4 60
6 8 32
4 8 58
5 9 2

Sample Output 2

No
No
Yes
No
Yes
No
No
No
Yes
Yes

update @ 2024/3/10 09:50:27