102237: [AtCoder]ABC223 H - Xor Query
Description
Score : $600$ points
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.
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
Input
题意翻译
- 给定一个长度为 $N$ 正整数序列 $A$，$Q$ 次询问。 - 每次询问给出三个正整数 $L$、$R$、$X$，请你判断能否从原序列 $A_L$，$A_{L+1}$，... ,$A_R$ 中选出若干个数使其异或和为 $X$。Output
Yes，如果可以从$A_{L_i}, A_{L_i + 1}, \dots, A_{R_i}$中选择一个或多个元素，使它们的异或值为$X_i$，否则为No。
样例输入1
5 2
3 1 4 1 5
1 3 7
2 5 7
样例输出1
Yes
No
在第一个查询中，您可以选择$A_1$和$A_3$，它们的异或值为$7$。
在第二个查询中，无法选择元素，使它们的异或值为$7$。
样例输入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
样例输出2
No
No
Yes
No
Yes
No
No
No
Yes
Yes