H - Xor Query
H - Xor Query
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:
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.
5 2
3 1 4 1 5
1 3 7
2 5 7
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 $7$.
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
No
No
Yes
No
Yes
No
No
No
Yes
Yes