Xor Query

AtCoder

IDabc223_h

Time3000ms

Memory256MB

Difficulty—

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: $N$ $Q$ $A_1$ $\ldots$ $A_N$ $L_1$ $R_1$ $X_1$ $\vdots$ $L_Q$ $R_Q$ $X_Q$ [samples]

Samples

Input #1

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 $7$.

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

Output #2

No
No
Yes
No
Yes
No
No
No
Yes
Yes

API Response (JSON)

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      "statement_type": "Markdown",
      "content": "Given is a sequence of $N$ positive integers $A = (A_1, \\dots, A_N)$.\nProcess $Q$ queries. In the $i$\\-th query $(1 \\leq i \\leq Q)$, determine whether it is possible to choose one or more elements fro...",
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