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D - Range Add Query

Score : $400$ points

Problem Statement

You are given an integer sequence of length $N$, $A = (A_1, A_2, \ldots, A_N)$, and a positive integer $K$.

For each $i = 1, 2, \ldots, Q$, determine whether a contiguous subsequence of $A$, $(A_{l_i}, A_{l_i+1}, \ldots, A_{r_i})$, is a good sequence.

Here, a sequence of length $n$, $X = (X_1, X_2, \ldots, X_n)$, is good if and only if there is a way to perform the operation below some number of times (possibly zero) to make all elements of $X$ equal $0$.

Choose an integer $i$ such that $1 \leq i \leq n-K+1$ and an integer $c$ (possibly negative). Add $c$ to each of the $K$ elements $X_{i}, X_{i+1}, \ldots, X_{i+K-1}$.

It is guaranteed that $r_i - l_i + 1 \geq K$ for every $i = 1, 2, \ldots, Q$.

Constraints

• $1 \leq N \leq 2 \times 10^5$
• $1 \leq K \leq \min\lbrace 10, N \rbrace$
• $-10^9 \leq A_i \leq 10^9$
• $1 \leq Q \leq 2 \times 10^5$
• $1 \leq l_i, r_i \leq N$
• $r_i-l_i+1 \geq K$
• All values in the input are integers.

Input

The input is given from Standard Input in the following format:

$N$ $K$

$A_1$ $A_2$ $\ldots$ $A_N$

$Q$

$l_1$ $r_1$

$l_2$ $r_2$

$\vdots$

$l_Q$ $r_Q$

Output

Print $Q$ lines. For each $i = 1, 2, \ldots, Q$, the $i$-th line should contain Yes if the sequence $(A_{l_i}, A_{l_i+1}, \ldots, A_{r_i})$ is good, and No otherwise.

7 3
3 -1 1 -2 2 0 5
2
1 6
2 7

Yes
No

The sequence $X \coloneqq (A_1, A_2, A_3, A_4, A_5, A_6) = (3, -1, 1, -2, 2, 0)$ is good. Indeed, you can do the following to make all elements equal $0$.

• First, do the operation with $i = 2, c = 4$, making $X = (3, 3, 5, 2, 2, 0)$.
• Next, do the operation with $i = 3, c = -2$, making $X = (3, 3, 3, 0, 0, 0)$.
• Finally, do the operation with $i = 1, c = -3$, making $X = (0, 0, 0, 0, 0, 0)$.

Thus, the first line should contain Yes.

On the other hand, for the sequence $(A_2, A_3, A_4, A_5, A_6, A_7) = (-1, 1, -2, 2, 0, 5)$, there is no way to make all elements equal $0$, so it is not a good sequence. Thus, the second line should contain No.

20 4
-19 -66 -99 16 18 33 32 28 26 11 12 0 -16 4 21 21 37 17 55 -19
5
13 16
4 11
3 12
13 18
4 10

No
Yes
No
Yes
No