P11843 [USACO25FEB] The Best Subsequence G

Farmer John has a binary string of length $N$ $(1 \leq N \leq 10^9)$, initially all zeros. He will first perform $M$ ($1 \leq M \leq 2 \cdot 10^5$) updates on the string, in order. Each update flips every character from $l$ to $r$. Specifically, flipping a character changes it from $0$ to $1$, or vice versa. Then, he asks you $Q$ ($1 \leq Q \leq 2 \cdot 10^5$) queries. For each query, he asks you to output the lexicographically greatest subsequence of length $k$ comprised of characters from the substring from $l$ to $r$. If your answer is a binary string $s_1s_2 \dots s_k$, then output $\sum_{i=0}^{k-1} 2^i \cdot s_{k-i}$ (that is, its value when interpreted as a binary number) modulo $10^9+7$. A subsequence is a string that can be derived from another string by deleting some or no characters without changing the order of the remaining characters. Recall that string $A$ is lexicographically greater than string $B$ of equal length if and only if at the first position $i$, if it exists, where $A_i \neq B_i$, we have $A_i > B_i$.

The first line contains $N$, $M$, and $Q$. The next $M$ lines contain two integers, $l$ and $r$ ($1 \leq l \leq r \leq N$) — the endpoints of each update. The next $Q$ lines contain three integers, $l$, $r$, and $k$ ($1 \leq l \leq r \leq N, 1 \leq k \leq r - l + 1$) — the endpoints of each query and the length of the subsequence.

Output $Q$ lines. The $i$th line should contain the answer for the $i$th query.

##### For Sample 1: After performing the $M$ operations, the string is $10101$. For the first query, there is only one subsequence of length $5$, $10101$, which is interpreted as $1 \cdot 2^4 + 0 \cdot 2^3 + 1 \cdot 2^2 + 0 \cdot 2^1 + 1 \cdot 2^0 = 21$. For the second query, there are $5$ unique subsequences of length $4$: $0101$, $1101$, $1001$, $1011$, $1010$. The lexicographically largest subsequence is $1101$, which is interpreted as $1 \cdot 2^3 + 1 \cdot 2^2 + 0 \cdot 2^1 + 1\cdot 2^0 = 13$. For the third query, the lexicographically largest sequence is $111$, which is interpreted as $7$. ##### For Sample 3: Make sure to output the answer modulo $10^9+7$. #### SCORING: - Input 4: $N \leq 10, Q \leq 1000$ - Input 5: $M \leq 10$ - Inputs 6-7: $N, Q \leq 1000$ - Inputs 8-12: $N \leq 2 \cdot 10^5$ - Inputs 13-20: No additional constraints.