Mana Collection

Note: The time limit for this problem is 5s, 2.5 times the default. The memory limit for this problem is 512MB, twice the default.

Bessie has recently taken an interest in magic and needs to collect mana for a very important spell. Bessie has $N$ ($1\le N\le 18$) mana pools, the $i$th of which accumulates $m_i$ mana per second ($1\le m_i\le 10^8$). The pools are linked by a collection of $M$ ($0\le M\le N(N-1)$) directed edges $(a_i,b_i,t_i)$, meaning that she can travel from $a_i$ to $b_i$ in $t_i$ seconds ($1\le a_i, b_i\le N$, $a_i\neq b_i$, $1\le t_i\le 10^9$). Whenever Bessie is present at a pool, she can collect all the mana stored at that location, emptying it. At time $0$, all mana pools are empty, and Bessie can select any pool to start at.

Answer $Q$ ($1\le Q\le 2\cdot 10^5$) queries, each specified by two integers $s$ and $e$ ($1\le s\le 10^9$, $1\le e\le N$). For each query, determine the maximum amount of mana Bessie can collect in $s$ seconds if she must be at mana pool $e$ at the end of the $s$th second.

First line contains $N$ and $M$.

Next line contains $m_1,m_2,\dots, m_N$.

Next $M$ lines contain $a_i,b_i,t_i$. No ordered pair $(a_i,b_i)$ appears more than once in the input.

Next line contains $Q$.

Next $Q$ lines contain two integers $s$ and $e$.

• Inputs 3-4: $N\le 10, Q\le 100$
• Inputs 5-9: $N\le 10$
• Inputs 10-14: $Q\le 100$
• Inputs 15-17: $N = 16$
• Inputs 18-20: $N = 17$
• Inputs 21-24: No additional constraints.

$Q$ lines, one for each query.

First query: Bessie takes 5 mana from pool 1 after 5 seconds.

Second query: Bessie takes 50 mana from pool 2 after 5 seconds.

Third query: Bessie takes 100 mana from pool 1 after 100 seconds.

Fourth query: Bessie takes 90 mana from pool 1 after 90 seconds and 1000 mana from pool 2 after 100 seconds.

An example where Bessie is able to collect much larger amounts of mana.