Watering the Plants

Bessie's garden has $N$ plants labeled $1$ through $N$ ($2\leq N\leq 5\cdot 10^5$) from left to right. Bessie knows that plant $i$ requires at least $w_i$ ($0\leq w_i \leq 10^6$) units of water.

Bessie has a very peculiar irrigation system with $N-1$ canals, numbered $1$ through $N-1$. Each canal $i$ has an associated unit cost $c_i$ ($1\le c_i\le 10^6$), such that Bessie can pay $c_i k$ to provide plants $i$ and $i+1$ each with $k$ units of water, where $k$ is a non-negative integer.

Bessie is busy and may not have time to use all the canals. For each $2\leq i \leq N$ compute the minimum cost required to water plants $1$ through $i$ using only the first $i-1$ canals.

The first line contains a single positive integer $N$.

The second line contains $N$ space-separated integers $w_1, \ldots, w_N$.

The third line contains $N-1$ space-separated integers $c_1, \ldots, c_{N-1}$.

• Input 4: $N \leq 200$, and all $w_i \leq 200$.
• Inputs 5-6: All $w_i \leq 200$.
• Inputs 7-10: $N \leq 5000$.
• Inputs 11-14: All $w_i$ and $c_i$ are generated independently and uniformly at random.
• Inputs 15-19: No additional constraints.

Output $N-1$ newline-separated integers. The $(i-1)$th integer should contain the minimum cost to water the first $i$ plants using the first $i-1$ canals.

The minimum cost to water the first $2$ plants using the first canal is to pay $30 \cdot 69 = 2070$ by using the first canal $69$ times.

The minimum cost to water the first $3$ plants is to use the first canal $39$ times and the second canal $33$ times, paying $39 \cdot 30 + 29 \cdot 33 = 2127$.