USACO 2025 January Contest, Platinum
Problem 3. Watering the Plants
Contest has ended.
Log in to allow submissions in analysis mode
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.
INPUT FORMAT (input arrives from the terminal / stdin):
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 $c_1, \ldots, c_{N-1}$.
OUTPUT FORMAT (print output to the terminal / stdout):
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.SAMPLE INPUT:
3 39 69 33 30 29
SAMPLE OUTPUT:
2070 2127
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 $39 \cdot 30 + 29 \cdot 33 = 2127$.
SAMPLE INPUT:
3 33 82 36 19 1
SAMPLE OUTPUT:
1558 676
SAMPLE INPUT:
8 35 89 44 1 35 3 62 50 7 86 94 62 63 9 49
SAMPLE OUTPUT:
623 4099 4114 6269 6272 6827 8827
SCORING:
- 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.
Problem credits: Benjamin Qi