USACO 2015 US Open, Silver

Problem 3. Bessie's Birthday Buffet

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For Bessie the cow’s birthday, Farmer John has given her free reign over one of his best fields to eat grass.

The field is covered in $N$ patches of grass ($1 \le N \le 1000$), conveniently numbered $1\ldots N$, that each have a distinct quality value. If Bessie eats grass of quality $Q$, she gains $Q$ units of energy. Each patch is connected to up to 10 neighboring patches via bi-directional paths, and it takes Bessie $E$ units of energy to move between adjacent patches ($1 \le E \le 1,000,000$). Bessie can choose to start grazing in any patch she wishes, and she wants to stop grazing once she has accumulated a maximum amount of energy.

Unfortunately, Bessie is a picky bovine, and once she eats grass of a certain quality, she’ll never eat grass at or below that quality level again! She is still happy to walk through patches without eating their grass; in fact, she might find it beneficial to walk through a patch of high-quality grass without eating it, only to return later for a tasty snack.

Please help determine the maximum amount of energy Bessie can accumulate.


The first line of input contains $N$ and $E$. Each of the remaining $N$ lines describe a patch of grass. They contain two integers $Q$ and $D$ giving the quality of the patch (in the range $1\ldots 1,000,000$) and its number of neighbors. The remaining $D$ numbers on the line specify the neighbors.

OUTPUT FORMAT (file buffet.out):

Please output the maximum amount of energy Bessie can accumulate.


5 2
4 1 2
1 3 1 3 4
6 2 2 5
5 2 2 5
2 2 3 4



Bessie starts at patch 4 gaining 5 units of energy from the grass there. She then takes the path to patch 5 losing 2 units of energy during her travel. She refuses to eat the lower quality grass at patch 5 and travels to patch 3 again losing 2 units of energy. Finally she eats the grass at patch 3 gaining 6 units of energy for a total of 7 energy.

Note tha the sample case above is different from test case 1 when you submit.

[Problem credits: Austin Anderson and Brian Dean, 2015]

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