USACO 2016 February Contest, Platinum

Problem 3. Circular Barn


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Being a fan of contemporary architecture, Farmer John has built a new barn in the shape of a perfect circle. Inside, the barn consists of a ring of $n$ rooms, numbered clockwise from $1 \ldots n$ around the perimeter of the barn ($3 \leq n \leq 1,000$). Each room has doors to its two neighboring rooms, and also a door opening to the exterior of the barn.

Farmer John wants exactly $r_i$ cows to end up in room $i$ ($1 \leq r_i \leq 1,000,000$). To herd the cows into the barn in an orderly fashion, he plans to unlock $k$ exterior doors ($1 \leq k \leq 7$), allowing the cows to enter through only those doors. Each cow then walks clockwise through the rooms until she reaches a suitable destination. Farmer John wants to unlock the exterior doors that will cause his cows to collectively walk a minimum total amount of distance after entering the barn (they can initially line up however they like outside the $k$ unlocked doors; this does not contribute to the total distance in question). Please determine the minimum total distance his cows will need to walk, if he chooses the best $k$ such doors to unlock.

INPUT FORMAT (file cbarn.in):

The first line of input contains $n$ and $k$. Each of the remaining $n$ lines contain $r_1 \ldots r_n$.

OUTPUT FORMAT (file cbarn.out):

Please write out the minimum amount of distance the cows need to travel.

SAMPLE INPUT:

6 2
2
5
4
2
6
2

SAMPLE OUTPUT:

14

Farmer John can unlock doors 2 and 5. 11 cows enter at door 2 and walk a total distance of 8 to get to rooms 2, 3, and 4. 10 cows enter at door 5 and walk a total distance of 6 to get to rooms 5, 6 and 1.

Problem credits: Brian Dean

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