USACO

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:

SAMPLE OUTPUT:

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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