USACO

USACO 2016 February Contest, Bronze

Problem 2. 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 each room $i$ ( $1 \leq r_i \leq 100$ ). To herd the cows into the barn in an orderly fashion, he plans to unlock the exterior door of a single room, allowing the cows to enter through that door. Each cow then walks clockwise through the rooms until she reaches a suitable destination. Farmer John wants to unlock the exterior door that will cause his cows to collectively walk a minimum total amount of distance. Please determine the minimum total distance his cows will need to walk, if he chooses the best such door to unlock. The distance walked by a single cow is the number of interior doors through which she passes.

INPUT FORMAT (file cbarn.in):

The first line of input contains

$n$ . Each of the remaining

$n$ lines contain

$r_1 \ldots r_n$ .

OUTPUT FORMAT (file cbarn.out):

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

SAMPLE INPUT:

SAMPLE OUTPUT:

In this example, the best solution is to let the cows enter through the door of the room that requires 7 cows.

Problem credits: Brian Dean

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