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Problem · Array

Find Minimum Cost

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There are n containers arranged in a circle. Container i initially contains products[i] products.

Redistribute products so every container contains the same number of products. Moving one product across one edge costs 1.

For the entire redistribution process, choose one global direction, either clockwise or counter-clockwise. Every product that is moved must travel only in that chosen direction around the circle. Products cannot independently choose shorter paths in the opposite direction. For example, if clockwise is chosen, moving a product from position 3 to position 2 costs n - 1, not 1.

Return the minimum total movement cost over the two possible global directions.

Examples

01 · Example 1

products = [3, 4, 6, 6, 6]
return = 7

🌷 Option 1 - Start at the 3^rd position and move clockwise. Collect one product each from the 3^rd, 4^th, and 5^th positions. Transfer the products from: - the 5^th position to the 1^st position, the cost is 1 - the 4^th position to the 1^st position, the cost is 2 - the 3^rd position to the 2^nd position, the cost is 4 Now each container has 5 units and the total cost is 1 + 2 + 4 = 7. 🌷 Option 2 - Start at the 5^th position and move anti-clockwise. Collect one product each from the 5^th, 4^th, and 3^rd positions. Transfer the products from: - the 3^rd position to the 1^st position, the cost is 2 - the 4^th position to the 1^st position, the cost is 3 - the 5^th position to the 2^nd position, the cost is 3 Now each container has 5 units and the total cost is 2 + 3 + 3 = 8.

02 · Example 2

products = [1, 11, 1, 1, 1]
return = 20

The final average is 3. Container 1 has 8 extra products. Moving clockwise, send 2 products to each of the next four containers, for cost 2*1 + 2*2 + 2*3 + 2*4 = 20.

Constraints

1 <= products.length <= 2 * 10^5
0 <= products[i] <= 10^9
The answer may exceed the 32-bit integer range.