|VK Cup 2016 - Round 3|
Radewoosh is playing a computer game. There are n levels, numbered 1 through n. Levels are divided into k regions (groups). Each region contains some positive number of consecutive levels.
The game repeats the the following process:
Given n, k and values t1, t2, ..., tn, your task is to split levels into regions. Each level must belong to exactly one region, and each region must contain non-empty consecutive set of levels. What is the minimum possible expected number of hours required to finish the game?
The first line of the input contains two integers n and k (1 ≤ n ≤ 200 000, 1 ≤ k ≤ min(50, n)) — the number of levels and the number of regions, respectively.
The second line contains n integers t1, t2, ..., tn (1 ≤ ti ≤ 100 000).
Print one real number — the minimum possible expected value of the number of hours spent to finish the game if levels are distributed between regions in the optimal way. Your answer will be considered correct if its absolute or relative error does not exceed 10 - 4.
Namely: let's assume that your answer is a, and the answer of the jury is b. The checker program will consider your answer correct if .
100 3 5 7
1 2 4 8 16 32
In the first sample, we are supposed to split 4 levels into 2 regions. It's optimal to create the first region with only one level (it must be the first level). Then, the second region must contain other three levels.
In the second sample, it's optimal to split levels into two regions with 3 levels each.