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E - Transformable Teacher

Score : $500$ points

Problem Statement

There are $N$ children in a kindergarten. The height of the $i$-th child is $H_i$. Here, $N$ is an odd number.

You, the teacher, and these children - $N+1$ people in total - will make $\large\frac{N+1}2$ pairs.

Your objective is to minimize the sum of the differences in height over those pairs. That is, you want to minimize $\displaystyle \sum_{i=1}^{(N+1)/2}|x_i-y_i|$, where $(x_i, y_i)$ is the pair of heights of people in the $i$-th pair.

You have $M$ forms that you can transform into. Your height in the $i$-th form is $W_i$.

Find the minimum possible sum of the differences in height over all pairs, achieved by optimally choosing your form and making pairs.

Constraints

• All values in input are integers.
• $1 \leq N, M \leq 2 \times 10^5$
• $N$ is an odd number.
• $1 \leq H_i \leq 10^9$
• $1 \leq W_i \leq 10^9$

Input

Input is given from Standard Input in the following format:

$N$ $M$

$H_1$ $\dots$ $H_N$

$W_1$ $\dots$ $W_M$

Output

Print the minimum possible sum of the differences in height over all pairs, achieved by optimally choosing your form and making pairs.

5 3
1 2 3 4 7
1 3 8

3

The minimum is minimized by choosing the form with the height $8$ and making pairs with heights $(1, 2)$, $(3, 4)$, and $(7, 8)$.

7 7
31 60 84 23 16 13 32
96 80 73 76 87 57 29

34

15 10
554 525 541 814 661 279 668 360 382 175 833 783 688 793 736
496 732 455 306 189 207 976 73 567 759

239