ICPC 2017 · Problem F · Posterize

Time limit: 2 seconds

Pixels in a digital picture can be represented with three integers in the range $0$ to $255$ that indicate the intensity of the red, green, and blue colors. To compress an image or to create an artistic effect, many photo-editing tools include a “posterize” operation which works as follows. Each color channel is examined separately; this problem focuses only on the red channel. Rather than allow all integers from $0$ to $255$ for the red channel, a posterized image allows at most $k$ integers from this range. Each pixel’s original red intensity is replaced with the nearest of the allowed integers. The photo-editing tool selects a set of $k$ integers that minimizes the sum of the squared errors introduced across all pixels in the original image. If there are $n$ pixels that have original red values $r_{1}, . . . , r_{n}$, and $k$ allowed integers $v_{1}, . . . , v_{k}$, the sum of squared errors is defined as

$n$ $X$

$i=1$ $min$ $_{1\le j\le k}(ri- vj)^{2}.$

Your task is to compute the minimum achievable sum of squared errors, given parameter $k$ and a description of the red intensities of an image’s pixels.

Input

The first line of the input contains two integers $d$ ($1\le d\le 256$), the number of distinct red values that occur in the original image, and $k$ ($1\le k \le d$), the number of distinct red values allowed in the posterized image. The remaining $d$ lines indicate the number of pixels of the image having various red values. Each such line contains two integers $r$ ($0 \le r \le 255$) and $p$ ($1 \le p \le 2^{26}$), where $r$ is a red intensity value and $p$ is the number of pixels having red intensity $r$. Those $d$ lines are given in increasing order of red value.

Output

Display the sum of the squared errors for an optimally chosen set of $k$ allowed integer values.

ACM-ICPC World Finals 2017 Problem F: Posterize

Sample Input 1

2 1
50 20000
150 10000

Sample Output 1

66670000

Sample Input 2

2 2
50 20000
150 10000

Sample Output 2

0

Sample Input 3

4 2
0 30000
25 30000
50 30000
255 30000

Sample Output 3

37500000

ACM-ICPC World Finals 2017 Problem F: Posterize