ACSL Practice 2009

You are asked to design a compression scheme for transmitting fax images of documents. After an image is scanned in using a scanner, you obtain a sequence of numbers where each number $x$ takes a value from $1$ to $256$. For example, an image may be converted into a sequence of the form

$${\displaystyle \begin{array}{rl}& 1,1,1,1,1,46,1,1,1,1,1,1,1,1,2,1,2,\\ & 1,1,1,25,26,25,25,255,256,256,256,255,\dots \end{array}}$$

Since it is not necessary to transmit the exact image, you decide to discretize each number to $4$ levels, namely $1,86,172$ and $256$, so that you can code each level using only $2$ bits as follows:

Level	$1$	$86$	$172$	$256$
Code	00	01	10	11

Then you realize that large areas of the same value (e.g. white) often occur in documents resulting in long sequences of very similar numbers. You decide to adopt a compression scheme as follows:

1. For the first number in the sequence, you will use the $2$ bits as described previously.
2. For all other numbers:
   • if the previous number is the same as the current number, you will transmit the bit 0.
   • otherwise, you will transmit the bit 1 followed by the $2$-bit code of the number. That is,
     • 1 followed by 00 for number $1$
     • 1 followed by 01 for number $86$
     • 1 followed by 10 for number $172$
     • 1 followed by 11 for number $256$

Finally, since you are not worried about a small amount of error, you realize that you can improve your scheme by ignoring isolated changes. For example, if the input sequence is $2,2,2,2,2,46,2,2$ and you transmit the sequence $1,1,1,1,1,86,1,1$, the total error will be $1+1+1+1+1+40+1+1=47$ and the string that needs to be transmitted is 0000001011000. If, instead you choose to transmit the sequence $1,1,1,1,1,1,1,1$, the total error will be $1+1+1+1+1+45+1+1=52$, but the string that needs to be transmitted would be 000000000 which is shorter.

You decide to measure the overall cost of a transmission by the following weighted sum

$${\displaystyle \text{cost}=\text{total error}+W\times (\text{total number of bits transmitted})}$$

where

1. the weight $W$ can be adjusted to give more emphasis to either the total error or the total number of bits transmitted
2. for an input sequence $x_1,x_2,\dots ,x_N$ and a transmitted sequence $y_1,y_2,\dots ,y_N$, both of length $N$, the total error is $\sum _{i=1}^N|x_i-y_i|=|x_1-y_1|+\cdots +|x_N-y_N|$.

Example. For the input sequence $2,2,2,2,2,46,2,2$ with $W=100$,

• we can transmit the sequence $y_1,\dots ,y_8=1,1,1,1,1,86,1,1$ by transmitting the string 0000001011000 with a cost of $47+100\times 13=1347$;
• or we can transmit the sequence $y_1,\dots ,y_8=1,1,1,1,1,1,1,1$ by transmitting the string 000000000 with a cost of $52+100\times 9=952$.

In this case, the second option has lower cost.

Input Specification

The input consists of several lines. The first line contains two integers $N$ $(1\le N\le 50)$ and $W$ $(1\le W\le 100)$ in this order. Integer $N$ denotes the length of the input sequence and integer $W$ is the weight. Each of the subsequent $N$ lines contains an integer $x$ $(1\le x\le 256)$ of the sequence.

Output Specification

The output contains a single integer which is the smallest possible cost required to transmit the input sequence with the given value of $W$.

Sample Input

8 100
2
2
2
2
2
46
2
2

Sample Output

952