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 takes a value from to . For example, an image may be converted into a sequence of the form
Since it is not necessary to transmit the exact image, you decide to discretize each number to levels, namely and , so that you can code each level using only bits as follows:
Level | ||||
---|---|---|---|---|
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:
- For the first number in the sequence, you will use the bits as described previously.
- 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 -bit code of the number. That is,1
followed by00
for number1
followed by01
for number1
followed by10
for number1
followed by11
for number
- if the previous number is the same as the current number, you
will transmit the bit
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 and you
transmit the sequence , the total error will be
and the string that needs to be
transmitted is 0000001011000
. If, instead you choose to transmit the
sequence , the total error will be
, 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
where
- the weight can be adjusted to give more emphasis to either the total error or the total number of bits transmitted
- for an input sequence and a transmitted sequence , both of length , the total error is .
Example. For the input sequence with ,
- we can transmit the sequence
by transmitting the string
0000001011000
with a cost of ; - or we can transmit the sequence
by transmitting the string
000000000
with a cost of .
In this case, the second option has lower cost.
Input Specification
The input consists of several lines. The first line contains two integers and in this order. Integer denotes the length of the input sequence and integer is the weight. Each of the subsequent lines contains an integer 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 .
Sample Input
8 100
2
2
2
2
2
46
2
2
Sample Output
952
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