시간 제한 메모리 제한 제출 정답 맞은 사람 정답 비율
2 초 128 MB 17 5 5 29.412%

문제

상근이는 팩스 이미지를 압축하는 계획을 만드려고 한다. 스캐너를 통해서 이미지를 입력받으면, 1과 256사이의 숫자로 이루어진 수열로 변환된다. 예를 들어, 아래와 같은 수열로 이미지를 변환할 수 있다.

1, 1, 1, 1, 1, 46, 1, 1, 1,1, 1, 1, 2, 1, 2, 1, 1, 25, 26, 25, 25, 255, 256, 256, 256, 255,...

이미지를 반드시 정확하게 변환할 필요는 없다. 따라서, 상근이는 수열의 각 숫자를 레벨 4개(1, 86, 172, 256)으로 나타내려고 한다. 그리고, 이 레벨은 다음과 같이 2-bit 코드로 나타낼 수 있다.

Level 1 86 172 256
Code 00 01 10 11

그림 안에는 같은 색으로 이루어진 영역(예를 들면 공백)이 있다. 이 영역을 변환하면 유사한 숫자들이 연속해서 나타난다. 이를 처리하기 위해서 다음과 같은 압축 방법을 사용하려고 한다.

  1. 수열의 첫 번째 숫자는 위의 표에 나온 2bit 코드로 변환한다.
  2. 그 다음에 나오는 다른 모든 숫자에 대해서는
    • 만약 수열의 바로 앞에 있는 숫자가 같은 숫자라면 0으로 변환한다.
    • 앞에 있는 숫자와 다른 숫자라면 1을 앞에 붙인 뒤 위 표의 2bit 코드로 변환한다.

상근이는 작은 양의 에러를 크게 신경쓰지 않기 때문에, 작은 양의 에러는 무시한다. 예를 들어, 입력된 수열이 2, 2, 2, 2, 2, 46, 2, 2 라고 했을 때, 이를 1, 1, 1, 1, 1, 86, 1, 1로 나타낸 다음 변환하면 0000001011000가 되고 전체 에러는 1+1+1+1+1+40+1+1=47 이 된다. 하지만 1, 1, 1, 1, 1, 1, 1, 1 로 나타낸 후 변환하면 전체 에러는 1+1+1+1+1+45+1+1=52가 되어 커지지만 코드가 000000000가 되어 더 짧아진다.

상근이는 변환 비용을 최소로 하는 방법을 찾으려고 한다. 변환 비용은 전체 에러 + W*(변환 코드의 길이)로 계산할 수 있다.

  1. W는 에러와 코드의 길이 중 어느 것을 중요시 하느냐에 따라 입력으로 주어진다.
  2. 이때 전체 에러는 입력 수열 x1, x2, , xN , 변환 수열 y1, y2, , yN에 대해서 다음과 같다. Σ|xi-yi|

수열이 2, 2, 2, 2, 2, 46, 2, 2 이고, W가 100인 경우에 y1,..., y8 = 1, 1, 1, 1, 1, 86, 1, 1 로 변환하면 코드는 0000001011000이 되고, 변환 비용은 47 + 100 * 13 = 1347이다. 하지만 y1,..., y8 = 1, 1, 1, 1, 1, 1, 1, 1로 변환하면 코드는 000000000이고 변환 비용은 52 + 100 * 9 = 952이다.

입력

첫째 줄에 수열의 길이 N(1 ≤ N ≤ 50), 가중치 W(1 ≤ W ≤ 100)가 주어진다. 다음 N개의 줄에는 수열을 이루는 숫자(≤256)들이 차례로 주어진다.

출력

첫줄에는 최소 변환 비용을 출력한다. 그리고 두 번째 줄에는 변환코드를 출력한다.

예제 입력 1

8 100
2
2
2
2
2
46
2
2

예제 출력 1

952
000000000
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\ub098\ud0c0\ub0b4\ub824\uace0 \ud55c\ub2e4. \uadf8\ub9ac\uace0, \uc774 \ub808\ubca8\uc740 \ub2e4\uc74c\uacfc \uac19\uc774 2-bit \ucf54\ub4dc\ub85c \ub098\ud0c0\ub0bc \uc218 \uc788\ub2e4.<\/p>\r\n\r\n<table class=\"table table-bordered\" style=\"width:40%\">\r\n\t<tbody>\r\n\t\t<tr>\r\n\t\t\t<th style=\"width:8%\">Level<\/th>\r\n\t\t\t<td style=\"width:8%\">1<\/td>\r\n\t\t\t<td style=\"width:8%\">86<\/td>\r\n\t\t\t<td style=\"width:8%\">172<\/td>\r\n\t\t\t<td style=\"width:8%\">256<\/td>\r\n\t\t<\/tr>\r\n\t\t<tr>\r\n\t\t\t<th style=\"width:8%\">Code<\/th>\r\n\t\t\t<td style=\"width:8%\">00<\/td>\r\n\t\t\t<td style=\"width:8%\">01<\/td>\r\n\t\t\t<td style=\"width:8%\">10<\/td>\r\n\t\t\t<td style=\"width:8%\">11<\/td>\r\n\t\t<\/tr>\r\n\t<\/tbody>\r\n<\/table>\r\n\r\n<p>\uadf8\ub9bc \uc548\uc5d0\ub294 \uac19\uc740 \uc0c9\uc73c\ub85c \uc774\ub8e8\uc5b4\uc9c4 \uc601\uc5ed(\uc608\ub97c \ub4e4\uba74 \uacf5\ubc31)\uc774 \uc788\ub2e4.&nbsp;\uc774 \uc601\uc5ed\uc744 \ubcc0\ud658\ud558\uba74 \uc720\uc0ac\ud55c \uc22b\uc790\ub4e4\uc774 \uc5f0\uc18d\ud574\uc11c \ub098\ud0c0\ub09c\ub2e4. \uc774\ub97c \ucc98\ub9ac\ud558\uae30 \uc704\ud574\uc11c \ub2e4\uc74c\uacfc \uac19\uc740 \uc555\ucd95 \ubc29\ubc95\uc744 \uc0ac\uc6a9\ud558\ub824\uace0 \ud55c\ub2e4.<\/p>\r\n\r\n<ol>\r\n\t<li>\uc218\uc5f4\uc758 \uccab \ubc88\uc9f8 \uc22b\uc790\ub294 \uc704\uc758 \ud45c\uc5d0 \ub098\uc628 2bit \ucf54\ub4dc\ub85c \ubcc0\ud658\ud55c\ub2e4.<\/li>\r\n\t<li>\uadf8 \ub2e4\uc74c\uc5d0 \ub098\uc624\ub294 \ub2e4\ub978 \ubaa8\ub4e0 \uc22b\uc790\uc5d0 \ub300\ud574\uc11c\ub294\r\n\t<ul>\r\n\t\t<li>\ub9cc\uc57d \uc218\uc5f4\uc758 \ubc14\ub85c \uc55e\uc5d0 \uc788\ub294 \uc22b\uc790\uac00 \uac19\uc740 \uc22b\uc790\ub77c\uba74 0\uc73c\ub85c \ubcc0\ud658\ud55c\ub2e4.<\/li>\r\n\t\t<li>\uc55e\uc5d0 \uc788\ub294 \uc22b\uc790\uc640 \ub2e4\ub978 \uc22b\uc790\ub77c\uba74 1\uc744 \uc55e\uc5d0 \ubd99\uc778 \ub4a4 \uc704 \ud45c\uc758 2bit \ucf54\ub4dc\ub85c \ubcc0\ud658\ud55c\ub2e4.<\/li>\r\n\t<\/ul>\r\n\t<\/li>\r\n<\/ol>\r\n\r\n<p>\uc0c1\uadfc\uc774\ub294 \uc791\uc740&nbsp;\uc591\uc758 \uc5d0\ub7ec\ub97c \ud06c\uac8c \uc2e0\uacbd\uc4f0\uc9c0 \uc54a\uae30 \ub54c\ubb38\uc5d0, \uc791\uc740 \uc591\uc758 \uc5d0\ub7ec\ub294 \ubb34\uc2dc\ud55c\ub2e4. \uc608\ub97c \ub4e4\uc5b4, \uc785\ub825\ub41c \uc218\uc5f4\uc774&nbsp;2, 2, 2, 2, 2, 46, 2, 2 \ub77c\uace0 \ud588\uc744 \ub54c, \uc774\ub97c 1, 1, 1, 1, 1, 86, 1, 1\ub85c \ub098\ud0c0\ub0b8 \ub2e4\uc74c&nbsp;\ubcc0\ud658\ud558\uba74 0000001011000\uac00 \ub418\uace0 \uc804\uccb4 \uc5d0\ub7ec\ub294 1+1+1+1+1+40+1+1=47 \uc774 \ub41c\ub2e4. \ud558\uc9c0\ub9cc 1, 1, 1, 1, 1, 1, 1, 1 \ub85c \ub098\ud0c0\ub0b8 \ud6c4 \ubcc0\ud658\ud558\uba74 \uc804\uccb4 \uc5d0\ub7ec\ub294 1+1+1+1+1+45+1+1=52\uac00 \ub418\uc5b4 \ucee4\uc9c0\uc9c0\ub9cc \ucf54\ub4dc\uac00 000000000\uac00 \ub418\uc5b4 \ub354 \uc9e7\uc544\uc9c4\ub2e4.<\/p>\r\n\r\n<p>\uc0c1\uadfc\uc774\ub294 \ubcc0\ud658 \ube44\uc6a9\uc744 \ucd5c\uc18c\ub85c \ud558\ub294 \ubc29\ubc95\uc744 \ucc3e\uc73c\ub824\uace0 \ud55c\ub2e4. \ubcc0\ud658 \ube44\uc6a9\uc740 \uc804\uccb4 \uc5d0\ub7ec + W*(\ubcc0\ud658 \ucf54\ub4dc\uc758 \uae38\uc774)\ub85c \uacc4\uc0b0\ud560 \uc218 \uc788\ub2e4.<\/p>\r\n\r\n<ol>\r\n\t<li>W\ub294 \uc5d0\ub7ec\uc640 \ucf54\ub4dc\uc758 \uae38\uc774 \uc911 \uc5b4\ub290 \uac83\uc744 \uc911\uc694\uc2dc \ud558\ub290\ub0d0\uc5d0 \ub530\ub77c \uc785\ub825\uc73c\ub85c \uc8fc\uc5b4\uc9c4\ub2e4.<\/li>\r\n\t<li>\uc774\ub54c \uc804\uccb4 \uc5d0\ub7ec\ub294 \uc785\ub825 \uc218\uc5f4 x<sub>1<\/sub>, x<sub>2<\/sub>, , x<sub>N<\/sub> , \ubcc0\ud658 \uc218\uc5f4 y<sub>1<\/sub>, y<sub>2<\/sub>, , y<sub>N<\/sub>\uc5d0 \ub300\ud574\uc11c \ub2e4\uc74c\uacfc&nbsp;\uac19\ub2e4.&nbsp;&Sigma;|x<sub>i<\/sub>-y<sub>i<\/sub>|<\/li>\r\n<\/ol>\r\n\r\n<p>\uc218\uc5f4\uc774 2, 2, 2, 2, 2, 46, 2, 2 \uc774\uace0,&nbsp;W\uac00 100\uc778 \uacbd\uc6b0\uc5d0&nbsp;y<sub>1<\/sub>,..., y<sub>8<\/sub> = 1, 1, 1, 1, 1, 86, 1, 1 \ub85c \ubcc0\ud658\ud558\uba74 \ucf54\ub4dc\ub294 0000001011000\uc774 \ub418\uace0,&nbsp;\ubcc0\ud658 \ube44\uc6a9\uc740 47 + 100 * 13 = 1347\uc774\ub2e4. \ud558\uc9c0\ub9cc y<sub>1<\/sub>,..., y<sub>8<\/sub> = 1, 1, 1, 1, 1, 1, 1, 1\ub85c \ubcc0\ud658\ud558\uba74 \ucf54\ub4dc\ub294 000000000\uc774\uace0 \ubcc0\ud658 \ube44\uc6a9\uc740 52 + 100 * 9 = 952\uc774\ub2e4.<\/p>\r\n","input":"<p>\uccab\uc9f8 \uc904\uc5d0&nbsp;\uc218\uc5f4\uc758 \uae38\uc774 N(1 &le;&nbsp;N &le; 50), \uac00\uc911\uce58 W(1 &le; W &le; 100)\uac00 \uc8fc\uc5b4\uc9c4\ub2e4. \ub2e4\uc74c N\uac1c\uc758 \uc904\uc5d0\ub294 \uc218\uc5f4\uc744 \uc774\ub8e8\ub294 \uc22b\uc790(&le;256)\ub4e4\uc774 \ucc28\ub840\ub85c \uc8fc\uc5b4\uc9c4\ub2e4.<\/p>\r\n","output":"<p>\uccab\uc904\uc5d0\ub294 \ucd5c\uc18c \ubcc0\ud658 \ube44\uc6a9\uc744 \ucd9c\ub825\ud55c\ub2e4. \uadf8\ub9ac\uace0 \ub450 \ubc88\uc9f8 \uc904\uc5d0\ub294 \ubcc0\ud658\ucf54\ub4dc\ub97c \ucd9c\ub825\ud55c\ub2e4.<\/p>\r\n","hint":"","original":"0","problem_lang_code":"\ud55c\uad6d\uc5b4"},{"problem_id":"2341","problem_lang":"1","title":"Fax","description":"<p>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&nbsp;<\/p>\r\n\r\n<p>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, &hellip;<\/p>\r\n\r\n<p>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<\/p>\r\n\r\n<table class=\"table table-bordered\" style=\"width:40%\">\r\n\t<tbody>\r\n\t\t<tr>\r\n\t\t\t<th style=\"width:8%\">Level<\/th>\r\n\t\t\t<td style=\"width:8%\">1<\/td>\r\n\t\t\t<td style=\"width:8%\">86<\/td>\r\n\t\t\t<td style=\"width:8%\">172<\/td>\r\n\t\t\t<td style=\"width:8%\">256<\/td>\r\n\t\t<\/tr>\r\n\t\t<tr>\r\n\t\t\t<th style=\"width:8%\">Code<\/th>\r\n\t\t\t<td style=\"width:8%\">00<\/td>\r\n\t\t\t<td style=\"width:8%\">01<\/td>\r\n\t\t\t<td style=\"width:8%\">10<\/td>\r\n\t\t\t<td style=\"width:8%\">11<\/td>\r\n\t\t<\/tr>\r\n\t<\/tbody>\r\n<\/table>\r\n\r\n<p>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:<\/p>\r\n\r\n<ol>\r\n\t<li>For the first number in the sequence, you will use the 2 bits as described previously.<\/li>\r\n\t<li>For all other numbers:\r\n\t<ul>\r\n\t\t<li>if the previous number is the same as the current number, you will transmit the bit 0.<\/li>\r\n\t\t<li>otherwise, you will transmit the bit 1 followed by the 2-bit code of the number. That is,\r\n\t\t<ul>\r\n\t\t\t<li>1 followed by 00 for number 1,<\/li>\r\n\t\t\t<li>1 followed by 01 for number 86,<\/li>\r\n\t\t\t<li>1 followed by 10 for number 172,<\/li>\r\n\t\t\t<li>1 followed by 11 for number 256.<\/li>\r\n\t\t<\/ul>\r\n\t\t<\/li>\r\n\t<\/ul>\r\n\t<\/li>\r\n<\/ol>\r\n\r\n<p>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.&nbsp;<\/p>\r\n\r\n<p>You decide to measure the overall cost of a transmission by the following weighted sum&nbsp;<\/p>\r\n\r\n<p>cost = total error + W &times; (total number of bits transmitted)<\/p>\r\n\r\n<p>where<\/p>\r\n\r\n<ol>\r\n\t<li>the weight W can be adjusted to give more emphasis to either the total error or the total number of bits transmitted<\/li>\r\n\t<li>for an input sequence x<sub>1<\/sub>, x<sub>2<\/sub>, &hellip;, x<sub>N<\/sub> and a transmitted sequence y<sub>1<\/sub>, y<sub>2<\/sub>, &hellip;, y<sub>N<\/sub>, both of length N, the total error is Summation (i from 1 to N) |x<sub>i<\/sub> - y<sub>i<\/sub>| = |x<sub>1<\/sub> - y<sub>1<\/sub>| + &hellip; + |x<sub>N<\/sub> - y<sub>N<\/sub>|.<\/li>\r\n<\/ol>\r\n\r\n<p>For the input sequence 2, 2, 2, 2, 2, 46, 2, 2 with W = 100,<\/p>\r\n\r\n<ul>\r\n\t<li>we can transmit the sequence y<sub>1<\/sub>, &hellip;, y<sub>8<\/sub> = 1, 1, 1, 1, 1, 86, 1, 1 by transmitting the string 0000001011000 with a cost of 47 + 100 &times; 13 = 1347;<\/li>\r\n\t<li>or we can transmit the sequence y<sub>1<\/sub>, &hellip;, y<sub>8<\/sub> = 1, 1, 1, 1, 1, 1, 1, 1 by transmitting the string 000000000 with a cost of 52 + 100 &times; 9 = 952.<\/li>\r\n<\/ul>\r\n\r\n<p>In this case, the second option has lower cost.&nbsp;<\/p>\r\n","input":"<p>The input consists of several lines. The first line contains two integers N (1 &le;&nbsp;N &le;&nbsp;50) and W (1 &le;&nbsp;W &le;&nbsp;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;&nbsp;x &le;&nbsp;256) of the sequence.<\/p>\r\n","output":"<p>The output contains a single integer which is the smallest possible cost required to transmit the input sequence with the given value of W.<\/p>\r\n","hint":"","original":"1","problem_lang_code":"\uc601\uc5b4"}]