6498번 - 엘리어스 감마 코드 다국어

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문제

엘리어스 감마 코드는 양의 정수로 이루어진 수열을 인코딩 하는데 사용하는 코드이다. 이 문제에서는 0도 인코딩할 수 있게 변형한 코드를 사용한다.

정수 n을 인코딩하는 과정은 다음과 같다.

  1. n의 비트의 개수를 k라고 한다.
  2. 0을 k-1개 쓰고, 그 뒤에 1을 쓴다.
  3. 그 다음 n을 이진수로 쓴다.

0부터 8까지 숫자를 인코딩한 결과는 아래와 같다.

숫자 이진수 비트의 수 Prefix 코드
0 0 1 1 10
1 1 1 1 11
2 10 2 01 0110
3 11 2 01 0111
4 100 3 001 001100
5 101 3 001 001101
6 110 3 001 001110
7 111 3 001 001111
8 1000 4 0001 00011000

정수 수열을 인코딩하려면, 수열의 각 숫자를 코드로 변환한 뒤, 수열에서 나온 순서와 동일하게 이어 붙이면 된다.

각각의 인코딩된 코드를 원래 숫자로 디코딩하려면, 이진수 표현 앞에 붙는 k개의 비트는 매우 중요하다. 인코딩된 수열을 읽을 때, 1을 읽기 전에 k-1개의 0을 읽었다면, 다음 k개의 비트가 인코딩된 숫자라는 뜻이다.

인코딩된 정수 수열의 길이를 줄이려면, 다음과 같은 두 가지 최적화를 생각해 볼 수 있다. 

  1. Prefix가 k비트를 나타내는데, 수열에 비트의 길이가 k인 숫자가 없는 경우가 있다. 이 경우에는, 이 prefix는 k+1비트를 나타내는 것으로 사용할 수 있다. 만약, k+1비트를 나타내는 prefix가 이미 있는 경우에는 k+2비트를 나타내는 prefix로 사용하고, 이와 같은 식으로 계속 한다.
  2. 비트의 길이가 k인 모든 숫자의 앞에 0을 붙이면, 그 숫자는 모두 k+1개 비트를 가지게 만들고 첫 번째 최적화를 사용하는 방법도 있다. 이 방법은 k비트 숫자가 매우 적고, k비트 이상 숫자가 많은 경우에 효과적이다.

정수 수열을 인코딩한 결과의 길이를 최소로 하려고 한다. 수열의 숫자가 주어지지 않으며, 수열에서 비트의 개수가 i인 수의 개수를 ci라고 한다.

예를 들어, c1=2, c2=4, c3=0, c4=1인 경우를 생각해보자. (수열 2,1,3,8,0,2,3이 이 경우에 해당한다) 최적화를 하지 않은 엘리어스 감마 코딩의 길이는 2×(1+1) + 4×(2+2) + 0×(3+3) + 1×(4+4) = 28이 된다. 1번 최적화를 사용해 prefix 001을 4비트 숫자를 나타내는 것으로 사용해 1비트를 줄일 수 있다. 또, 최적화 2번을 사용해 1비트 앞에 0을 붙여 2비트로 만들 수 있다. 여기서 다시 최적화 1번을 사용해 prefix 1은 2비트 숫자에 사용하고, prefix 01은 4비트 숫자에 사용하면, 인코딩된 문자열의 길이는 6×(1+2) + 1×(2+4) = 24가 된다.

두 최적화 방법은 여러 번 사용할 수 있다. 두 방법을 적절히 사용해 가능한 길이중 최솟값을 구하는 프로그램을 작성하시오.

입력

입력은 여러 개의 테스트 케이스로 이루어져 있다. 각 테스트 케이스의 첫째 줄에는 n이 주어진다. (1 ≤ n ≤ 128) 둘째 줄에는 c1부터 cn까지 값이 주어진다. (0 ≤ ci ≤ 10000) 입력은 n=0으로 끝난다.

출력

각 테스트 케이스 마다 입력으로 주어진 수열의 최소 엘리어스 감마 인코딩 길이를 출력한다.

예제 입력 1 

4
2 4 0 1
5
9 4 2 4 3
11
44 56 96 26 73 80 77 50 33 16 78
0

예제 출력 1 

24
99
5494
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\uacb0\uacfc\ub294 \uc544\ub798\uc640 \uac19\ub2e4.<\/p>\r\n\r\n<table class=\"table table-bordered\" style=\"width:50%\">\r\n\t<thead>\r\n\t\t<tr>\r\n\t\t\t<th>\uc22b\uc790<\/th>\r\n\t\t\t<th>\uc774\uc9c4\uc218<\/th>\r\n\t\t\t<th>\ube44\ud2b8\uc758 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\uc218\uc5f4\uc744 \uc778\ucf54\ub529\ud558\ub824\uba74, \uc218\uc5f4\uc758 \uac01 \uc22b\uc790\ub97c \ucf54\ub4dc\ub85c \ubcc0\ud658\ud55c \ub4a4, \uc218\uc5f4\uc5d0\uc11c \ub098\uc628 \uc21c\uc11c\uc640 \ub3d9\uc77c\ud558\uac8c \uc774\uc5b4 \ubd99\uc774\uba74 \ub41c\ub2e4.<\/p>\r\n\r\n<p>\uac01\uac01\uc758 \uc778\ucf54\ub529\ub41c \ucf54\ub4dc\ub97c \uc6d0\ub798 \uc22b\uc790\ub85c \ub514\ucf54\ub529\ud558\ub824\uba74, \uc774\uc9c4\uc218 \ud45c\ud604 \uc55e\uc5d0 \ubd99\ub294 k\uac1c\uc758 \ube44\ud2b8\ub294 \ub9e4\uc6b0 \uc911\uc694\ud558\ub2e4. \uc778\ucf54\ub529\ub41c \uc218\uc5f4\uc744 \uc77d\uc744 \ub54c, 1\uc744 \uc77d\uae30 \uc804\uc5d0 k-1\uac1c\uc758 0\uc744 \uc77d\uc5c8\ub2e4\uba74, \ub2e4\uc74c k\uac1c\uc758 \ube44\ud2b8\uac00 \uc778\ucf54\ub529\ub41c \uc22b\uc790\ub77c\ub294 \ub73b\uc774\ub2e4.<\/p>\r\n\r\n<p>\uc778\ucf54\ub529\ub41c \uc815\uc218 \uc218\uc5f4\uc758 \uae38\uc774\ub97c \uc904\uc774\ub824\uba74, \ub2e4\uc74c\uacfc \uac19\uc740 \ub450 \uac00\uc9c0 \ucd5c\uc801\ud654\ub97c \uc0dd\uac01\ud574 \ubcfc \uc218 \uc788\ub2e4.&nbsp;<\/p>\r\n\r\n<ol>\r\n\t<li>Prefix\uac00 k\ube44\ud2b8\ub97c \ub098\ud0c0\ub0b4\ub294\ub370, \uc218\uc5f4\uc5d0 \ube44\ud2b8\uc758 \uae38\uc774\uac00 k\uc778 \uc22b\uc790\uac00 \uc5c6\ub294 \uacbd\uc6b0\uac00 \uc788\ub2e4. \uc774 \uacbd\uc6b0\uc5d0\ub294, \uc774 prefix\ub294 k+1\ube44\ud2b8\ub97c \ub098\ud0c0\ub0b4\ub294 \uac83\uc73c\ub85c \uc0ac\uc6a9\ud560 \uc218 \uc788\ub2e4. \ub9cc\uc57d, k+1\ube44\ud2b8\ub97c \ub098\ud0c0\ub0b4\ub294 prefix\uac00 \uc774\ubbf8 \uc788\ub294 \uacbd\uc6b0\uc5d0\ub294 k+2\ube44\ud2b8\ub97c \ub098\ud0c0\ub0b4\ub294 prefix\ub85c \uc0ac\uc6a9\ud558\uace0, \uc774\uc640 \uac19\uc740 \uc2dd\uc73c\ub85c \uacc4\uc18d \ud55c\ub2e4.<\/li>\r\n\t<li>\ube44\ud2b8\uc758 \uae38\uc774\uac00 k\uc778 \ubaa8\ub4e0 \uc22b\uc790\uc758 \uc55e\uc5d0 0\uc744 \ubd99\uc774\uba74, \uadf8 \uc22b\uc790\ub294 \ubaa8\ub450 k+1\uac1c \ube44\ud2b8\ub97c \uac00\uc9c0\uac8c \ub9cc\ub4e4\uace0 \uccab \ubc88\uc9f8 \ucd5c\uc801\ud654\ub97c \uc0ac\uc6a9\ud558\ub294 \ubc29\ubc95\ub3c4 \uc788\ub2e4. \uc774 \ubc29\ubc95\uc740 k\ube44\ud2b8 \uc22b\uc790\uac00 \ub9e4\uc6b0 \uc801\uace0, k\ube44\ud2b8 \uc774\uc0c1 \uc22b\uc790\uac00 \ub9ce\uc740 \uacbd\uc6b0\uc5d0 \ud6a8\uacfc\uc801\uc774\ub2e4.<\/li>\r\n<\/ol>\r\n\r\n<p>\uc815\uc218 \uc218\uc5f4\uc744 \uc778\ucf54\ub529\ud55c \uacb0\uacfc\uc758 \uae38\uc774\ub97c \ucd5c\uc18c\ub85c \ud558\ub824\uace0 \ud55c\ub2e4. \uc218\uc5f4\uc758 \uc22b\uc790\uac00 \uc8fc\uc5b4\uc9c0\uc9c0 \uc54a\uc73c\uba70, \uc218\uc5f4\uc5d0\uc11c \ube44\ud2b8\uc758 \uac1c\uc218\uac00 i\uc778 \uc218\uc758 \uac1c\uc218\ub97c c<sub>i<\/sub>\ub77c\uace0 \ud55c\ub2e4.<\/p>\r\n\r\n<p>\uc608\ub97c \ub4e4\uc5b4, c<sub>1<\/sub>=2, c<sub>2<\/sub>=4, c<sub>3<\/sub>=0, c<sub>4<\/sub>=1\uc778 \uacbd\uc6b0\ub97c \uc0dd\uac01\ud574\ubcf4\uc790. (\uc218\uc5f4 2,1,3,8,0,2,3\uc774 \uc774 \uacbd\uc6b0\uc5d0 \ud574\ub2f9\ud55c\ub2e4) \ucd5c\uc801\ud654\ub97c \ud558\uc9c0 \uc54a\uc740 \uc5d8\ub9ac\uc5b4\uc2a4 \uac10\ub9c8 \ucf54\ub529\uc758 \uae38\uc774\ub294 2&times;(1+1) + 4&times;(2+2) + 0&times;(3+3) + 1&times;(4+4) = 28\uc774 \ub41c\ub2e4. 1\ubc88 \ucd5c\uc801\ud654\ub97c \uc0ac\uc6a9\ud574 prefix 001\uc744 4\ube44\ud2b8 \uc22b\uc790\ub97c \ub098\ud0c0\ub0b4\ub294 \uac83\uc73c\ub85c \uc0ac\uc6a9\ud574 1\ube44\ud2b8\ub97c \uc904\uc77c \uc218 \uc788\ub2e4. \ub610, \ucd5c\uc801\ud654 2\ubc88\uc744 \uc0ac\uc6a9\ud574 1\ube44\ud2b8 \uc55e\uc5d0 0\uc744 \ubd99\uc5ec 2\ube44\ud2b8\ub85c \ub9cc\ub4e4 \uc218 \uc788\ub2e4. \uc5ec\uae30\uc11c \ub2e4\uc2dc \ucd5c\uc801\ud654 1\ubc88\uc744 \uc0ac\uc6a9\ud574 prefix 1\uc740 2\ube44\ud2b8 \uc22b\uc790\uc5d0 \uc0ac\uc6a9\ud558\uace0, prefix 01\uc740 4\ube44\ud2b8 \uc22b\uc790\uc5d0 \uc0ac\uc6a9\ud558\uba74, \uc778\ucf54\ub529\ub41c \ubb38\uc790\uc5f4\uc758 \uae38\uc774\ub294 6&times;(1+2) + 1&times;(2+4) = 24\uac00 \ub41c\ub2e4.<\/p>\r\n\r\n<p>\ub450 \ucd5c\uc801\ud654 \ubc29\ubc95\uc740 \uc5ec\ub7ec \ubc88 \uc0ac\uc6a9\ud560 \uc218 \uc788\ub2e4. \ub450 \ubc29\ubc95\uc744 \uc801\uc808\ud788 \uc0ac\uc6a9\ud574 \uac00\ub2a5\ud55c \uae38\uc774\uc911 \ucd5c\uc19f\uac12\uc744 \uad6c\ud558\ub294 \ud504\ub85c\uadf8\ub7a8\uc744 \uc791\uc131\ud558\uc2dc\uc624.<\/p>\r\n","input":"<p>\uc785\ub825\uc740 \uc5ec\ub7ec \uac1c\uc758 \ud14c\uc2a4\ud2b8 \ucf00\uc774\uc2a4\ub85c \uc774\ub8e8\uc5b4\uc838 \uc788\ub2e4. \uac01 \ud14c\uc2a4\ud2b8 \ucf00\uc774\uc2a4\uc758 \uccab\uc9f8 \uc904\uc5d0\ub294 n\uc774 \uc8fc\uc5b4\uc9c4\ub2e4. (1 &le; n &le; 128) \ub458\uc9f8 \uc904\uc5d0\ub294 c<sub>1<\/sub>\ubd80\ud130 c<sub>n<\/sub>\uae4c\uc9c0 \uac12\uc774 \uc8fc\uc5b4\uc9c4\ub2e4. (0 &le; c<sub>i<\/sub> &le; 10000) \uc785\ub825\uc740 n=0\uc73c\ub85c \ub05d\ub09c\ub2e4.<\/p>\r\n","output":"<p>\uac01 \ud14c\uc2a4\ud2b8 \ucf00\uc774\uc2a4 \ub9c8\ub2e4 \uc785\ub825\uc73c\ub85c \uc8fc\uc5b4\uc9c4 \uc218\uc5f4\uc758 \ucd5c\uc18c \uc5d8\ub9ac\uc5b4\uc2a4 \uac10\ub9c8 \uc778\ucf54\ub529 \uae38\uc774\ub97c \ucd9c\ub825\ud55c\ub2e4.<\/p>\r\n","hint":"","original":"0","html_title":"0","problem_lang_tcode":"Korean"},{"problem_id":"6498","problem_lang":"1","title":"Elias gamma coding","description":"<p>The Elias gamma code is a simple code which can be used to encode a sequence of positive integers. We will use a modified code which is also able to encode zeros. To encode an integer n, do the following:<\/p>\r\n\r\n<ol>\r\n\t<li>Let k be the number of bits of n<\/li>\r\n\t<li>Write k-1 zeros followed by a 1<\/li>\r\n\t<li>Write n in binary<\/li>\r\n<\/ol>\r\n\r\n<table class=\"table table-bordered\" style=\"width:60%\">\r\n\t<thead>\r\n\t\t<tr>\r\n\t\t\t<th>Number<\/th>\r\n\t\t\t<th>Binary<\/th>\r\n\t\t\t<th>Number of bits<\/th>\r\n\t\t\t<th>Prefix<\/th>\r\n\t\t\t<th>Code<\/th>\r\n\t\t<\/tr>\r\n\t<\/thead>\r\n\t<tbody>\r\n\t\t<tr>\r\n\t\t\t<td>0<\/td>\r\n\t\t\t<td>0<\/td>\r\n\t\t\t<td>1<\/td>\r\n\t\t\t<td>1<\/td>\r\n\t\t\t<td>10<\/td>\r\n\t\t<\/tr>\r\n\t\t<tr>\r\n\t\t\t<td>1<\/td>\r\n\t\t\t<td>1<\/td>\r\n\t\t\t<td>1<\/td>\r\n\t\t\t<td>1<\/td>\r\n\t\t\t<td>11<\/td>\r\n\t\t<\/tr>\r\n\t\t<tr>\r\n\t\t\t<td>2<\/td>\r\n\t\t\t<td>10<\/td>\r\n\t\t\t<td>2<\/td>\r\n\t\t\t<td>01<\/td>\r\n\t\t\t<td>0110<\/td>\r\n\t\t<\/tr>\r\n\t\t<tr>\r\n\t\t\t<td>3<\/td>\r\n\t\t\t<td>11<\/td>\r\n\t\t\t<td>2<\/td>\r\n\t\t\t<td>01<\/td>\r\n\t\t\t<td>0111<\/td>\r\n\t\t<\/tr>\r\n\t\t<tr>\r\n\t\t\t<td>4<\/td>\r\n\t\t\t<td>100<\/td>\r\n\t\t\t<td>3<\/td>\r\n\t\t\t<td>001<\/td>\r\n\t\t\t<td>001100<\/td>\r\n\t\t<\/tr>\r\n\t\t<tr>\r\n\t\t\t<td>5<\/td>\r\n\t\t\t<td>101<\/td>\r\n\t\t\t<td>3<\/td>\r\n\t\t\t<td>001<\/td>\r\n\t\t\t<td>001101<\/td>\r\n\t\t<\/tr>\r\n\t\t<tr>\r\n\t\t\t<td>6<\/td>\r\n\t\t\t<td>110<\/td>\r\n\t\t\t<td>3<\/td>\r\n\t\t\t<td>001<\/td>\r\n\t\t\t<td>001110<\/td>\r\n\t\t<\/tr>\r\n\t\t<tr>\r\n\t\t\t<td>7<\/td>\r\n\t\t\t<td>111<\/td>\r\n\t\t\t<td>3<\/td>\r\n\t\t\t<td>001<\/td>\r\n\t\t\t<td>001111<\/td>\r\n\t\t<\/tr>\r\n\t\t<tr>\r\n\t\t\t<td>8<\/td>\r\n\t\t\t<td>1000<\/td>\r\n\t\t\t<td>4<\/td>\r\n\t\t\t<td>0001<\/td>\r\n\t\t\t<td>00011000<\/td>\r\n\t\t<\/tr>\r\n\t<\/tbody>\r\n<\/table>\r\n\r\n<p>A sequence of integers is encoded by writing the codes of the individual integers of the sequence in the same order as the integers appear in the sequence. The prefix of k additional bits before the binary representation of each integer is needed to be able to decode the encoded integers. So when reading the encoding of a sequence of integers, if we read k-1 zeros followed by a one, it means that there are k bits following which are the binary representation of the next encoded integer.<\/p>\r\n\r\n<p>If we want to shorten the length of the encoding of a sequence of integers, there may be still some room for improvement; we will consider the following two optimizations:<\/p>\r\n\r\n<ol>\r\n\t<li>If there is a prefix which indicates that k bits are following, but there is no integer in the sequence with k bits, we can use this prefix to indicate that k+1 bits are following. If there already was a prefix which indicates that k+1 bits are following, this prefix is not needed anymore, and it can be used to indicate that k+2 bits are following, and so on.<\/li>\r\n\t<li>We can add a leading zero to the binary representation of all integers in the sequence with k bits, which then become integers with k+1 bits, and then the first optimization can be used. This optimization seems especially useful if there are few integers with k bits, but many integers with more than k bits.<\/li>\r\n<\/ol>\r\n\r\n<p>When we are minimizing the length of the encoding of a sequence of integers, we only care about how many integers in the sequence have a certain number of bits. Let c<sub>i<\/sub> denote the number of integers in a sequence with i bits.<\/p>\r\n\r\n<p>Let us look at the following example: c<sub>1<\/sub> = 2, c<sub>2<\/sub> = 4, c<sub>3<\/sub> = 0, c<sub>4<\/sub> = 1 (which, for example, could correspond to a sequence 2, 1, 3, 8, 0, 2, 3). With the original elias gamma coding, the encoding of the sequence would have length 2 &times; (1 + 1) + 4 &times; (2 + 2) + 0 &times; (3 + 3) + 1 &times; (4 + 4) = 28. By using optimization 1 we can save 1 bit by using prefix 001 for the integer with 4 bits. Then, we could use optimization 2 and add leading zeros to the integers with 1 bit, making them use 2 bits. Then, we use optimization 1 and use prefix 1 for the integers with 2 bits, prefix 01 for the integer with 4 bits, and we get the new length of 6 &times; (1 + 2) + 1 &times; (2 + 4) = 24.<\/p>\r\n\r\n<p>Both optimizations can possibly be used several times. Note that for the second optimization, it is not easy to decide when and how to use it. The goal is to combine these two optimizations in the best possible way, that means we want to find an encoding of a given sequence of integers that has minimum length among all encodings using elias gamma coding with any combination of these two optimizations.<\/p>\r\n","input":"<p>The input file contains several test cases. Each test case starts with a line containing an integer n, (1 &le; n &le; 128). The next line contains the values c<sub>1<\/sub>, ..., c<sub>n<\/sub> (0 &le; c<sub>i<\/sub> &le; 10000). Input is terminated by n=0.<\/p>\r\n\r\n<p>&nbsp;<\/p>\r\n","output":"<p>For each test case print one line with the minimum length of an encoding of the given input sequence.<\/p>\r\n\r\n<p>&nbsp;<\/p>\r\n","hint":"","original":"1","html_title":"0","problem_lang_tcode":"English"}]

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