시간 제한 메모리 제한 제출 정답 맞은 사람 정답 비율
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문제

과학자들은 화성에서 특이한 박테리아를 발견했고, 열심히 조사하고 있다. 그들은 박테리아의 개수가 2의 제곱수인 것을 알아챘다. 그것은 이들이 1개의 박테리아에서 시작되고, 각 박테리아는 매 세대 2개의 박테리아로 분열하기 때문이다(원래의 박테리아는 없어지는 꼴이다).

즉, 1세대는 하나의 박테리아, 2세대는 첫 세대의 박테리아가 둘로 나누어진 두 박테리아, 3세대는 또 그것이 나누어진 네 개의 박테리아, 4세대는 여덟 개, 이런 식으로 K+1세대에는 2k개의 박테리아가 있는 것을 알게 되었다.

과학자들은 그들이 찾아낸 박테리아를 1부터 2K까지 다음과 같이 번호 붙였다.

  • 이전의 K세대의 박테리아들의 후손은: {1, 2}, {3, 4}, {5, 6}, ..., {2K-1, 2K}
  • 더욱 전의 (K-1)세대의 박테리아들의 후손은: {1, 2, 3, 4}, {5, 6, 7, 8}, ..., {2K-3, 2K-2, 2K-1, 2K}
  • 훨씬 더욱 전의 (K-2)세대의 박테리아들의 후손은: {1, 2, 3, 4, 5, 6, 7, 8}, ..., {2K-7, 2K-6, 2K-5, 2K-4, 2K-3, 2K-2, 2K-1, 2K}
  • ...
  • 2세대의 두 박테리아의 후손은: {1, 2, ..., 2K-1}와 {2K-1 + 1, 2K-1 + 2, ..., 2K}

중괄호는 이들이 한 박테리아의 후손들이라는 뜻이다. 즉 어떤 조상 박테리아의 후손도 연속된 번호를 가지고 있도록 번호붙여졌다.

이때 박테리아의 순서를 바꾸더라도 여전히 위의 규칙을 만족하게 할 수 있는데, 과학자들은 그런 순서들 중 '순열의 길이'를 최소화하는 규칙을 찾으려 한다. '순열의 길이'는 이웃한 박테리아 사이의 거리를 모두 합한 것으로 정의된다.

정확히는, 모든 두 박테리아 사이에는 반발력이 존재하는데, 이 반발력은 수치로 나타내어진다. 두 박테리아 사이의 반발력은 두 박테리아가 서로 이웃해 있을 때 떨어져야 하는 최소 길이이다(두 박테리아가 이웃하지 않는다면 반발력은 아무 효과도 미치지 않는다). 모든 박테리아 쌍들의 반발력이 주어질 때, 이들을 앞서 말한 규칙을 지키면서 나열할 수 있는 최소의 길이를 구하시오.

입력

입력의 첫 줄에는 문제에서 언급한 양수 K가 주어진다(1 ≤ K ≤ 9).

그 후의 2K개의 줄에는 [0, 106] 범위에 있는 2K개의 자연수가 주어진다. 2K × 2K개의 숫자는 박테리아 쌍들 사이의 반발력을 뜻한다: 여기서 (m, n)에 있는 숫자는 박테리아 m과 n 사이의 반발력을 뜻한다. (m, n)과 (n, m)에는 같은 숫자가 들어가고, m = n일 때는 0이다.

출력

조건을 만족하면서 박테리아를 나열하는 최소의 길이만을 출력하라.

예제 입력 1

2
0 7 2 1
7 0 4 3
2 4 0 5
1 3 5 0

예제 출력 1

13
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K&nbsp;&le;&nbsp;9).<\/p>\r\n\r\n<p>\uadf8 \ud6c4\uc758 2<sup>K<\/sup>\uac1c\uc758 \uc904\uc5d0\ub294 [0, 10<sup>6<\/sup>] \ubc94\uc704\uc5d0 \uc788\ub294 2<sup>K<\/sup>\uac1c\uc758 \uc790\uc5f0\uc218\uac00 \uc8fc\uc5b4\uc9c4\ub2e4. 2<sup>K<\/sup> &times; 2<sup>K<\/sup>\uac1c\uc758 \uc22b\uc790\ub294 \ubc15\ud14c\ub9ac\uc544 \uc30d\ub4e4 \uc0ac\uc774\uc758 \ubc18\ubc1c\ub825\uc744 \ub73b\ud55c\ub2e4: \uc5ec\uae30\uc11c (m, n)\uc5d0 \uc788\ub294 \uc22b\uc790\ub294 \ubc15\ud14c\ub9ac\uc544 m\uacfc n \uc0ac\uc774\uc758 \ubc18\ubc1c\ub825\uc744 \ub73b\ud55c\ub2e4. (m, n)\uacfc (n, m)\uc5d0\ub294 \uac19\uc740 \uc22b\uc790\uac00 \ub4e4\uc5b4\uac00\uace0, m = n\uc77c \ub54c\ub294 0\uc774\ub2e4.<\/p>\r\n","output":"<p>\uc870\uac74\uc744 \ub9cc\uc871\ud558\uba74\uc11c \ubc15\ud14c\ub9ac\uc544\ub97c \ub098\uc5f4\ud558\ub294 \ucd5c\uc18c\uc758 \uae38\uc774\ub9cc\uc744 \ucd9c\ub825\ud558\ub77c.<\/p>\r\n","hint":"","original":"0","problem_lang_code":"\ud55c\uad6d\uc5b4"},{"problem_id":"2794","problem_lang":"1","title":"MARS","description":"<p>Scientists have discovered some strange bacteria on Mars and are now busy studying them. They have noticed that the number of bacteria is a power of 2, since each bacterium on Mars splits into two new bacteria (dying in the process), and it all started from a single bacterium.<\/p>\r\n\r\n<p>Thus, in the first generation there was a single bacterium. It split into two bacteria of the second generation, which split into four bacteria of the third generation, and so on &ndash; until the 2<sup>K<\/sup> bacteria of generation K+1 that the scientists have discovered. They have numbered the bacteria using numbers from 1 to 2<sup>K<\/sup> in the following way:<\/p>\r\n\r\n<ul>\r\n\t<li>descendants of bacteria of the previous (K<sup>th<\/sup>) generation are, in this order: {1, 2}, {3, 4}, {5, 6}, ..., {2<sup>K<\/sup>-1, 2<sup>K<\/sup>}<\/li>\r\n\t<li>descendants of bacteria of the older ((K-1)<sup>th<\/sup>) generation are, in this order: {1, 2, 3, 4}, {5, 6, 7, 8}, ..., {2<sup>K<\/sup>-3, 2<sup>K<\/sup>-2, 2<sup>K<\/sup>-1, 2<sup>K<\/sup>}<\/li>\r\n\t<li>descendants of bacteria of the even older ((K-2)<sup>th<\/sup>) generation are, in this order: {1, 2, 3, 4, 5, 6, 7, 8}, ..., {2<sup>K<\/sup>-7, 2<sup>K<\/sup>-6, 2<sup>K<\/sup>-5, 2<sup>K<\/sup>-4, 2<sup>K<\/sup>-3, 2<sup>K<\/sup>-2, 2<sup>K<\/sup>-1, 2<sup>K<\/sup>}<\/li>\r\n\t<li>...<\/li>\r\n\t<li>descendants of the two bacteria of the second generation are, in this order: {1, 2, ..., 2<sup>K-1<\/sup>} and {2<sup>K-1<\/sup> + 1, 2<sup>K-1<\/sup> + 2, ..., 2<sup>K<\/sup>}<\/li>\r\n<\/ul>\r\n\r\n<p>where curly braces denote a set of descendants of a single bacteria. That is, the 2<sup>K<\/sup> bacteria of the current generation were numbered such that descendants of any older bacterium have consecutive numbers.<\/p>\r\n\r\n<p>Notice that there exist many different permutations of these bacteria which still satisfy the rule that descendants of any older bacterium have consecutive sequence numbers. Scientists want to arrange the bacteria into such a sequence which also has the minimum possible length. The length of a bacteria sequence is the sum of distances between all neighbouring bacteria pairs.<\/p>\r\n\r\n<p>Specifically, there is a certain quantifiable repulsion between every two bacteria, which is the minimum distance between them if they are next to each other in the sequence. (Repulsion plays no role between bacteria that are not immediate neighbours in the sequence.) Given the repulsion values for all bacteria pairs, find the minimum possible length of a bacteria sequence (permutation) satisfying the descendant rule given above.<\/p>\r\n","input":"<p>The first line of input contains the positive integer K (1 &le; K &le; 9) from the problem statement.Each of the following 2<sup>K<\/sup> lines contains 2<sup>K<\/sup> integers from the interval [0, 10<sup>6<\/sup>]. These 2K &times; 2<sup>K<\/sup> numbers represent repulsion between bacteria pairs: the number in row m and column n is the repulsion between bacteria m and n. This number will, of course, be equal to the number in row n and column m. For m = n the number will be 0.<\/p>\r\n","output":"<p style=\"margin-left: 40px;\">The first and only line of output should contain the minimum possible length of a bacteria sequence satisfying the constraints.<\/p>\r\n","hint":"","original":"1","problem_lang_code":"\uc601\uc5b4"}]