시간 제한 메모리 제한 제출 정답 맞은 사람 정답 비율
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문제

힙은 자료의 추가, 우선순위가 제일 높은 자료의 삭제가 가능한 자료구조이다. 이와 같은 힙에는 두 종류가 있는데, 각각 최소-힙, 최대-힙이다. 이 문제에서는 최대-힙을 다루기로 하자.

이와 같은 최대-힙을 이용하면 O(n log n)정렬인 힙 정렬을 할 수 있다. 우리가 다루기로 한 최대-힙을 이용하면 오름차순 정렬을 할 수 있다. 힙 정렬은 크게 두 개의 단계로 나뉘는데, 첫 번째 단계는 주어진 자료들로 힙을 구성하는 단계이고, 두 번째 단계는 이렇게 구성된 힙에서 최댓값을 계속 제거하는 단계이다.

예를 들어 (5, 4, 2, 1, 3)과 같은 힙을 살펴보자. 이 힙에서 최댓값을 삭제하면 (3, 4, 2, 1)이 되고, 힙의 조건을 맞추기 위해 Swap을 한 번 하면 (4, 3, 2, 1)의 힙을 얻는다. 이 힙에서 최댓값을 삭제하면 (1, 3, 2)이 되고, 힙의 조건을 맞추기 위해 Swap을 한 번 하면 (3, 1, 2)가 된다. 다음 단계에서는 (2, 1), (1)이 되고 힙 정렬이 종료된다. 즉, 힙이 (5, 4, 2, 1, 3)과 같이 구성되어 있었다면, 정렬을 위해 Swap을 두 번 사용하게 된다. 하지만 (5, 4, 3, 2, 1)과 같은 힙은 총 네 번의 Swap을 해야 한다.

n이 주어졌을 때, 1부터 n까지의 수를 한 번씩 사용하여 만들 수 있는 힙들 중에서, 위와 같은 Swap 회수가 최대가 되도록 하는 힙을 출력하는 프로그램을 작성하시오.

입력

첫째 줄에 n이 주어진다. (1 ≤ n ≤ 100,000)

출력

첫째 줄에 n개의 정수로 힙을 출력한다.

예제 입력 1

6

예제 출력 1

6 5 3 2 4 1
[{"problem_id":"2220","problem_lang":"0","title":"\ud799 \uc815\ub82c","description":"<p>\ud799\uc740 \uc790\ub8cc\uc758 \ucd94\uac00, \uc6b0\uc120\uc21c\uc704\uac00 \uc81c\uc77c \ub192\uc740 \uc790\ub8cc\uc758 \uc0ad\uc81c\uac00 \uac00\ub2a5\ud55c \uc790\ub8cc\uad6c\uc870\uc774\ub2e4. \uc774\uc640 \uac19\uc740 \ud799\uc5d0\ub294 \ub450 \uc885\ub958\uac00 \uc788\ub294\ub370, \uac01\uac01 \ucd5c\uc18c-\ud799, \ucd5c\ub300-\ud799\uc774\ub2e4. \uc774 \ubb38\uc81c\uc5d0\uc11c\ub294 \ucd5c\ub300-\ud799\uc744 \ub2e4\ub8e8\uae30\ub85c \ud558\uc790.<\/p>\r\n\r\n<p>\uc774\uc640 \uac19\uc740 \ucd5c\ub300-\ud799\uc744 \uc774\uc6a9\ud558\uba74 O(n log n)\uc815\ub82c\uc778 \ud799 \uc815\ub82c\uc744 \ud560 \uc218 \uc788\ub2e4. \uc6b0\ub9ac\uac00 \ub2e4\ub8e8\uae30\ub85c \ud55c \ucd5c\ub300-\ud799\uc744 \uc774\uc6a9\ud558\uba74 \uc624\ub984\ucc28\uc21c \uc815\ub82c\uc744 \ud560 \uc218 \uc788\ub2e4. \ud799 \uc815\ub82c\uc740 \ud06c\uac8c \ub450 \uac1c\uc758 \ub2e8\uacc4\ub85c \ub098\ub258\ub294\ub370, \uccab \ubc88\uc9f8 \ub2e8\uacc4\ub294 \uc8fc\uc5b4\uc9c4 \uc790\ub8cc\ub4e4\ub85c \ud799\uc744 \uad6c\uc131\ud558\ub294 \ub2e8\uacc4\uc774\uace0, \ub450 \ubc88\uc9f8 \ub2e8\uacc4\ub294 \uc774\ub807\uac8c \uad6c\uc131\ub41c \ud799\uc5d0\uc11c \ucd5c\ub313\uac12\uc744 \uacc4\uc18d \uc81c\uac70\ud558\ub294 \ub2e8\uacc4\uc774\ub2e4.<\/p>\r\n\r\n<p>\uc608\ub97c \ub4e4\uc5b4 (5, 4, 2, 1, 3)\uacfc \uac19\uc740 \ud799\uc744 \uc0b4\ud3b4\ubcf4\uc790. \uc774 \ud799\uc5d0\uc11c \ucd5c\ub313\uac12\uc744 \uc0ad\uc81c\ud558\uba74 (3, 4, 2, 1)\uc774 \ub418\uace0, \ud799\uc758 \uc870\uac74\uc744 \ub9de\ucd94\uae30 \uc704\ud574 Swap\uc744 \ud55c \ubc88 \ud558\uba74 (4, 3, 2, 1)\uc758 \ud799\uc744 \uc5bb\ub294\ub2e4. \uc774 \ud799\uc5d0\uc11c \ucd5c\ub313\uac12\uc744 \uc0ad\uc81c\ud558\uba74 (1, 3, 2)\uc774 \ub418\uace0, \ud799\uc758 \uc870\uac74\uc744 \ub9de\ucd94\uae30 \uc704\ud574 Swap\uc744 \ud55c \ubc88 \ud558\uba74 (3, 1, 2)\uac00 \ub41c\ub2e4. \ub2e4\uc74c \ub2e8\uacc4\uc5d0\uc11c\ub294 (2, 1), (1)\uc774 \ub418\uace0 \ud799 \uc815\ub82c\uc774 \uc885\ub8cc\ub41c\ub2e4. \uc989, \ud799\uc774 (5, 4, 2, 1, 3)\uacfc \uac19\uc774 \uad6c\uc131\ub418\uc5b4 \uc788\uc5c8\ub2e4\uba74, \uc815\ub82c\uc744 \uc704\ud574 Swap\uc744 \ub450 \ubc88 \uc0ac\uc6a9\ud558\uac8c \ub41c\ub2e4. \ud558\uc9c0\ub9cc (5, 4, 3, 2, 1)\uacfc \uac19\uc740 \ud799\uc740 \ucd1d \ub124 \ubc88\uc758 Swap\uc744 \ud574\uc57c \ud55c\ub2e4.<\/p>\r\n\r\n<p>n\uc774 \uc8fc\uc5b4\uc84c\uc744 \ub54c, 1\ubd80\ud130 n\uae4c\uc9c0\uc758 \uc218\ub97c \ud55c \ubc88\uc529 \uc0ac\uc6a9\ud558\uc5ec \ub9cc\ub4e4 \uc218 \uc788\ub294 \ud799\ub4e4 \uc911\uc5d0\uc11c, \uc704\uc640 \uac19\uc740 Swap \ud68c\uc218\uac00 \ucd5c\ub300\uac00 \ub418\ub3c4\ub85d \ud558\ub294 \ud799\uc744 \ucd9c\ub825\ud558\ub294 \ud504\ub85c\uadf8\ub7a8\uc744 \uc791\uc131\ud558\uc2dc\uc624.<\/p>\r\n","input":"<p>\uccab\uc9f8 \uc904\uc5d0 n\uc774 \uc8fc\uc5b4\uc9c4\ub2e4. (1 &le; n &le; 100,000)<\/p>\r\n","output":"<p>\uccab\uc9f8 \uc904\uc5d0 n\uac1c\uc758 \uc815\uc218\ub85c \ud799\uc744 \ucd9c\ub825\ud55c\ub2e4.<\/p>\r\n","hint":"","original":"0","html_title":"0","problem_lang_tcode":"Korean"},{"problem_id":"2220","problem_lang":"1","title":"Heapsort","description":"<p>A well known algorithm called <em>heapsort<\/em>&nbsp;is a deterministic sorting algorithm taking $O(n \\log n)$ time and $O(1)$ additional memory. Let us describe ascending sorting&nbsp;of an array of different integer numbers.<\/p>\r\n\r\n<p>The algorithm consists of two phases. In the first phase,&nbsp;called <em>heapification<\/em>, the array of integers to be sorted is converted to a <em>heap<\/em>. An array $a[1\\ldots n]$ of integers&nbsp;is called a heap if for all $1 \\le i \\le n$ the following <em>heap conditions<\/em>&nbsp;are satisfied:<\/p>\r\n\r\n<ul>\r\n\t<li>if $2i \\le n$ then $a[i] &gt; a[2i]$;<\/li>\r\n\t<li>if $2i + 1 \\le n$ then $a[i] &gt; a[2i + 1]$.<\/li>\r\n<\/ul>\r\n\r\n<p>We can interpret an array as a binary tree, considering children of element $a[i]$ to be $a[2i]$ and $a[2i + 1]$. In this case the parent of $a[i]$ is $a[i \\text{ div } 2]$, where $i \\text{ div } 2 = \\lfloor i \/ 2 \\rfloor$.&nbsp;In terms of trees the property of being a heap means that for each node its value is greater than the values of its children.<\/p>\r\n\r\n<p>In the second phase the heap is turned into a sorted array. Because of the heap condition the greatest element&nbsp;in the heapified array is $a[1]$. Let us exchange it with $a[n]$, now the greatest element of the array is at its correct position in the sorted array. This is called <em>extract-max<\/em>. Now let us consider the part of the array $a[1\\ldots n-1]$. It may be not a heap because the heap condition may fail for $i=1$.&nbsp;If it is so (that is, either $a[2]$ or $a[3]$, or both are greater&nbsp;than $a[1]$) let us exchange the greatest child of $a[1]$ with it, restoring&nbsp;the heap condition for $i=1$. Now it is possible that the heap condition&nbsp;fails for the position that now contains the former value of $a[1]$. Apply the same procedure to it, exchanging it with its greatest child. Proceeding so we convert the whole array $a[1\\ldots n-1]$ to a heap.&nbsp;This procedure is called <em>sifting down<\/em>.&nbsp;After converting the part $a[1\\ldots n-1]$ to a heap by sifting, we&nbsp;apply extract-max again, putting second greatest element of the array to $a[n-1]$, and so on.<\/p>\r\n\r\n<p>For example, let us see how the heap $a=(5, 4, 2, 1, 3)$ is converted to a sorted array. Let us make the first extract-max. After that the array&nbsp;turns to $(3, 4, 2, 1, \\mathbf{5})$. Heap condition fails for $a[1] = 3$ because its child $a[2] = 4$ is greater than it. Let us sift it down, exchanging $a[1]$ and $a[2]$. Now the array is $(4, 3, 2, 1, \\mathbf{5})$. The heap condition is satisfied for all elements, so sifting is over. Let us make extract-max again. Now the array turns to $(1, 3, 2, \\mathbf{4}, \\mathbf{5})$. Again&nbsp;the heap condition fails for $a[1]$; exchanging it with its greatest child we get the array $(3, 1, 2, \\mathbf{4}, \\mathbf{5})$ which is the correct heap. So we make extract-max and get&nbsp;$(2, 1, \\mathbf{3}, \\mathbf{4}, \\mathbf{5})$. This time the heap condition is satisfied for all elements, so we make extract-max, getting&nbsp;$(1, \\mathbf{2}, \\mathbf{3}, \\mathbf{4}, \\mathbf{5})$. The leading part of the array is a heap, and the last extract-max finally gives $(\\mathbf{1}, \\mathbf{2}, \\mathbf{3}, \\mathbf{4}, \\mathbf{5})$.<\/p>\r\n\r\n<p>It is known that heapification can be done in $O(n)$ time. Therefore, the most time consuming operation in heapsort algorithm is sifting, which takes $O(n\\log n)$ time.<\/p>\r\n\r\n<p>In this problem you have to find a heapified array containing different numbers from $1$ to $n$, such that when converting it to a sorted array, the total number of exchanges in all sifting operations is maximal possible. In the example above the number of exchanges is $1+1+0+0+0 = 2$, which is not the maximum. $(5, 4, 3, 2, 1)$ gives the maximal number of 4 exchanges for $n=5$.<\/p>\r\n","input":"<p>Input file contains $n$ ($1 \\le n \\le 100\\,000$).<\/p>\r\n","output":"<p>Output the array containing $n$ different integer&nbsp;numbers from 1 to $n$, such that it is a heap, and when converting it to a sorted array, the total number of exchanges in sifting operations is maximal possible. Separate numbers by spaces.<\/p>\r\n","hint":"","original":"1","html_title":"0","problem_lang_tcode":"English"}]