시간 제한 메모리 제한 제출 정답 맞은 사람 정답 비율
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문제

꿍은 자료구조를 공부할때 각 정렬마다 비교횟수를 구해보라는 문제를 볼때마다 화가난 나머지 세상에서 제일 거지같은 문제를 만들어보고자 다음과 같은 문제를 만들었다.

N과 X가 주어질 때, 삽입정렬이 퀵정렬보다 비교를 최대 X번 더 하는 {1,2,...,N} 으로 구성된 순열의 개수를 구해야 한다. 단, 숫자가 매우 커질수 있으므로 출력값은 1234567로 나눈 나머지를 출력한다.

아래 코드는 삽입정렬 코드이며 비교횟수도 계산해준다.

procedure insertionSort(int N, array A[1..N]) defined as:
    A[0] := -Infinity
    for i := 2 to N do:
        j := i
        Increment(comparison_count)
        while A[j - 1] > A[j] do:
        SWAP(A[j - 1], A[j])
        j := j - 1
        Increment(comparison_count)
    end while
end for

아래코드는 퀵정렬 코드다. 만약 정렬하려는 배열의 길이가 L이라면, partition 알고리즘은 L-1번의 비교를 한다.

procedure quickSort(list A) defined as:
    list less, greater
    if length(A) <= 1 then
        return A

    pivot := A[1]
    for i := 2 to length(A) do:
        Increment(comparison_count)
        if A[i] < pivot then append A[i] to less
                        else append A[i] to greater
        end if
    end for
    return concatenate(quickSort(less), pivot, quickSort(greater))

예를 들어, 순열 (3,1,4,2)를 생각하자. 삽입정렬의 비교횟수는 총 6회로 i=2일때 2번, i=3일때 1번, i=4일때 3번의 비교가 이루어진다. 반면 퀵정렬의 비교횟수는 총 4회로 3이 pivot일때 3회의 비교를 한 후 (1,2)와 (4)로 partition이 이루어진다. 이후 (1,2)에서 1번의 비교가 이루어지므로 총 4회의 비교가 이루어진다.

입력

입력의 첫번째 줄에 두 개의 정수 N과 X가 주어지며 각각의 범위는 1 < N < 32, 1 ≤ X ≤ N^2 이다.

출력

삽입정렬이 퀵정렬보다 최대 X번 비교를 더 하게되는 순열의 개수를 1234567로 나눈 나머지를 출력한다.

예제 입력 1

3 1

예제 출력 1

2

예제 입력 2

6 2

예제 출력 2

719

예제 입력 3

21 3

예제 출력 3

660773

힌트

N=3 일때 가능한 6개의 순열과 각 경우의 정렬별 비교횟수는 아래와 같다.

1 2 3 - NI = 2, NQ = 3
1 3 2 - NI = 3, NQ = 3
2 1 3 - NI = 3, NQ = 2
2 3 1 - NI = 4, NQ = 2
3 1 2 - NI = 4, NQ = 3
3 2 1 - NI = 5, NQ = 3

이 문제의 데이터는 총 10개이고, 다음과 같다.

  • N = 5, X = 1
  • N = 12, X = 2
  • N = 18, X = 3
  • N = 20, X = 1
  • N = 22, X = 5
  • N = 23, X = 3
  • N = 24, X = 4
  • N = 29, X = 2
  • N = 30, X = 5
  • N = 31, X = 1
[{"problem_id":"3322","problem_lang":"0","title":"\uc815\ub82c\uc758 \uc2dc\uac04\ubcf5\uc7a1\ub3c4","description":"<p>\uafcd\uc740 \uc790\ub8cc\uad6c\uc870\ub97c \uacf5\ubd80\ud560\ub54c \uac01 \uc815\ub82c\ub9c8\ub2e4 \ube44\uad50\ud69f\uc218\ub97c \uad6c\ud574\ubcf4\ub77c\ub294 \ubb38\uc81c\ub97c \ubcfc\ub54c\ub9c8\ub2e4 \ud654\uac00\ub09c \ub098\uba38\uc9c0 \uc138\uc0c1\uc5d0\uc11c \uc81c\uc77c \uac70\uc9c0\uac19\uc740 \ubb38\uc81c\ub97c \ub9cc\ub4e4\uc5b4\ubcf4\uace0\uc790 \ub2e4\uc74c\uacfc \uac19\uc740 \ubb38\uc81c\ub97c \ub9cc\ub4e4\uc5c8\ub2e4.<\/p>\r\n\r\n<p>N\uacfc X\uac00 \uc8fc\uc5b4\uc9c8 \ub54c, \uc0bd\uc785\uc815\ub82c\uc774 \ud035\uc815\ub82c\ubcf4\ub2e4 \ube44\uad50\ub97c \ucd5c\ub300 X\ubc88 \ub354 \ud558\ub294&nbsp;{1,2,...,N} \uc73c\ub85c \uad6c\uc131\ub41c \uc21c\uc5f4\uc758 \uac1c\uc218\ub97c \uad6c\ud574\uc57c \ud55c\ub2e4. \ub2e8, \uc22b\uc790\uac00 \ub9e4\uc6b0 \ucee4\uc9c8\uc218 \uc788\uc73c\ubbc0\ub85c \ucd9c\ub825\uac12\uc740 1234567\ub85c \ub098\ub208 \ub098\uba38\uc9c0\ub97c \ucd9c\ub825\ud55c\ub2e4.<\/p>\r\n\r\n<p>\uc544\ub798 \ucf54\ub4dc\ub294 \uc0bd\uc785\uc815\ub82c \ucf54\ub4dc\uc774\uba70 \ube44\uad50\ud69f\uc218\ub3c4 \uacc4\uc0b0\ud574\uc900\ub2e4.<\/p>\r\n\r\n<pre>\r\nprocedure insertionSort(int N, array A[1..N]) defined as:\r\n    A[0] := -Infinity\r\n    for i := 2 to N do:\r\n        j := i\r\n        <strong>Increment(comparison_count)<\/strong>\r\n        while A[j - 1] &gt; A[j] do:\r\n        SWAP(A[j - 1], A[j])\r\n        j := j - 1\r\n        <strong>Increment(comparison_count)<\/strong>\r\n    end while\r\nend for<\/pre>\r\n\r\n<p>\uc544\ub798\ucf54\ub4dc\ub294 \ud035\uc815\ub82c \ucf54\ub4dc\ub2e4. \ub9cc\uc57d \uc815\ub82c\ud558\ub824\ub294&nbsp;\ubc30\uc5f4\uc758 \uae38\uc774\uac00 L\uc774\ub77c\uba74, partition \uc54c\uace0\ub9ac\uc998\uc740 L-1\ubc88\uc758 \ube44\uad50\ub97c \ud55c\ub2e4.<\/p>\r\n\r\n<pre>\r\nprocedure quickSort(list A) defined as:\r\n    list less, greater\r\n    if length(A) &lt;= 1 then\r\n        return A\r\n\r\n    pivot := A[1]\r\n    for i := 2 to length(A) do:\r\n        <strong>Increment(comparison_count)<\/strong>\r\n        if A[i] &lt; pivot then append A[i] to less\r\n                        else append A[i] to greater\r\n        end if\r\n    end for\r\n    return concatenate(quickSort(less), pivot, quickSort(greater))<\/pre>\r\n\r\n<p>\uc608\ub97c \ub4e4\uc5b4, \uc21c\uc5f4 (3,1,4,2)\ub97c \uc0dd\uac01\ud558\uc790. \uc0bd\uc785\uc815\ub82c\uc758 \ube44\uad50\ud69f\uc218\ub294 \ucd1d 6\ud68c\ub85c i=2\uc77c\ub54c 2\ubc88, i=3\uc77c\ub54c 1\ubc88, i=4\uc77c\ub54c 3\ubc88\uc758 \ube44\uad50\uac00 \uc774\ub8e8\uc5b4\uc9c4\ub2e4. \ubc18\uba74 \ud035\uc815\ub82c\uc758 \ube44\uad50\ud69f\uc218\ub294 \ucd1d 4\ud68c\ub85c 3\uc774 pivot\uc77c\ub54c 3\ud68c\uc758 \ube44\uad50\ub97c \ud55c \ud6c4 (1,2)\uc640 (4)\ub85c partition\uc774 \uc774\ub8e8\uc5b4\uc9c4\ub2e4. \uc774\ud6c4 (1,2)\uc5d0\uc11c 1\ubc88\uc758 \ube44\uad50\uac00 \uc774\ub8e8\uc5b4\uc9c0\ubbc0\ub85c \ucd1d 4\ud68c\uc758 \ube44\uad50\uac00 \uc774\ub8e8\uc5b4\uc9c4\ub2e4.<\/p>\r\n","input":"<p>\uc785\ub825\uc758 \uccab\ubc88\uc9f8 \uc904\uc5d0 \ub450 \uac1c\uc758 \uc815\uc218 N\uacfc X\uac00 \uc8fc\uc5b4\uc9c0\uba70 \uac01\uac01\uc758 \ubc94\uc704\ub294 1 &lt; N &lt; 32, 1 &le; X &le; N^2 \uc774\ub2e4.<\/p>\r\n","output":"<p>\uc0bd\uc785\uc815\ub82c\uc774 \ud035\uc815\ub82c\ubcf4\ub2e4 \ucd5c\ub300 X\ubc88 \ube44\uad50\ub97c \ub354 \ud558\uac8c\ub418\ub294 \uc21c\uc5f4\uc758 \uac1c\uc218\ub97c 1234567\ub85c \ub098\ub208 \ub098\uba38\uc9c0\ub97c \ucd9c\ub825\ud55c\ub2e4.<\/p>\r\n","hint":"<p>N=3 \uc77c\ub54c \uac00\ub2a5\ud55c 6\uac1c\uc758 \uc21c\uc5f4\uacfc \uac01 \uacbd\uc6b0\uc758 \uc815\ub82c\ubcc4 \ube44\uad50\ud69f\uc218\ub294 \uc544\ub798\uc640 \uac19\ub2e4.<\/p>\r\n\r\n<pre>\r\n1 2 3 - NI = 2, NQ = 3\r\n1 3 2 - NI = 3, NQ = 3\r\n2 1 3 - NI = 3, NQ = 2\r\n2 3 1 - NI = 4, NQ = 2\r\n3 1 2 - NI = 4, NQ = 3\r\n3 2 1 - NI = 5, NQ = 3<\/pre>\r\n\r\n<p>\uc774 \ubb38\uc81c\uc758 \ub370\uc774\ud130\ub294 \ucd1d 10\uac1c\uc774\uace0, \ub2e4\uc74c\uacfc \uac19\ub2e4.<\/p>\r\n\r\n<ul>\r\n\t<li>N = 5, X = 1<\/li>\r\n\t<li>N = 12, X = 2<\/li>\r\n\t<li>N = 18, X = 3<\/li>\r\n\t<li>N = 20, X = 1<\/li>\r\n\t<li>N = 22, X = 5<\/li>\r\n\t<li>N = 23, X = 3<\/li>\r\n\t<li>N = 24, X = 4<\/li>\r\n\t<li>N = 29, X = 2<\/li>\r\n\t<li>N = 30, X = 5<\/li>\r\n\t<li>N = 31, X = 1<\/li>\r\n<\/ul>\r\n","original":"0","problem_lang_code":"\ud55c\uad6d\uc5b4"},{"problem_id":"3322","problem_lang":"1","title":"sorting","description":"<p>For given N and X, determine the number of permutations of {1, 2, &hellip;, N} on which Insertion Sort makes at most X times the number of comparisons that Quick Sort makes. Since the number can be pretty big, we ask you to output it modulo 1234567.<\/p>\r\n\r\n<p>The following is our implementation of Insertion Sort, which also counts the number of comparisons it makes:&nbsp;<\/p>\r\n\r\n<pre>\r\nprocedure insertionSort(int N, array A[1..N]) defined as:\r\n    A[0] := -Infinity\r\n    for i := 2 to N do:\r\n        j := i\r\n        <strong>Increment(comparison_count)<\/strong>\r\n        while A[j - 1] &gt; A[j] do:\r\n        SWAP(A[j - 1], A[j])\r\n        j := j - 1\r\n        <strong>Increment(comparison_count)<\/strong>\r\n    end while\r\nend for<\/pre>\r\n\r\n<p>The following is our implementation of Quick Sort. If L is the length of the list we are sorting in a recursive step, the partition algorithm makes L-1 comparisons.<\/p>\r\n\r\n<pre>\r\nprocedure quickSort(list A) defined as:\r\n    list less, greater\r\n    if length(A) &lt;= 1 then\r\n        return A\r\n\r\n    pivot := A[1]\r\n    for i := 2 to length(A) do:\r\n        <strong>Increment(comparison_count)<\/strong>\r\n        if A[i] &lt; pivot then append A[i] to less\r\n                        else append A[i] to greater\r\n        end if\r\n    end for\r\n    return concatenate(quickSort(less), pivot, quickSort(greater))<\/pre>\r\n\r\n<p>For example, let us consider the permutation (3, 1, 4, 2).<\/p>\r\n\r\n<p>The number of comparisons for Insertion Sort is 6: 2 comparisons for i=2, 1 for i=3, and 3 for i=4.&nbsp;<\/p>\r\n\r\n<p>The number of comparisons for Quick Sort is 4. In the first iteration 3 is the pivot. It takes 3 comparisons to partition (1,4,2) into (1,2) and (4). To further sort (1,2) it takes 1 more comparison.<\/p>\r\n","input":"<p>The first line contains 2 integers, separated by a space: N and X.<\/p>\r\n\r\n<ul>\r\n\t<li>In all inputs files, 1 &lt; N &lt; 32.<\/li>\r\n\t<li>In all inputs files, 1 &le; X &le; N<sup>2<\/sup><\/li>\r\n<\/ul>\r\n","output":"<p>The number of permutations for which Insertion Sort is at most X times slower than Quick Sort. The number should be printed modulo 1234567.&nbsp;<\/p>\r\n","hint":"<p>For the 6 possible permutations we have NI and NQ, the number of comparisons for Insertion Sort and Quick Sort:&nbsp;<\/p>\r\n\r\n<pre>\r\n1 2 3 - NI = 2, NQ = 3\r\n1 3 2 - NI = 3, NQ = 3\r\n2 1 3 - NI = 3, NQ = 2\r\n2 3 1 - NI = 4, NQ = 2\r\n3 1 2 - NI = 4, NQ = 3\r\n3 2 1 - NI = 5, NQ = 3 <\/pre>\r\n","original":"1","problem_lang_code":"\uc601\uc5b4"}]