시간 제한 메모리 제한 제출 정답 맞은 사람 정답 비율
2 초 128 MB 106 14 10 12.346%

문제

흰색과 검은색으로 이루어진 이미지는 쿼드트리를 이용하여 표현할 수 있다. 쿼드트리의 루트는 이미지의 전체를 나타낸다. 만약 전체 이미지가 한 색깔로 이루어져 있으면, 그 색깔에 해당하는 정점을 루트로 한다. 그 외의 경우에는 그림을 넷으로 나누고, 각각의 나눠진 그림을 루트의 네 자식 정점으로 표현하면 된다. 이 때 첫 번째 자식이 왼쪽 위의 그림을, 두 번째 자식이 오른쪽 위의 그림을, 세 번째 자식이 왼쪽 아래의 그림을, 네 번째 자식이 오른쪽 아래의 그림을 표현하게 된다. 각각의 작은 이미지를 표현할 때도 같은 방법을 재귀적으로 적용한다.

위의 예는 4×4 크기의 이미지와 그에 해당하는 쿼드트리이다. 위의 예에서는 이미지의 크기(사각형의 변의 길이)가 2의 k승이라서 상관이 없었지만, 그 외의 경우에는 문제가 된다. 만약에 이미지의 크기가 2의 k승이 아닐 경우에는, 이미지의 크기를 오른쪽과 아래쪽으로 가장 가까운 2의 k승으로 늘린 뒤, 흰색으로 나머지 공간을 채우면 된다. 이 때 이미지가 정사각형 모양이 되도록 한다. 예를 들어 5×13의 이미지는 16×16 크기의 이미지가 되고, 원래의 이미지가 왼쪽 위에 들어가고 나머지 공간은 흰색으로 채우면 된다.

쿼드트리 자체가 이미지를 압축적으로 표현하는 방법이지만, 우리는 압축된 쿼드트리를 이용하여 좀 더 효율적으로 표현하려 한다. 만약 쿼드트리에서 같은 모양의 부분 트리가 여러 개 나올 경우, 이를 한 번만 저장한 뒤 나머지는 포인터로 연결할 수 있다. 위의 예는 다음과 같은 압축된 쿼드트리로 표현할 수 있다.

원래의 쿼드트리는 17개의 정점을 갖지만, 압축된 쿼드트리는 12개의 정점을 갖는다. 12개의 정점을 갖는 압축된 쿼드트리가 몇 개 더 있기는 하지만, 11개 이하의 정점을 갖는 압축된 쿼드트리는 존재하지 않는다. 따라서 위의 경우가 가장 효율적(정점 개수가 최소)이다.

주의할 것은, B나 W로 표현되는 정점은 포인터가 아니가 값 자체이므로 이를 연결할 경우는 압축하는 것이 아니다. 즉, 정점이 하나인 경우(높이가 1인 경우)에는 포인터를 이용하여 연결할 필요가 없다. 또, 실제 이미지의 크기가 다르더라도 트리의 모양이 같다면, 같은 쿼드트리로 표현될 수 있는데, 이런 경우에도 포인터로 연결할 수 있다.

이미지가 주어졌을 때, 쿼드트리의 정점의 개수와 가장 효율적인 압축된 쿼드트리의 정점의 개수를 구하는 프로그램을 작성하시오.

입력

첫째 줄에 이미지의 크기를 나타내는 두 정수 n(1≤n≤128), m(1≤m≤128)이 주어진다. 이는 이미지의 크기가 n×m이라는 의미이다. 다음 n개의 줄에는 m개의 숫자로 이미지가 주어진다. 흰색은 0으로, 검은색은 1로 표현된다.

출력

첫째 줄에 쿼드트리의 정점의 개수와 가장 효율적인 압축된 쿼드트리의 정점의 개수를 출력한다.

예제 입력 1

4 4
1011
0111
1010
0111

예제 출력 1

17 12
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\uc5f0\uacb0\ud560 \ud544\uc694\uac00 \uc5c6\ub2e4. \ub610, \uc2e4\uc81c \uc774\ubbf8\uc9c0\uc758 \ud06c\uae30\uac00 \ub2e4\ub974\ub354\ub77c\ub3c4 \ud2b8\ub9ac\uc758 \ubaa8\uc591\uc774 \uac19\ub2e4\uba74, \uac19\uc740 \ucffc\ub4dc\ud2b8\ub9ac\ub85c \ud45c\ud604\ub420 \uc218 \uc788\ub294\ub370, \uc774\ub7f0 \uacbd\uc6b0\uc5d0\ub3c4 \ud3ec\uc778\ud130\ub85c \uc5f0\uacb0\ud560 \uc218 \uc788\ub2e4.<\/p>\r\n<p>\uc774\ubbf8\uc9c0\uac00 \uc8fc\uc5b4\uc84c\uc744 \ub54c, \ucffc\ub4dc\ud2b8\ub9ac\uc758 \uc815\uc810\uc758 \uac1c\uc218\uc640 \uac00\uc7a5 \ud6a8\uc728\uc801\uc778 \uc555\ucd95\ub41c \ucffc\ub4dc\ud2b8\ub9ac\uc758 \uc815\uc810\uc758 \uac1c\uc218\ub97c \uad6c\ud558\ub294 \ud504\ub85c\uadf8\ub7a8\uc744 \uc791\uc131\ud558\uc2dc\uc624.<\/p>","input":"<p>\uccab\uc9f8 \uc904\uc5d0 \uc774\ubbf8\uc9c0\uc758 \ud06c\uae30\ub97c \ub098\ud0c0\ub0b4\ub294 \ub450 \uc815\uc218 n(1&le;n&le;128), m(1&le;m&le;128)\uc774 \uc8fc\uc5b4\uc9c4\ub2e4. \uc774\ub294 \uc774\ubbf8\uc9c0\uc758 \ud06c\uae30\uac00 n&times;m\uc774\ub77c\ub294 \uc758\ubbf8\uc774\ub2e4. \ub2e4\uc74c n\uac1c\uc758 \uc904\uc5d0\ub294 m\uac1c\uc758 \uc22b\uc790\ub85c \uc774\ubbf8\uc9c0\uac00 \uc8fc\uc5b4\uc9c4\ub2e4. \ud770\uc0c9\uc740 0\uc73c\ub85c, \uac80\uc740\uc0c9\uc740 1\ub85c \ud45c\ud604\ub41c\ub2e4.<\/p>","output":"<p>\uccab\uc9f8 \uc904\uc5d0 \ucffc\ub4dc\ud2b8\ub9ac\uc758 \uc815\uc810\uc758 \uac1c\uc218\uc640 \uac00\uc7a5 \ud6a8\uc728\uc801\uc778 \uc555\ucd95\ub41c \ucffc\ub4dc\ud2b8\ub9ac\uc758 \uc815\uc810\uc758 \uac1c\uc218\ub97c \ucd9c\ub825\ud55c\ub2e4.<\/p>","hint":"","original":"0","problem_lang_code":"\ud55c\uad6d\uc5b4"},{"problem_id":"1998","problem_lang":"1","title":"Squadtrees","description":"<p>Quadtrees are data structures used to store digital images. For our purposes, the images will be simple bitmaps, where every pixel is either a 1 (black) or a 0 (white). A quadtree representation of a bitmap is obtained as follows: first, associate the root node with the entire image. If the entire image is either all 1&#39;s or all 0&#39;s, then store that value in the node and you&#39;re done. Otherwise divide the region into four equal size quadrants, add four children to the root, and assign each child one of the four regions in the following order: the first child gets the upper left quadrant, the second the upper right, the third the lower left and the fourth the lower right. Then recursively apply the above rules to each of the children. For example, the 4x4 image on the left would be represented by the quadtree on the right:<\/p>\r\n\r\n<p><img alt=\"\" src=\"\/upload\/images\/qdt1.png\" style=\"height:115px; width:375px\" \/><\/p>\r\n\r\n<p>Note that this procedure only works as stated if the image is a square and has a side length equal to a power of 2. For those images which do not meet those requirements, we pad the rows and columns with 0&#39;s (on the right and on the bottom, respectively) until we have a bitmap of the appropriate size. For example, a 5x13 image would be converted to a 16x16 bitmap (with the original image residing in the upper left portion, and the remainder of the image filled with 0&#39;s).&nbsp;<\/p>\r\n\r\n<p>While quadtrees can result in a significant savings in space over the original image, even more savings can be achieved if we identify repeated subtrees. For example, in the tree above, the first and third subtrees of the root are identical, so we could replace the root of the third subtree with a reference to the first subtree, obtaining something that symbolically looks like the following:<\/p>\r\n\r\n<p><img alt=\"\" src=\"\/upload\/images\/qdt2.png\" style=\"height:104px; width:282px\" \/><\/p>\r\n\r\n<p>We will call these compressed quadtrees super quadtrees, or squadtrees. For our purposes the use of a reference saves space only when the tree it replaces has height of at least 1. Thus, while we could replace 5 of the nodes which contain a B with references to the first node with a B, this would not in practice save any space in the compression. Using this rule, our squadtree contains only 12 nodes, as opposed to 17 in the original quadtree. Your job for this problem is to take a set of images and determine the number of nodes in both the resulting quadtrees and squadtrees.<\/p>\r\n","input":"<p>Input will consist of multiple problem instances. Each instance will start with a single line containing two integers n and m, indicating the number of rows and columns in the image. The maximum values for these integers is 128. The next n lines will each contain m characters representing the image to process. A black pixel will be represented by a `1&#39;, and a white pixel will be represented by a `0&#39;. The input line 0 0 will terminate input and should not be processed.<\/p>\r\n","output":"<p>For each problem instance, output two integers separated by a single space. The first value is the number of nodes in the quadtree for the problem instance, and the second is the number of nodes in the squadtree.<\/p>\r\n","hint":"","original":"1","problem_lang_code":"\uc601\uc5b4"}]