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문제

파급효과가 가능한 퍼즐인지 판별하여라.

파급효과는 여러개의 폴리오미노로 구성된 직사각형의 그리드에서 발생한다. 폴리오미노는 하나 이상의 같은 크기의 정사각형을 이어붙여 만든 도형이다.

우리는 그리드 안의 모든 칸에 자연수를 배치해야한다. 이 때 다음과 같은 규칙을 따르며, 일부 자연수는 미리 주어질 수 있다.

  • 1 ~ (폴리노미오 안의 칸 수) 의 자연수들은 해당 폴리노미오 안에 반드시 한번씩만 배치되어야 한다.
  • 하나의 줄이나 행에 동일한 숫자가 배치되기 위해선 두 숫자 사이에 해당 숫자만큼의 다른 칸이 존재해야 한다.
    예를 들어, '1'이 배치된 칸은 서로 인접할 수 없으며, 다른 숫자가 쓰여진 칸이 두 '1' 사이에 하나 이상 존재해야 한다.
    같은 줄이나 행에 '3'이 배치되기 위해선 다른 숫자가 쓰여진 칸이 사이에 세개 이상 존재해야 한다.

이 문제에서 폴리노미오의 크기는 최대 8이다.

 

입력

입력의 첫째 줄에는 테스트 케이스 T가 주어진다. (T<=100)

각각의 테스트 케이스의 첫째 줄에는 두 정수 R, C 가 주어진다. (4<=R, C<=15)

이후 R개 줄에는 그리드에 쓰여진 숫자들을 나태는 C개의 숫자 d_i로 구성된 문자열이 입력된다. (1<=d_i<=8)

이후 R개 줄에는 퍼즐의 각 칸이 폴리노미오로 연결된 방향을 나타내는 C개의 숫자가 입력된다. (이는 RxC 의 테이블이며, 이를 "descr" 라고 하자. 0<=descr(r,c)<=15)

descr(r,c)의 값은 이웃된 칸과의 연결 여부에 따라 다음과 같이 결정된다.

descr(r,c)=0
if(connected((r,c),(r-1,c)) descr(r,c)+=1; (UP)
if(connected((r,c),(r,c+1)) descr(r,c)+=2; (RIGHT)
if(connected((r,c),(r+1,c)) descr(r,c)+=4; (DOWN)
if(connected((r,c),(r,c-1)) descr(r,c)+=8; (LEFT)

예를 들어, 크기가 1인 폴리노미오의 칸은 주변의 어떠한 칸과도 이어져있지 않으므로 0의 값을 가지며, 위쪽의 칸과 아래의 칸에 이어져있는 칸들은 5의 값을 가지게 된다.

출력

각각의 테스트 케이스에 대해 한 "valid" 또는 "invalid" 를 출력하여라.

예제 입력 1

2
6 6
241321
312432
131243
423121
214312
141231
2 12 4 4 6 8
4 5 1 5 5 4
5 1 0 5 1 5
3 8 4 1 4 1
2 10 9 2 9 4
0 2 10 10 8 1
6 6
421321
312432
131243
423121
214312
141231
2 12 4 4 6 8
4 5 1 5 5 4
5 1 0 5 1 5
3 8 4 1 4 1
2 10 9 2 9 4
0 2 10 10 8 1

예제 출력 1

valid
invalid
[{"problem_id":"1896","problem_lang":"0","title":"\ud30c\uae09\ud6a8\uacfc","description":"<p><img src=\"https:\/\/onlinejudgeimages.s3-ap-northeast-1.amazonaws.com\/userupload\/jb_cubedr\/20160121\/25de801384dd4133869fac0efbc6636b.png\" style=\"float:right; height:346px; line-height:20.8px; opacity:0.9; width:347px\" \/>\ud30c\uae09\ud6a8\uacfc\uac00 \uac00\ub2a5\ud55c \ud37c\uc990\uc778\uc9c0 \ud310\ubcc4\ud558\uc5ec\ub77c.<\/p>\r\n\r\n<p>\ud30c\uae09\ud6a8\uacfc\ub294 \uc5ec\ub7ec\uac1c\uc758 \ud3f4\ub9ac\uc624\ubbf8\ub178\ub85c \uad6c\uc131\ub41c&nbsp;\uc9c1\uc0ac\uac01\ud615\uc758 \uadf8\ub9ac\ub4dc\uc5d0\uc11c \ubc1c\uc0dd\ud55c\ub2e4. \ud3f4\ub9ac\uc624\ubbf8\ub178\ub294 \ud558\ub098 \uc774\uc0c1\uc758 \uac19\uc740 \ud06c\uae30\uc758 \uc815\uc0ac\uac01\ud615\uc744 \uc774\uc5b4\ubd99\uc5ec \ub9cc\ub4e0 \ub3c4\ud615\uc774\ub2e4.<\/p>\r\n\r\n<p>\uc6b0\ub9ac\ub294 \uadf8\ub9ac\ub4dc \uc548\uc758 \ubaa8\ub4e0 \uce78\uc5d0 \uc790\uc5f0\uc218\ub97c \ubc30\uce58\ud574\uc57c\ud55c\ub2e4. \uc774 \ub54c \ub2e4\uc74c\uacfc \uac19\uc740 \uaddc\uce59\uc744 \ub530\ub974\uba70, \uc77c\ubd80 \uc790\uc5f0\uc218\ub294 \ubbf8\ub9ac \uc8fc\uc5b4\uc9c8 \uc218 \uc788\ub2e4.<\/p>\r\n\r\n<ul>\r\n\t<li>1 ~ (\ud3f4\ub9ac\ub178\ubbf8\uc624 \uc548\uc758 \uce78 \uc218) \uc758 \uc790\uc5f0\uc218\ub4e4\uc740 \ud574\ub2f9 \ud3f4\ub9ac\ub178\ubbf8\uc624 \uc548\uc5d0 \ubc18\ub4dc\uc2dc \ud55c\ubc88\uc529\ub9cc \ubc30\uce58\ub418\uc5b4\uc57c \ud55c\ub2e4.<\/li>\r\n\t<li>\ud558\ub098\uc758 \uc904\uc774\ub098 \ud589\uc5d0 \ub3d9\uc77c\ud55c \uc22b\uc790\uac00 \ubc30\uce58\ub418\uae30 \uc704\ud574\uc120 \ub450 \uc22b\uc790 \uc0ac\uc774\uc5d0 \ud574\ub2f9 \uc22b\uc790\ub9cc\ud07c\uc758 \ub2e4\ub978 \uce78\uc774 \uc874\uc7ac\ud574\uc57c \ud55c\ub2e4.<br \/>\r\n\t\uc608\ub97c \ub4e4\uc5b4, &#39;1&#39;\uc774 \ubc30\uce58\ub41c \uce78\uc740 \uc11c\ub85c \uc778\uc811\ud560 \uc218 \uc5c6\uc73c\uba70, \ub2e4\ub978 \uc22b\uc790\uac00 \uc4f0\uc5ec\uc9c4 \uce78\uc774 \ub450 &#39;1&#39; \uc0ac\uc774\uc5d0 \ud558\ub098 \uc774\uc0c1 \uc874\uc7ac\ud574\uc57c \ud55c\ub2e4.<br \/>\r\n\t\uac19\uc740 \uc904\uc774\ub098 \ud589\uc5d0 &#39;3&#39;\uc774 \ubc30\uce58\ub418\uae30 \uc704\ud574\uc120 \ub2e4\ub978 \uc22b\uc790\uac00 \uc4f0\uc5ec\uc9c4 \uce78\uc774 \uc0ac\uc774\uc5d0 \uc138\uac1c \uc774\uc0c1 \uc874\uc7ac\ud574\uc57c \ud55c\ub2e4.<\/li>\r\n<\/ul>\r\n\r\n<p>\uc774 \ubb38\uc81c\uc5d0\uc11c \ud3f4\ub9ac\ub178\ubbf8\uc624\uc758 \ud06c\uae30\ub294 \ucd5c\ub300 8\uc774\ub2e4.<\/p>\r\n\r\n<p>&nbsp;<\/p>\r\n","input":"<p>\uc785\ub825\uc758 \uccab\uc9f8 \uc904\uc5d0\ub294 \ud14c\uc2a4\ud2b8 \ucf00\uc774\uc2a4 T\uac00 \uc8fc\uc5b4\uc9c4\ub2e4. (T&lt;=100)<\/p>\r\n\r\n<p>\uac01\uac01\uc758 \ud14c\uc2a4\ud2b8 \ucf00\uc774\uc2a4\uc758 \uccab\uc9f8 \uc904\uc5d0\ub294 \ub450 \uc815\uc218 R, C \uac00 \uc8fc\uc5b4\uc9c4\ub2e4. (4&lt;=R, C&lt;=15)<\/p>\r\n\r\n<p>\uc774\ud6c4 R\uac1c \uc904\uc5d0\ub294 \uadf8\ub9ac\ub4dc\uc5d0 \uc4f0\uc5ec\uc9c4 \uc22b\uc790\ub4e4\uc744 \ub098\ud0dc\ub294&nbsp;C\uac1c\uc758 \uc22b\uc790 d_i\ub85c \uad6c\uc131\ub41c \ubb38\uc790\uc5f4\uc774 \uc785\ub825\ub41c\ub2e4. (1&lt;=d_i&lt;=8)<\/p>\r\n\r\n<p>\uc774\ud6c4 R\uac1c \uc904\uc5d0\ub294 \ud37c\uc990\uc758 \uac01 \uce78\uc774 \ud3f4\ub9ac\ub178\ubbf8\uc624\ub85c \uc5f0\uacb0\ub41c \ubc29\ud5a5\uc744 \ub098\ud0c0\ub0b4\ub294 C\uac1c\uc758 \uc22b\uc790\uac00 \uc785\ub825\ub41c\ub2e4.&nbsp;(\uc774\ub294 RxC \uc758 \ud14c\uc774\ube14\uc774\uba70, \uc774\ub97c &quot;descr&quot; \ub77c\uace0 \ud558\uc790. 0&lt;=descr(r,c)&lt;=15)<\/p>\r\n\r\n<p>descr(r,c)\uc758 \uac12\uc740 \uc774\uc6c3\ub41c \uce78\uacfc\uc758 \uc5f0\uacb0 \uc5ec\ubd80\uc5d0 \ub530\ub77c \ub2e4\uc74c\uacfc \uac19\uc774 \uacb0\uc815\ub41c\ub2e4.<\/p>\r\n\r\n<pre>\r\ndescr(r,c)=0\r\nif(connected((r,c),(r-1,c)) descr(r,c)+=1; (UP)\r\nif(connected((r,c),(r,c+1)) descr(r,c)+=2; (RIGHT)\r\nif(connected((r,c),(r+1,c)) descr(r,c)+=4; (DOWN)\r\nif(connected((r,c),(r,c-1)) descr(r,c)+=8; (LEFT)<\/pre>\r\n\r\n<p>\uc608\ub97c \ub4e4\uc5b4, \ud06c\uae30\uac00 1\uc778 \ud3f4\ub9ac\ub178\ubbf8\uc624\uc758 \uce78\uc740 \uc8fc\ubcc0\uc758 \uc5b4\ub5a0\ud55c \uce78\uacfc\ub3c4 \uc774\uc5b4\uc838\uc788\uc9c0 \uc54a\uc73c\ubbc0\ub85c 0\uc758 \uac12\uc744 \uac00\uc9c0\uba70, \uc704\ucabd\uc758 \uce78\uacfc \uc544\ub798\uc758 \uce78\uc5d0 \uc774\uc5b4\uc838\uc788\ub294 \uce78\ub4e4\uc740 5\uc758 \uac12\uc744 \uac00\uc9c0\uac8c \ub41c\ub2e4.<\/p>\r\n","output":"<p>\uac01\uac01\uc758 \ud14c\uc2a4\ud2b8 \ucf00\uc774\uc2a4\uc5d0 \ub300\ud574 \ud55c &quot;valid&quot; \ub610\ub294 &quot;invalid&quot; \ub97c \ucd9c\ub825\ud558\uc5ec\ub77c.<\/p>\r\n","hint":"","original":"0","problem_lang_code":"\ud55c\uad6d\uc5b4"},{"problem_id":"1896","problem_lang":"1","title":"Ripple Effect","description":"<p><img alt=\"\" src=\"https:\/\/onlinejudgeimages.s3-ap-northeast-1.amazonaws.com\/problem\/1896\/1.png\" style=\"float:right; height:346px; width:347px\" \/>Check if a solution to a Ripple Effect puzzle is valid.<\/p>\r\n\r\n<p>Ripple Effect is played on a rectangular grid divided into polyominoes. A polyomino is a plane geometric figure formed by joining one or more equal squares edge to edge.<\/p>\r\n\r\n<p>The solver must place one positive integer into each cell of the grid - some of which may be given in advance - according to these rules:<\/p>\r\n\r\n<ul>\r\n\t<li>Every integer from 1 to the quantity of cells in a polyomino must appear exactly once in that polyomino.<\/li>\r\n\t<li>If two identical numbers appear in the same row or column, at least that many cells with other numbers must separate them.<br \/>\r\n\tFor example, two cells both containing &#39;1&#39; may not be orthogonally adjacent, but must have at least one cell between them with a different number.<br \/>\r\n\tTwo cells marked &#39;3&#39; in the same row or column must have at least three cells with other numbers between them in that row or column, and so on.<\/li>\r\n<\/ul>\r\n\r\n<p>In this problem polyominoes will be of size at most 8.<\/p>\r\n","input":"<p>The input starts with a line containing an integer T (T &lt;= 100), the number of test cases.<\/p>\r\n\r\n<p>Each test case starts with two integers on a line, R and C (4&lt;=R,C&lt;=15).<\/p>\r\n\r\n<p>R lines follow, each containing a string of C digits d_i(1&lt;=d_i&lt;=8), the description of the solution.<\/p>\r\n\r\n<p>Next R lines will contain C integers (RxC table, we will call it &quot;descr&quot;), each describing the corresponding cell in the puzzle (0&lt;=descr(r,c)&lt;=15).<\/p>\r\n\r\n<p>The value of descr(r,c) is determined by the values of connections with neighbouring cells, in the following manner:<\/p>\r\n\r\n<pre>\r\ndescr(r,c)=0\r\nif(connected((r,c),(r-1,c)) descr(r,c)+=1; (UP)\r\nif(connected((r,c),(r,c+1)) descr(r,c)+=2; (RIGHT)\r\nif(connected((r,c),(r+1,c)) descr(r,c)+=4; (DOWN)\r\nif(connected((r,c),(r,c-1)) descr(r,c)+=8; (LEFT)\r\n<\/pre>\r\n\r\n<p>For example, a polyomino of size 1 (single square not connected to others) will have the value of 0 and a square that is connected only to the squares above and below will have the value of 5.<\/p>\r\n\r\n<p>See sample input for clarification.<\/p>\r\n","output":"<p>For each test case print either &quot;valid&quot; or &quot;invalid&quot; on a single line.<\/p>\r\n","hint":"","original":"1","problem_lang_code":"\uc601\uc5b4"}]