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문제

평소 그래프 이론에 관심이 많았던 주연이는 최근에 방향그래프에서 transitive closure를 계산하는 방법중 하나인 Warshall algorithm을 공부하였다. 주연이는 최근에 사귀게 된 남자친구 윤호가 얼마나 똑똑한지 시험하기 위해 transitive closure를 계산하는 방법을 윤호에게 물어보기로 결심했다. 질문의 내용은 아래와 같다.

"윤호오빠, ...

방향그래프 G = <V, E> (2<=|V|<=2,500. 1<=|E|<=10,000)가 주어졌을 때, G의 transitive closure는 0, 1로 구성된 V×V크기의 행렬 R로 표현할 수 있다. 만약 G의 두 정점 A와 B에 대해, A에서 B로 가는 경로가 존재하면 행렬R의 A행 B열의 값을 1로, 경로가 존재하지 않으면 0으로 채워넣는 것이다.

주어진 그래프 G에 대해 행렬 R을 구하여라. 단, R의 크기가 매우 커질 수 있으므로 그래프 G의 서로 다른 두 정점 X에서 Y에 대해, X에서 Y로 가는 경로가 존재하는 정점 쌍 (X, Y)의 개수를 출력하도록 한다. 즉, 그래프 G에 대한 행렬R에서 (행≠열)인 모든 셀에 대해 '1'로 채워진 셀의 개수를 출력한다.

오빠 할 수 있지 ?"

자신의 밑천이 주연이에게 드러날까 두려운 윤호는 여러분들에게 도움을 요청했다. 윤호와 주연이 사이를 갈라놓을지 말지는 여러분들의 손에 달렸다.

입력

입력의 첫 줄에는 테스트 케이스의 수 T (T<=10)가 주어진다.

각 테스트 케이스마다 첫 번째 줄에 정점과 간선의 수 N, M (2<=|V|<=2,500. 1<=|E|<=10,000)이 주어지고, 다음 M개의 줄에 각각 간선정보 A, B (1<=A, B<=N)가 주어진다. A에서 B로 가는 간선이 존재한다는 의미이다. 동일한 간선에 대한 입력이 중복으로 주어지지 않는다.

출력

각 테스트 케이스마다 한 줄에 X≠Y인 정점 X, Y에 대해 X에서 Y로 가는 경로가 존재하는 정점쌍 (X, Y)의 개수를 출력한다.

예제 입력 1

2
3 2
1 2
2 3
4 4
1 2
2 3
2 4
4 1

예제 출력 1

3
9

힌트

첫 번째 예제에 대해 주어진 그래프 G의 transitive closure를 계산한 행렬 R은 아래와 같다.

0 1 1
0 0 1
0 0 0

위 행렬에서 X≠Y인 정점쌍 (X, Y)는 3개 존재한다.

[{"problem_id":"7827","problem_lang":"0","title":"Transitive Closure","description":"<p>\ud3c9\uc18c \uadf8\ub798\ud504 \uc774\ub860\uc5d0 \uad00\uc2ec\uc774 \ub9ce\uc558\ub358 \uc8fc\uc5f0\uc774\ub294 \ucd5c\uadfc\uc5d0 \ubc29\ud5a5\uadf8\ub798\ud504\uc5d0\uc11c <a href=\"https:\/\/en.wikipedia.org\/wiki\/Transitive_closure\">transitive closure<\/a>\ub97c \uacc4\uc0b0\ud558\ub294 \ubc29\ubc95\uc911 \ud558\ub098\uc778 <a href=\"https:\/\/en.wikipedia.org\/wiki\/Floyd%E2%80%93Warshall_algorithm\">Warshall algorithm<\/a>\uc744 \uacf5\ubd80\ud558\uc600\ub2e4. \uc8fc\uc5f0\uc774\ub294 \ucd5c\uadfc\uc5d0 \uc0ac\uadc0\uac8c \ub41c \ub0a8\uc790\uce5c\uad6c \uc724\ud638\uac00 \uc5bc\ub9c8\ub098 \ub611\ub611\ud55c\uc9c0 \uc2dc\ud5d8\ud558\uae30 \uc704\ud574 transitive closure\ub97c \uacc4\uc0b0\ud558\ub294 \ubc29\ubc95\uc744 \uc724\ud638\uc5d0\uac8c \ubb3c\uc5b4\ubcf4\uae30\ub85c \uacb0\uc2ec\ud588\ub2e4. \uc9c8\ubb38\uc758 \ub0b4\uc6a9\uc740 \uc544\ub798\uc640 \uac19\ub2e4.<\/p>\r\n\r\n<p>&quot;\uc724\ud638\uc624\ube60, ...<\/p>\r\n\r\n<p>\ubc29\ud5a5\uadf8\ub798\ud504 G = &lt;V, E&gt; (2&lt;=|V|&lt;=2,500. 1&lt;=|E|&lt;=10,000)\uac00 \uc8fc\uc5b4\uc84c\uc744 \ub54c, G\uc758 transitive closure\ub294 0, 1\ub85c \uad6c\uc131\ub41c V&times;V\ud06c\uae30\uc758 \ud589\ub82c R\ub85c \ud45c\ud604\ud560 \uc218 \uc788\ub2e4. \ub9cc\uc57d G\uc758 \ub450 \uc815\uc810 A\uc640 B\uc5d0 \ub300\ud574, A\uc5d0\uc11c B\ub85c \uac00\ub294 \uacbd\ub85c\uac00 \uc874\uc7ac\ud558\uba74 \ud589\ub82cR\uc758 A\ud589 B\uc5f4\uc758 \uac12\uc744 1\ub85c, \uacbd\ub85c\uac00 \uc874\uc7ac\ud558\uc9c0 \uc54a\uc73c\uba74 0\uc73c\ub85c \ucc44\uc6cc\ub123\ub294 \uac83\uc774\ub2e4.<\/p>\r\n\r\n<p>\uc8fc\uc5b4\uc9c4 \uadf8\ub798\ud504 G\uc5d0 \ub300\ud574 \ud589\ub82c R\uc744 \uad6c\ud558\uc5ec\ub77c. \ub2e8, R\uc758 \ud06c\uae30\uac00 \ub9e4\uc6b0 \ucee4\uc9c8 \uc218 \uc788\uc73c\ubbc0\ub85c \uadf8\ub798\ud504 G\uc758 <strong>\uc11c\ub85c \ub2e4\ub978<\/strong> \ub450 \uc815\uc810 X\uc5d0\uc11c Y\uc5d0 \ub300\ud574, X\uc5d0\uc11c Y\ub85c \uac00\ub294 \uacbd\ub85c\uac00 \uc874\uc7ac\ud558\ub294 \uc815\uc810 \uc30d (X, Y)\uc758 \uac1c\uc218\ub97c \ucd9c\ub825\ud558\ub3c4\ub85d \ud55c\ub2e4. \uc989, \uadf8\ub798\ud504 G\uc5d0 \ub300\ud55c \ud589\ub82cR\uc5d0\uc11c (\ud589&ne;\uc5f4)\uc778 \ubaa8\ub4e0 \uc140\uc5d0 \ub300\ud574 &#39;1&#39;\ub85c \ucc44\uc6cc\uc9c4 \uc140\uc758 \uac1c\uc218\ub97c \ucd9c\ub825\ud55c\ub2e4.<\/p>\r\n\r\n<p>\uc624\ube60 \ud560 \uc218 \uc788\uc9c0 ?&quot;<\/p>\r\n\r\n<p>\uc790\uc2e0\uc758 \ubc11\ucc9c\uc774 \uc8fc\uc5f0\uc774\uc5d0\uac8c \ub4dc\ub7ec\ub0a0\uae4c \ub450\ub824\uc6b4 \uc724\ud638\ub294 \uc5ec\ub7ec\ubd84\ub4e4\uc5d0\uac8c \ub3c4\uc6c0\uc744 \uc694\uccad\ud588\ub2e4. \uc724\ud638\uc640 \uc8fc\uc5f0\uc774 \uc0ac\uc774\ub97c \uac08\ub77c\ub193\uc744\uc9c0 \ub9d0\uc9c0\ub294 \uc5ec\ub7ec\ubd84\ub4e4\uc758 \uc190\uc5d0 \ub2ec\ub838\ub2e4.<\/p>\r\n\r\n<p><img src=\"\/userupload\/Hibbah\/201504\/d9b29c14ef1d7d5d60b352a4b115913e.jpg\" style=\"height:217.469879518072px; width:200px\" \/><\/p>\r\n","input":"<p>\uc785\ub825\uc758 \uccab \uc904\uc5d0\ub294 \ud14c\uc2a4\ud2b8 \ucf00\uc774\uc2a4\uc758 \uc218 T (T&lt;=10)\uac00 \uc8fc\uc5b4\uc9c4\ub2e4.<\/p>\r\n\r\n<p>\uac01 \ud14c\uc2a4\ud2b8 \ucf00\uc774\uc2a4\ub9c8\ub2e4 \uccab \ubc88\uc9f8 \uc904\uc5d0 \uc815\uc810\uacfc \uac04\uc120\uc758 \uc218 N, M (2&lt;=|V|&lt;=2,500. 1&lt;=|E|&lt;=10,000)\uc774 \uc8fc\uc5b4\uc9c0\uace0, \ub2e4\uc74c M\uac1c\uc758 \uc904\uc5d0 \uac01\uac01 \uac04\uc120\uc815\ubcf4 A, B (1&lt;=A, B&lt;=N)\uac00 \uc8fc\uc5b4\uc9c4\ub2e4. A\uc5d0\uc11c B\ub85c \uac00\ub294 \uac04\uc120\uc774 \uc874\uc7ac\ud55c\ub2e4\ub294 \uc758\ubbf8\uc774\ub2e4. \ub3d9\uc77c\ud55c \uac04\uc120\uc5d0 \ub300\ud55c \uc785\ub825\uc774 \uc911\ubcf5\uc73c\ub85c \uc8fc\uc5b4\uc9c0\uc9c0 \uc54a\ub294\ub2e4.<\/p>\r\n","output":"<p>\uac01 \ud14c\uc2a4\ud2b8 \ucf00\uc774\uc2a4\ub9c8\ub2e4 \ud55c \uc904\uc5d0 X&ne;Y\uc778 \uc815\uc810 X, Y\uc5d0 \ub300\ud574 X\uc5d0\uc11c Y\ub85c \uac00\ub294 \uacbd\ub85c\uac00 \uc874\uc7ac\ud558\ub294 \uc815\uc810\uc30d (X, Y)\uc758 \uac1c\uc218\ub97c \ucd9c\ub825\ud55c\ub2e4.<\/p>\r\n","hint":"<p>\uccab \ubc88\uc9f8 \uc608\uc81c\uc5d0 \ub300\ud574 \uc8fc\uc5b4\uc9c4 \uadf8\ub798\ud504 G\uc758 transitive closure\ub97c \uacc4\uc0b0\ud55c \ud589\ub82c R\uc740 \uc544\ub798\uc640 \uac19\ub2e4.<\/p>\r\n\r\n<pre>\r\n0 1 1\r\n0 0 1\r\n0 0 0\r\n<\/pre>\r\n\r\n<p>\uc704 \ud589\ub82c\uc5d0\uc11c X&ne;Y\uc778 \uc815\uc810\uc30d (X, Y)\ub294 3\uac1c \uc874\uc7ac\ud55c\ub2e4.<\/p>\r\n","original":"0","problem_lang_code":"\ud55c\uad6d\uc5b4"},{"problem_id":"7827","problem_lang":"1","title":"Transitive Closure","description":"<p>In almost all ICPC trainings, one of the basic items to be taught (or even better, assumed) is how to compute the transitive closure of a directed graph. Since this concept is so elementary, there is a professor that suggests to solve this problem using Warshall&#39;s algorithm. As one of the bonus tasks of this ICPC, we would like to examine whether the participants have indeed mastered this technique.<\/p>\r\n\r\n<p>Given a directed graph G = &lt;V, E&gt; with N = |V| vertices (2 &le; N &le; 2,500) and M = |E| edges (1 &le; M &le; 10,000), its transitive closure is the binary relation R on V such that two vertices, A and B, are related in R if there is a directed path from A to B, where a directed path is a sequence of edges of the graph C<sub>0<\/sub>C<sub>1<\/sub>, C<sub>1<\/sub>C<sub>2<\/sub>, ..., C<sub>K-1<\/sub>C<sub>K<\/sub> such that A = C<sub>0<\/sub> and B = C<sub>K<\/sub>. Your task is to compute R.<\/p>\r\n\r\n<p>Because R might be very large, we\u201fve decided to simplify the output. Calculate the number of ordered pairs of (X,Y) where X is not equal to Y such that there is a path from vertex X to vertex Y in graph G.<\/p>\r\n","input":"<p>The first line of input contains an integer T (T &le; 10), denoting the number of testcases. The following lines describe each test case of the following format.<\/p>\r\n\r\n<p>The first line of a testcase consists of two integers N and M. Each of the following M lines contains 2 integers, A and B, denoting that there is a directed edge from A to B. The vertices are numbered from 1 to N, and there will be no repeated edges within a description of a graph.<\/p>\r\n","output":"<p>For each case, output an integer denoting the number of ordered pairs of (X,Y) where X is not equal to Y such that there is a path from vertex X to vertex Y.<\/p>\r\n","hint":"<p>Explanation for the 1st sample input.<\/p>\r\n\r\n<p>The binary relation R is:<\/p>\r\n\r\n<pre>\r\n0 1 1\r\n0 0 1\r\n0 0 0<\/pre>\r\n\r\n<p>There are 3 ordered pairs of (X,Y) such that R<sub>X,Y<\/sub> is true.<\/p>\r\n","original":"1","problem_lang_code":"\uc601\uc5b4"}]