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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.

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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","html_title":"0","problem_lang_tcode":"Korean"},{"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","html_title":"0","problem_lang_tcode":"English"}]

출처

ICPC > Regionals > Asia Pacific > Indonesia > Jakarta > ACM-ICPC Regional Jakarta 2010 F번

  • 문제를 번역한 사람: Hibbah