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문제

방향 그래프 G = (V, E)가 주어져 있다.

임의의 노드 u, v ∈ V에 대해서 u에서 v로 E에 포함된 간선만을 이용해 갈 수 있는 경로가 있으면 uv로 표현한다.

만약 어떤 노드 vV가 자신으로부터 도달할 수 있는 모든 노드로부터 돌아오는 경로가 있다면, 즉 다음 조건을 만족한다면 노드 vV를 싱크라고 부른다.

  • 조건: ∀wV, (vw) ⇒ (wv)

또한, 그래프 G의 싱크를 모아놓은 집합을 bottom(G)로 표현한다.

  • bottom(G) = {v ∈ V: ∀wV, (vw) ⇒ (wv)}

주어진 그래프 G=(V, E)의 bottom(G)를 구하시오.

입력

입력은 여러 개의 테스트 케이스로 구분되어 있다.

각 테스트 케이스의 첫 줄에는 노드의 수 n (1 ≤ n ≤ 5,000)과 음이 아닌 정수 m (1 ≤ m ≤ 100,000)이 주어진다. V = {1, 2, ..., n}이고, 간선의 수 |E|=m임을 의미한다.

그 다음부터는 각 간선을 나타내는 m쌍의 수 v1 w1 v2 w2 ... vm wm이 공백으로 구분되어 주어진다. 이는 (vi, wi) ∈ E를 의미한다.

마지막 줄에 0이 주어지고, 이 경우는 처리하지 않고 프로그램을 종료시켜야 한다.

출력

각 테스트 케이스마다 한 줄에 걸쳐 bottom(G)의 모든 노드를 출력한다. 노드는 공백으로 구분해야 하며, 오름차순이어야 한다. 만약, bottom(G)가 공집합이면 빈 줄을 출력한다.

예제 입력 1

3 3
1 3 2 3 3 1
2 1
1 2
0

예제 출력 1

1 3
2

예제 입력 2

2 1
2 1
2 0

5 5
1 2 2 3 3 1 5 4 4 3
5 5
1 2 2 3 3 1 3 4 4 5
5 1
5 1
5 6
1 2 2 3 3 1 3 4 4 5 5 3
0

예제 출력 2

1
1 2
1 2 3
5
1 2 3 4
1 2 3 4 5

힌트

[{"problem_id":"6543","problem_lang":"0","title":"\uadf8\ub798\ud504\uc758 \uc2f1\ud06c","description":"<p>\ubc29\ud5a5 \uadf8\ub798\ud504&nbsp;<em>G<\/em> = (<em>V<\/em>, <em>E<\/em>)\uac00 \uc8fc\uc5b4\uc838 \uc788\ub2e4.<\/p>\r\n\r\n<p>\uc784\uc758\uc758 \ub178\ub4dc <em>u<\/em>, <em>v<\/em>&nbsp;&isin; <em>V<\/em>\uc5d0 \ub300\ud574\uc11c <em>u<\/em>\uc5d0\uc11c <em>v<\/em>\ub85c E\uc5d0 \ud3ec\ud568\ub41c \uac04\uc120\ub9cc\uc744 \uc774\uc6a9\ud574 \uac08 \uc218 \uc788\ub294 \uacbd\ub85c\uac00 \uc788\uc73c\uba74 <em>u<\/em>&rarr;<em>v<\/em>\ub85c \ud45c\ud604\ud55c\ub2e4.<\/p>\r\n\r\n<p>\ub9cc\uc57d \uc5b4\ub5a4 \ub178\ub4dc <em>v<\/em> &isin; <em>V<\/em>\uac00 \uc790\uc2e0\uc73c\ub85c\ubd80\ud130 \ub3c4\ub2ec\ud560 \uc218 \uc788\ub294 \ubaa8\ub4e0 \ub178\ub4dc\ub85c\ubd80\ud130 \ub3cc\uc544\uc624\ub294 \uacbd\ub85c\uac00 \uc788\ub2e4\uba74, \uc989 \ub2e4\uc74c \uc870\uac74\uc744 \ub9cc\uc871\ud55c\ub2e4\uba74 \ub178\ub4dc <em>v<\/em> &isin; <em>V<\/em>\ub97c \uc2f1\ud06c\ub77c\uace0 \ubd80\ub978\ub2e4.<\/p>\r\n\r\n<ul>\r\n\t<li>\uc870\uac74:&nbsp;&forall;<em>w<\/em> &isin; <em>V<\/em>, (<em>v<\/em>&rarr;<em>w<\/em>) &rArr; (<em>w<\/em>&rarr;<em>v<\/em>)<\/li>\r\n<\/ul>\r\n\r\n<p>\ub610\ud55c, \uadf8\ub798\ud504 <em>G<\/em>\uc758 \uc2f1\ud06c\ub97c \ubaa8\uc544\ub193\uc740 \uc9d1\ud569\uc744 bottom(G)\ub85c \ud45c\ud604\ud55c\ub2e4.<\/p>\r\n\r\n<ul>\r\n\t<li>bottom(<em>G<\/em>) = {<em>v<\/em>&nbsp;&isin; <em>V<\/em>:&nbsp;&forall;<em>w<\/em> &isin; <em>V<\/em>, (<em>v<\/em>&rarr;<em>w<\/em>) &rArr; (<em>w<\/em>&rarr;<em>v<\/em>)}<\/li>\r\n<\/ul>\r\n\r\n<p>\uc8fc\uc5b4\uc9c4 \uadf8\ub798\ud504 <em>G<\/em>=(<em>V<\/em>, <em>E<\/em>)\uc758 bottom(<em>G<\/em>)\ub97c \uad6c\ud558\uc2dc\uc624.<\/p>\r\n","input":"<p>\uc785\ub825\uc740 \uc5ec\ub7ec \uac1c\uc758 \ud14c\uc2a4\ud2b8 \ucf00\uc774\uc2a4\ub85c \uad6c\ubd84\ub418\uc5b4 \uc788\ub2e4.<\/p>\r\n\r\n<p>\uac01 \ud14c\uc2a4\ud2b8 \ucf00\uc774\uc2a4\uc758 \uccab \uc904\uc5d0\ub294 \ub178\ub4dc\uc758 \uc218 <em>n<\/em> (1 &le; <em>n<\/em> &le; 5,000)\uacfc \uc74c\uc774 \uc544\ub2cc \uc815\uc218 <em>m<\/em> (1 &le; m &le; 100,000)\uc774 \uc8fc\uc5b4\uc9c4\ub2e4. <em>V<\/em> = {1, 2, ..., <em>n<\/em>}\uc774\uace0, \uac04\uc120\uc758 \uc218 |<em>E<\/em>|=<em>m<\/em>\uc784\uc744 \uc758\ubbf8\ud55c\ub2e4.<\/p>\r\n\r\n<p>\uadf8 \ub2e4\uc74c\ubd80\ud130\ub294 \uac01 \uac04\uc120\uc744 \ub098\ud0c0\ub0b4\ub294 <em>m<\/em>\uc30d\uc758 \uc218 <em>v<\/em><sub>1<\/sub> <em>w<\/em><sub>1<\/sub> <em>v<\/em><sub>2<\/sub> <em>w<\/em><sub>2<\/sub> ... <em>v<sub>m<\/sub><\/em> <em>w<sub>m<\/sub><\/em>\uc774 \uacf5\ubc31\uc73c\ub85c \uad6c\ubd84\ub418\uc5b4 \uc8fc\uc5b4\uc9c4\ub2e4. \uc774\ub294 (<em>v<sub>i<\/sub><\/em>, <em>w<sub>i<\/sub><\/em>)&nbsp;&isin; <em>E<\/em>\ub97c \uc758\ubbf8\ud55c\ub2e4.<\/p>\r\n\r\n<p>\ub9c8\uc9c0\ub9c9 \uc904\uc5d0 0\uc774 \uc8fc\uc5b4\uc9c0\uace0, \uc774 \uacbd\uc6b0\ub294 \ucc98\ub9ac\ud558\uc9c0 \uc54a\uace0 \ud504\ub85c\uadf8\ub7a8\uc744 \uc885\ub8cc\uc2dc\ucf1c\uc57c \ud55c\ub2e4.<\/p>\r\n","output":"<p>\uac01 \ud14c\uc2a4\ud2b8 \ucf00\uc774\uc2a4\ub9c8\ub2e4 \ud55c \uc904\uc5d0 \uac78\uccd0 bottom(<em>G<\/em>)\uc758 \ubaa8\ub4e0 \ub178\ub4dc\ub97c \ucd9c\ub825\ud55c\ub2e4. \ub178\ub4dc\ub294 \uacf5\ubc31\uc73c\ub85c \uad6c\ubd84\ud574\uc57c \ud558\uba70, \uc624\ub984\ucc28\uc21c\uc774\uc5b4\uc57c \ud55c\ub2e4. \ub9cc\uc57d, bottom(<em>G<\/em>)\uac00 \uacf5\uc9d1\ud569\uc774\uba74 \ube48 \uc904\uc744 \ucd9c\ub825\ud55c\ub2e4.<\/p>\r\n","hint":"<p><img alt=\"\" src=\"\/upload\/images2\/bottom.gif\" style=\"height:190px; width:159px\" \/><\/p>\r\n","original":"0","html_title":"0","problem_lang_tcode":"Korean"},{"problem_id":"6543","problem_lang":"1","title":"The Bottom of a Graph","description":"<p>We will use the following (standard) definitions from graph theory. Let <em>V<\/em> be a nonempty and finite set, its elements being called vertices (or nodes). Let <em>E<\/em> be a subset of the Cartesian product <em>V&times;V<\/em>, its elements being called edges. Then <em>G=(V,E)<\/em> is called a directed graph.<\/p>\r\n\r\n<p>Let <em>n<\/em> be a positive integer, and let <em>p=(e<sub>1<\/sub>,...,e<sub>n<\/sub>)<\/em> be a sequence of length <em>n<\/em> of edges <em>e<sub>i<\/sub>&isin;E<\/em> such that <em>e<sub>i<\/sub>=(v<sub>i<\/sub>,v<sub>i+1<\/sub>)<\/em> for a sequence of vertices <em>(v<sub>1<\/sub>,...,v<sub>n+1<\/sub>)<\/em>. Then <em>p<\/em> is called a path from vertex <em>v<sub>1<\/sub><\/em> to vertex <em>v<sub>n+1<\/sub><\/em> in <em>G<\/em> and we say that <em>v<sub>n+1<\/sub><\/em> is reachable from <em>v<sub>1<\/sub><\/em>, writing <em>(v<sub>1<\/sub>&rarr;v<sub>n+1<\/sub>)<\/em>.<\/p>\r\n\r\n<p>Here are some new definitions. A node <em>v<\/em> in a graph <em>G=(V,E)<\/em> is called a sink, if for every node <em>w<\/em> in <em>G<\/em> that is reachable from <em>v<\/em>, <em>v<\/em> is also reachable from <em>w<\/em>. The bottom of a graph is the subset of all nodes that are sinks, i.e., <em>bottom(G)={v&isin;V|&forall;w&isin;V:(v&rarr;w)&rArr;(w&rarr;v)}<\/em>. You have to calculate the bottom of certain graphs.<\/p>\r\n","input":"<p>The input contains several test cases, each of which corresponds to a directed graph <em>G<\/em>. Each test case starts with an integer number <em>v<\/em>, denoting the number of vertices of <em>G=(V,E)<\/em>, where the vertices will be identified by the integer numbers in the set <em>V={1,...,v}<\/em>. You may assume that 1 &le; <em>v<\/em> &le;&nbsp;5000, 1 &le; <em>e<\/em> &le; 100,000. That is followed by a non-negative integer <em>e<\/em> and, thereafter, <em>e<\/em> pairs of vertex identifiers <em>v<sub>1<\/sub>,w<sub>1<\/sub>,...,v<sub>e<\/sub>,w<sub>e<\/sub><\/em> with the meaning that <em>(v<sub>i<\/sub>,w<sub>i<\/sub>)&isin;E<\/em>. There are no edges other than specified by these pairs. The last test case is followed by a zero.<\/p>\r\n","output":"<p>For each test case output the bottom of the specified graph on a single line. To this end, print the numbers of all nodes that are sinks in sorted order separated by a single space character. If the bottom is empty, print an empty line.<\/p>\r\n","hint":"<p><img alt=\"\" src=\"\/upload\/images2\/bottom.gif\" style=\"height:190px; width:159px\" \/><\/p>\r\n","original":"1","html_title":"0","problem_lang_tcode":"English"}]

출처

Contest > University of Ulm Local Contest > University of Ulm Local Contest 2003 B번

  • 문제의 오타를 찾은 사람: kdr06006