33807번 - Minimum Spanning Arborescence 서브태스크다국어

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문제

$r$을 루트로 하는 arborescence $G = (V, E)$란 다음 조건을 만족하는 방향 그래프를 뜻한다.

  • 다른 모든 정점 $v$에 대해, $r$에서 $v$로 가는 경로가 정확히 하나 존재한다.

DAG(Directed Acyclic Graph)란 사이클이 없는 유향 그래프를 뜻한다. DAG의 각 간선에 가중치가 부여되었을 때, DAG 위에서의 minimum spanning arborescence 란 다음과 같다.

  • DAG위에서의 $1$을 루트로 한 arborescence이다.
  • DAG 상의 모든 정점을 포함한다.
  • 사용된 간선들의 가중치 합이 최소이다. 그런 방법이 여럿 존재한다면 모두 minimum spanning arborescence이다.

정점 $N$개, 유향 간선 $M$개로 이루어진 DAG가 주어진다. $M$개의 간선에 각각 $1$이상 $K$이하의 가중치를 붙이는 $K^M$가지의 경우에 대해, minimum spanning arborescence를 구성하는 간선의 가중치 합의 기댓값을 $998\,244\,353$으로 나눈 나머지를 구하여라. 중복 간선이 있을 수 있음에 유의하라.

입력

첫 줄에 세 정수 $N, M, K$가 주어진다.

다음 $M$개의 줄에 걸쳐 $i+1$번째 줄에 간선을 뜻하는 두 정수 $a_i, b_i$가 주어진다. 이는 $a_i$에서 $b_i$로 가는 유향 간선이 존재함을 뜻한다.

출력

기댓값을 $998\,244\,353$으로 나눈 나머지를 출력한다.

즉, 기댓값이 서로소인 두 양의 정수 $a$, $b$에 대해 기약분수 $\frac{a}{b}$의 형태로 표현될 때, $b \cdot k \equiv a \pmod{998\,244\,353}$을 만족하는 유일한 정수 $k$ ($0 \le k < 998\,244\,353$)를 출력한다.

제한

  • $1 \leq N, M, K \leq 10^5 $
  • $ N-1 \leq M $
  • 모든 $1 \le i \le N$에 대하여 $ 1 \leq a_i < b_i \leq N$
  • 기존 DAG에서 1번 정점에서 다른 모든 정점에 도달할 수 있다.

서브태스크

번호배점제한
15

$M = N-1$

211

$M \leq 10, K \leq 2$

337

$N, M, K \leq 1000$

447

추가적인 제약 조건이 없다.

예제 입력 1 

10 15 3
1 2
1 3
3 4
4 5
2 6
2 7
1 8
1 9
6 10
4 9
1 8
8 9
1 2
8 10
4 10

예제 출력 1 

110916055

예제 입력 2 

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

예제 출력 2 

3

예제 입력 3 

10 9 100
1 2
1 3
3 4
3 5
5 6
2 7
2 8
6 9
7 10

예제 출력 3 

499122631

예제 입력 4 

11 10 2
1 2
1 3
2 4
4 5
3 6
1 7
5 8
8 9
2 10
3 11

예제 출력 4 

15

예제 입력 5 

5 7 2
1 2
2 3
1 4
1 5
1 5
2 4
1 4

예제 출력 5 

623902726
[{"problem_id":"33807","problem_lang":"0","title":"Minimum Spanning Arborescence","description":"<p>$r$\uc744 \ub8e8\ud2b8\ub85c \ud558\ub294 arborescence $G = (V, E)$\ub780 \ub2e4\uc74c \uc870\uac74\uc744 \ub9cc\uc871\ud558\ub294 \ubc29\ud5a5 \uadf8\ub798\ud504\ub97c \ub73b\ud55c\ub2e4.<\/p>\r\n\r\n<ul>\r\n\t<li>\ub2e4\ub978 \ubaa8\ub4e0 \uc815\uc810 $v$\uc5d0 \ub300\ud574, $r$\uc5d0\uc11c $v$\ub85c \uac00\ub294 \uacbd\ub85c\uac00 \uc815\ud655\ud788 \ud558\ub098 \uc874\uc7ac\ud55c\ub2e4.<\/li>\r\n<\/ul>\r\n\r\n<p>DAG(Directed Acyclic Graph)\ub780 \uc0ac\uc774\ud074\uc774 \uc5c6\ub294 \uc720\ud5a5 \uadf8\ub798\ud504\ub97c \ub73b\ud55c\ub2e4. DAG\uc758 \uac01 \uac04\uc120\uc5d0 \uac00\uc911\uce58\uac00 \ubd80\uc5ec\ub418\uc5c8\uc744 \ub54c, DAG \uc704\uc5d0\uc11c\uc758 minimum spanning arborescence \ub780 \ub2e4\uc74c\uacfc \uac19\ub2e4.<\/p>\r\n\r\n\r\n<ul>\r\n\t<li>DAG\uc704\uc5d0\uc11c\uc758 <span style=\"color:#c0392b;\"><strong>$1$<\/strong><\/span>\uc744 \ub8e8\ud2b8\ub85c \ud55c arborescence\uc774\ub2e4.<\/li>\r\n\t<li>DAG \uc0c1\uc758 \ubaa8\ub4e0 \uc815\uc810\uc744 \ud3ec\ud568\ud55c\ub2e4.<\/li>\r\n\t<li>\uc0ac\uc6a9\ub41c \uac04\uc120\ub4e4\uc758 \uac00\uc911\uce58 \ud569\uc774 \ucd5c\uc18c\uc774\ub2e4. \uadf8\ub7f0 \ubc29\ubc95\uc774 \uc5ec\ub7ff \uc874\uc7ac\ud55c\ub2e4\uba74 \ubaa8\ub450 minimum spanning arborescence\uc774\ub2e4.<\/li>\r\n<\/ul>\r\n\r\n<p>\uc815\uc810 $N$\uac1c, \uc720\ud5a5 \uac04\uc120 $M$\uac1c\ub85c \uc774\ub8e8\uc5b4\uc9c4 DAG\uac00 \uc8fc\uc5b4\uc9c4\ub2e4. $M$\uac1c\uc758 \uac04\uc120\uc5d0 \uac01\uac01 $1$\uc774\uc0c1 $K$\uc774\ud558\uc758 \uac00\uc911\uce58\ub97c \ubd99\uc774\ub294 $K^M$\uac00\uc9c0\uc758 \uacbd\uc6b0\uc5d0 \ub300\ud574, minimum spanning arborescence\ub97c \uad6c\uc131\ud558\ub294 \uac04\uc120\uc758 \uac00\uc911\uce58 \ud569\uc758 \uae30\ub313\uac12\uc744 $998\\,244\\,353$\uc73c\ub85c \ub098\ub208 \ub098\uba38\uc9c0\ub97c \uad6c\ud558\uc5ec\ub77c. \uc911\ubcf5 \uac04\uc120\uc774 \uc788\uc744 \uc218 \uc788\uc74c\uc5d0 \uc720\uc758\ud558\ub77c.<\/p>\r\n","input":"<p>\uccab \uc904\uc5d0 \uc138 \uc815\uc218 $N, M, K$\uac00 \uc8fc\uc5b4\uc9c4\ub2e4.<\/p>\r\n\r\n<p>\ub2e4\uc74c $M$\uac1c\uc758 \uc904\uc5d0 \uac78\uccd0 $i+1$\ubc88\uc9f8 \uc904\uc5d0 \uac04\uc120\uc744 \ub73b\ud558\ub294 \ub450 \uc815\uc218 $a_i, b_i$\uac00 \uc8fc\uc5b4\uc9c4\ub2e4. \uc774\ub294 $a_i$\uc5d0\uc11c $b_i$\ub85c \uac00\ub294 \uc720\ud5a5 \uac04\uc120\uc774 \uc874\uc7ac\ud568\uc744 \ub73b\ud55c\ub2e4.<\/p>\r\n","output":"<p>\uae30\ub313\uac12\uc744 $998\\,244\\,353$\uc73c\ub85c \ub098\ub208 \ub098\uba38\uc9c0\ub97c \ucd9c\ub825\ud55c\ub2e4.<\/p>\r\n\r\n<p>\uc989, \uae30\ub313\uac12\uc774 \uc11c\ub85c\uc18c\uc778 \ub450 \uc591\uc758 \uc815\uc218 $a$, $b$\uc5d0 \ub300\ud574 \uae30\uc57d\ubd84\uc218 $\\frac{a}{b}$\uc758 \ud615\ud0dc\ub85c \ud45c\ud604\ub420 \ub54c, $b \\cdot k \\equiv a \\pmod{998\\,244\\,353}$\uc744 \ub9cc\uc871\ud558\ub294 \uc720\uc77c\ud55c \uc815\uc218 $k$ ($0 \\le k &lt; 998\\,244\\,353$)\ub97c \ucd9c\ub825\ud55c\ub2e4.<\/p>\r\n","hint":"","original":"1","html_title":"0","problem_lang_tcode":"Korean","limit":"<ul>\r\n\t<li>$1 \\leq N, M, K \\leq 10^5 $<\/li>\r\n\t<li>$ N-1 \\leq M $<\/li>\r\n\t<li>\ubaa8\ub4e0 $1 \\le i \\le N$\uc5d0 \ub300\ud558\uc5ec $ 1 \\leq a_i &lt; b_i \\leq N$<\/li>\r\n\t<li>\uae30\uc874 DAG\uc5d0\uc11c 1\ubc88 \uc815\uc810\uc5d0\uc11c \ub2e4\ub978 \ubaa8\ub4e0 \uc815\uc810\uc5d0 \ub3c4\ub2ec\ud560 \uc218 \uc788\ub2e4.<\/li>\r\n<\/ul>\r\n","subtask1":"<p>$M = N-1$<\/p>\r\n","subtask2":"<p>$M \\leq 10, K \\leq 2$<\/p>\r\n","subtask3":"<p>$N, M, K \\leq 1000$<\/p>\r\n","subtask4":"<p>\ucd94\uac00\uc801\uc778 \uc81c\uc57d \uc870\uac74\uc774 \uc5c6\ub2e4.<\/p>\r\n"},{"problem_id":"33807","problem_lang":"1","title":"Minimum Spanning Arborescence","description":"<p>An arborescence $G = (V, E)$ rooted at $r$ is a directed graph that satisfies the following condition:<\/p>\r\n\r\n<ul>\r\n\t<li>For every other vertex $v$, there exists exactly one directed path from $r$ to $v$.<\/li>\r\n<\/ul>\r\n\r\n<p>A DAG (Directed Acyclic Graph) is a directed graph with no cycles. When each edge in the DAG is assigned a weight, a minimum spanning arborescence<b> <\/b>is defined as follows:<\/p>\r\n\r\n<ul>\r\n\t<li>It is an arborescence rooted at vertex <span style=\"color:#c0392b;\"><strong>$1$<\/strong><\/span>.<\/li>\r\n\t<li>It spans all vertices in the DAG.<\/li>\r\n\t<li>Among all such arborescences, the sum of the weights of the edges used is minimized. If there are multiple such arborescences, all of them are considered minimum spanning arborescences.<\/li>\r\n<\/ul>\r\n\r\n<p>You are given a DAG with $N$ vertices and $M$ directed edges. Each of the $M$ edges can be assigned a weight between $1$ and $K$, resulting in $K^M$ possible assignments. For each possible assignment of weights, consider the minimum spanning arborescence rooted at vertex $1$. Compute the expected total weight of the edges used in such arborescences, and output the result modulo $998\\,244\\,353$. Note that multiple edges between the same pair of vertices may exist.<\/p>\r\n","input":"<p>The first line contains three integers $N$, $M$, and $K$.<\/p>\r\n\r\n<p>Each of the following $M$ lines contains two integers $a_i$ and $b_i$ indicating a directed edge&nbsp;from vertex $a_i$ to vertex $b_i$.<\/p>\r\n","output":"<p>Output the expected value modulo $998\\,244\\,353$.<\/p>\r\n\r\n<p>That is, if the expected value can be expressed as a reduced fraction $\\frac{a}{b}$ for two positive coprime integers $a$ and $b$, output the unique integer $k$ such that $0 \\le k &lt; 998\\,244\\,353$ and $b \\cdot k \\equiv a \\pmod{998\\,244\\,353}$.<\/p>\r\n","hint":"","original":"0","html_title":"0","problem_lang_tcode":"English","limit":"<ul>\r\n\t<li>$1 \\leq N, M, K \\leq 10^5 $<\/li>\r\n\t<li>$ N-1 \\leq M $<\/li>\r\n\t<li>$ 1 \\leq a_i &lt; b_i \\leq N$ for every&nbsp; $1 \\le i \\le N$<\/li>\r\n\t<li>In the given DAG, vertex $1$ can reach every other vertex.<\/li>\r\n<\/ul>\r\n","subtask1":"<p>$M = N - 1$<\/p>\r\n","subtask2":"<p>$M \\le 10$, $K \\le 2$<\/p>\r\n","subtask3":"<p>$N, M, K \\le 1,000$<\/p>\r\n","subtask4":"<p>No additional constraints.<\/p>\r\n"}]

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