시간 제한 메모리 제한 제출 정답 맞은 사람 정답 비율
10 초 128 MB 16 1 1 16.667%

문제

최흉최악의 해커 yum3은 팰린드롬에 집착하는 경향이 있다(정작 자신의 이름은 팰린드롬이 아니다).

어느날 yum3은 도시 s에서 출발하여 도시 t로 여행을 떠나게 되었다. 이 세계의 도시와 도시 사이에 있는 모든 길에는 알파벳 대문자가 하나 적혀 있다. 같은 두 도시 사이에 길이 여러 개 존재할 수는 있지만, 시작점과 도착점이 같은 길은 없다.

yum3은 도시 t에 도착할 때까지는 현재 도시에서 인접한 길 중 하나로 균등한 확률로 이동하며, 같은 도시나 길을 여러 번 지나도 상관없다. 단, 도시 t에 도착하거나 도시 t로 갈 수 없는 어떤 도시에 이르게 되면 여행을 즉시 멈춘다. 그리고 t에 도착했으면서 출발지부터 지나온 길에 적혀있던 글자들을 순서대로 나열하여, 만약 그것이 팰린드롬이면 lucky하다고 느낀다. 만약 t에 도착할 수 없게 되거나 경로가 팰린드롬을 이루지 않으면 lucky하지 못하다.

yum3이 lucky한 여행을 할 수 있을 확률은 얼마일까?

입력

첫째 줄에 테스트 케이스의 개수 T(≤ 100)가 주어진다. 이어서 각 테스트 케이스가 여러 줄에 걸쳐 주어진다.

각 테스트 케이스는 빈 줄로 시작하고, 이어서 한 줄에 도시의 개수 n, 길의 개수 m이 주어진다(2 ≤ n ≤ 12, 0 ≤ m ≤1000). 이어서 m개의 줄에 각 길의 정보 정수 u, v(0 ≤ u, v < n, u ≠ v), 알파벳 대문자 w가 주어지는데 이는 도시 u에서 도시 v로 가는 일방향 길이 존재하며 그 길에 적힌 글자가 w라는 의미이다. 이어서 하나의 줄에 질문의 개수를 의미하는 정수 q(≤ 150)가 주어지고, 이어서 q개의 줄에 질문 s, t(0 ≤ s, t < n, s ≠ t)가 주어지는데 이는 출발지가 도시 s, 도착지가 도시 t라는 의미이다.

출력

각 테스트 케이스마다 첫째 줄에 양식에 맞게 테스트 케이스 번호를 출력하고, 이어서 각 줄마다 질문에 대한 답을 출력한다. 각 값은 정답과 10-4 미만의 오차가 나면 정답으로 인정된다.

예제 입력 1

2

4 3
0 1 A
1 2 A
2 3 A
2
0 3
2 0


5 4
1 2 B
2 3 D
2 4 A
2 0 B
2
1 3
1 0

예제 출력 1

Case 1:
1.000000
0.000000
Case 2:
0.000000
0.333333
[{"problem_id":"9621","problem_lang":"0","title":"\ud330\ub9b0\ub4dc\ub86c \uc5ec\ud589","description":"<p>\ucd5c\ud749\ucd5c\uc545\uc758 \ud574\ucee4 yum3\uc740 \ud330\ub9b0\ub4dc\ub86c\uc5d0 \uc9d1\ucc29\ud558\ub294 \uacbd\ud5a5\uc774 \uc788\ub2e4(\uc815\uc791 \uc790\uc2e0\uc758 \uc774\ub984\uc740 \ud330\ub9b0\ub4dc\ub86c\uc774 \uc544\ub2c8\ub2e4).<\/p>\r\n\r\n<p>\uc5b4\ub290\ub0a0 yum3\uc740 \ub3c4\uc2dc s\uc5d0\uc11c \ucd9c\ubc1c\ud558\uc5ec \ub3c4\uc2dc t\ub85c \uc5ec\ud589\uc744 \ub5a0\ub098\uac8c \ub418\uc5c8\ub2e4. \uc774 \uc138\uacc4\uc758 \ub3c4\uc2dc\uc640 \ub3c4\uc2dc \uc0ac\uc774\uc5d0 \uc788\ub294 \ubaa8\ub4e0 \uae38\uc5d0\ub294 \uc54c\ud30c\ubcb3 \ub300\ubb38\uc790\uac00 \ud558\ub098 \uc801\ud600 \uc788\ub2e4. \uac19\uc740 \ub450 \ub3c4\uc2dc \uc0ac\uc774\uc5d0 \uae38\uc774 \uc5ec\ub7ec \uac1c \uc874\uc7ac\ud560 \uc218\ub294 \uc788\uc9c0\ub9cc, \uc2dc\uc791\uc810\uacfc \ub3c4\ucc29\uc810\uc774 \uac19\uc740 \uae38\uc740 \uc5c6\ub2e4.<\/p>\r\n\r\n<p>yum3\uc740 \ub3c4\uc2dc t\uc5d0 \ub3c4\ucc29\ud560 \ub54c\uae4c\uc9c0\ub294 \ud604\uc7ac \ub3c4\uc2dc\uc5d0\uc11c \uc778\uc811\ud55c \uae38 \uc911 \ud558\ub098\ub85c \uade0\ub4f1\ud55c \ud655\ub960\ub85c \uc774\ub3d9\ud558\uba70, \uac19\uc740 \ub3c4\uc2dc\ub098 \uae38\uc744 \uc5ec\ub7ec \ubc88 \uc9c0\ub098\ub3c4 \uc0c1\uad00\uc5c6\ub2e4. \ub2e8, \ub3c4\uc2dc t\uc5d0 \ub3c4\ucc29\ud558\uac70\ub098 \ub3c4\uc2dc t\ub85c \uac08 \uc218 \uc5c6\ub294 \uc5b4\ub5a4 \ub3c4\uc2dc\uc5d0 \uc774\ub974\uac8c \ub418\uba74 \uc5ec\ud589\uc744 \uc989\uc2dc \uba48\ucd98\ub2e4. \uadf8\ub9ac\uace0 t\uc5d0 \ub3c4\ucc29\ud588\uc73c\uba74\uc11c \ucd9c\ubc1c\uc9c0\ubd80\ud130 \uc9c0\ub098\uc628 \uae38\uc5d0 \uc801\ud600\uc788\ub358 \uae00\uc790\ub4e4\uc744 \uc21c\uc11c\ub300\ub85c \ub098\uc5f4\ud558\uc5ec, \ub9cc\uc57d \uadf8\uac83\uc774 \ud330\ub9b0\ub4dc\ub86c\uc774\uba74 lucky\ud558\ub2e4\uace0 \ub290\ub080\ub2e4. \ub9cc\uc57d t\uc5d0 \ub3c4\ucc29\ud560 \uc218 \uc5c6\uac8c \ub418\uac70\ub098 \uacbd\ub85c\uac00 \ud330\ub9b0\ub4dc\ub86c\uc744 \uc774\ub8e8\uc9c0 \uc54a\uc73c\uba74 lucky\ud558\uc9c0 \ubabb\ud558\ub2e4.<\/p>\r\n\r\n<p>yum3\uc774 lucky\ud55c \uc5ec\ud589\uc744 \ud560 \uc218 \uc788\uc744 \ud655\ub960\uc740 \uc5bc\ub9c8\uc77c\uae4c?<\/p>\r\n","input":"<p>\uccab\uc9f8 \uc904\uc5d0 \ud14c\uc2a4\ud2b8 \ucf00\uc774\uc2a4\uc758 \uac1c\uc218 T(&le; 100)\uac00 \uc8fc\uc5b4\uc9c4\ub2e4. \uc774\uc5b4\uc11c \uac01 \ud14c\uc2a4\ud2b8 \ucf00\uc774\uc2a4\uac00 \uc5ec\ub7ec \uc904\uc5d0 \uac78\uccd0 \uc8fc\uc5b4\uc9c4\ub2e4.<\/p>\r\n\r\n<p>\uac01 \ud14c\uc2a4\ud2b8 \ucf00\uc774\uc2a4\ub294 \ube48 \uc904\ub85c \uc2dc\uc791\ud558\uace0, \uc774\uc5b4\uc11c \ud55c \uc904\uc5d0 \ub3c4\uc2dc\uc758 \uac1c\uc218 n, \uae38\uc758 \uac1c\uc218 m\uc774 \uc8fc\uc5b4\uc9c4\ub2e4(2 &le; n &le; 12, 0 &le; m &le;1000). \uc774\uc5b4\uc11c m\uac1c\uc758 \uc904\uc5d0 \uac01 \uae38\uc758 \uc815\ubcf4 \uc815\uc218 u, v(0 &le; u, v &lt; n, u &ne; v), \uc54c\ud30c\ubcb3 \ub300\ubb38\uc790 w\uac00 \uc8fc\uc5b4\uc9c0\ub294\ub370 \uc774\ub294 \ub3c4\uc2dc u\uc5d0\uc11c \ub3c4\uc2dc v\ub85c \uac00\ub294 \uc77c\ubc29\ud5a5 \uae38\uc774 \uc874\uc7ac\ud558\uba70 \uadf8 \uae38\uc5d0 \uc801\ud78c \uae00\uc790\uac00 w\ub77c\ub294 \uc758\ubbf8\uc774\ub2e4. \uc774\uc5b4\uc11c \ud558\ub098\uc758 \uc904\uc5d0 \uc9c8\ubb38\uc758 \uac1c\uc218\ub97c \uc758\ubbf8\ud558\ub294 \uc815\uc218 q(&le; 150)\uac00 \uc8fc\uc5b4\uc9c0\uace0, \uc774\uc5b4\uc11c q\uac1c\uc758 \uc904\uc5d0 \uc9c8\ubb38 s, t(0 &le; s, t &lt; n, s &ne; t)\uac00 \uc8fc\uc5b4\uc9c0\ub294\ub370 \uc774\ub294 \ucd9c\ubc1c\uc9c0\uac00 \ub3c4\uc2dc s, \ub3c4\ucc29\uc9c0\uac00 \ub3c4\uc2dc t\ub77c\ub294 \uc758\ubbf8\uc774\ub2e4.<\/p>\r\n","output":"<p>\uac01 \ud14c\uc2a4\ud2b8 \ucf00\uc774\uc2a4\ub9c8\ub2e4 \uccab\uc9f8 \uc904\uc5d0 \uc591\uc2dd\uc5d0 \ub9de\uac8c \ud14c\uc2a4\ud2b8 \ucf00\uc774\uc2a4 \ubc88\ud638\ub97c \ucd9c\ub825\ud558\uace0, \uc774\uc5b4\uc11c \uac01 \uc904\ub9c8\ub2e4 \uc9c8\ubb38\uc5d0 \ub300\ud55c \ub2f5\uc744 \ucd9c\ub825\ud55c\ub2e4. \uac01 \uac12\uc740 \uc815\ub2f5\uacfc 10<sup>-4<\/sup> \ubbf8\ub9cc\uc758 \uc624\ucc28\uac00 \ub098\uba74 \uc815\ub2f5\uc73c\ub85c \uc778\uc815\ub41c\ub2e4.<\/p>\r\n","hint":"","original":"0","problem_lang_code":"\ud55c\uad6d\uc5b4"},{"problem_id":"9621","problem_lang":"1","title":"Traveling Fool","description":"<p>You must have heard the &lsquo;Traveling Salesman Problem&rsquo;. Here we are talking about another problem named &lsquo;Traveling Fool Problem&rsquo;. Assume that there are n cities connected by m one way roads. Each road is labeled by an uppercase English letter (i.e &lsquo;A&rsquo; to &lsquo;Z&rsquo;). There can be multiple roads between two cities but no roads will start and end at the same city.&nbsp;<\/p>\r\n\r\n<p>The traveling fool starts his journey from city s and he continues his journey until he reaches t, or he reaches a city from which t is unreachable. If he is in city u, he can choose any road that starts from u with equal probability.&nbsp;<\/p>\r\n\r\n<p>He may visit same city\/road more than once, but once he reaches t, he immediately stops his journey and remembers the road-labels he found in his path in the same order the roads were visited. If the road-labels in the path he traveled form a palindrome, he finds himself lucky. If he is unable to reach t or the road-labels don&rsquo;t form a valid palindrome, he finds himself unlucky.&nbsp;<\/p>\r\n\r\n<p>Given the cities, roads, s and t, can you find the probability of Mr Traveling Fool being lucky?&nbsp;<\/p>\r\n","input":"<p>Input starts with an integer T (&le; 100), denoting the number of test cases.&nbsp;<\/p>\r\n\r\n<p>Each case starts with a blank line. Next line contains two integers n (2 &le; n &le; 12) and m (0 &le; m &le;1000). Each of the next m lines contains two integers, u v (0 &le; u, v &lt; n, u &ne; v) and an uppercase English letter w, meaning that there is a one-way road from city u to city v and the road label is w. Next line contains an integer (1 &le; q &le; 150) denoting the number of queries. Each of the next q lines contains two integers denoting s t (0 &le; s, t &lt; n, s &ne; t).&nbsp;<\/p>\r\n","output":"<p>For each case, print the case number first. Then for each query, print the probability as stated. Errors less than 10<sup>-4<\/sup> will be ignored.&nbsp;<\/p>\r\n","hint":"","original":"1","problem_lang_code":"\uc601\uc5b4"}]