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문제

상현이는 어떤 기밀 단체의 요원이다. 상현이는 매일 아침 기차를 타고 출근한다. 어느 날 출근을 하던 상현이는 무언가 불합리하다는 것을 알아챘다. 상현이는 매일 요금을 내고 기차를 타지만, 실제로 승무원이 티켓을 확인하는 경우는 드물었기 때문이다. 결국 몇 년에 걸쳐, 상현이는 각 기차마다 어느 정도의 확률로 티켓을 확인받게 되는지에 대한 정보를 모두 작성하는 데 성공했다.

하지만 무임승차 벌금은 일반적으로 실제 요금보다 많기 때문에, 상현이는 모든 역에서 요금을 내지 않기보다는 요금과 벌금, 티켓을 확인받을 확률을 계산하여 기댓값이 가장 작은 방법으로 출근을 하기로 했다. 아마 어떤 역에서는 요금을 내고 탑승하는 것이 나을 것이고, 어떤 역에서는 무임승차를 하는 것이 나을 것이다.

기차 티켓은 기본 s원에 출발지 A와 도착지 B 사이의 최단거리에 비례하여 1 킬로미터당 p의 요금이 추가된다. 당연히 이 티켓으로는 A와 B 사이를 최단거리로 운행하는 기차만 탈 수 있다. 만일 기차가 운행하던 도중 티켓을 체크받았는데 티켓을 소지하지 않았다면, 기본 y원의 벌금에 지금 탑승한 기차가 마지막으로 방문한 도시에서부터 기차의 이번 정차역까지의 거리에 비례하여 1 킬로미터당 역시 p의 추가 벌금을 지불하게 된다. 일반적으로 무임승차가 적발될 경우 기차에서 즉시 내려야 하지만, 상현이는 기밀 요원이기에 변장술에 능하다. 따라서 적발되더라도 벌금을 지불한 뒤 계속 기차를 탈 수 있다.

이제 상현이가 통근 요금의 기댓값이 가장 적도록 출근할 수 있는 경로를 찾아 줄 프로그램을 작성하면 된다.

위는 세 번째 예제이다. 도시 1에서 도시 4까지 가는 최적의 경로는 아래와 같다.

도시 1에서 도시 2까지는 20의 요금을 지불하고 티켓을 구입(10 + 1 × 10), 2에서 3까지 가는 경로에선 무임승차를 하며(이때의 벌금의 기댓값은 0.1 × (100 + 1 × 120) = 22), 그리고 3에서 4까지는 20의 요금을 지불하고 티켓을 구입한다(10 + 1 × 10). 이와 같이 출근할 경우, 총 요금의 기댓값은 62이며(20 + 22 + 20) 이 경우가 최소의 기댓값으로 출근하는 방법이다.

입력

첫 줄에 테스트 케이스의 수 T가 주어진다. (T≤100)

각 테스트 케이스는 아래와 같이 구성되어 있다

  • 첫 줄에 공백으로 구분된 7개의 정수가 주어진다.
    1. n ( 2 ≤ n ≤ 200 ) : 기차의 운행 경로에 놓인 도시의 수
    2. m ( 1 ≤ m ≤ n(n-1) / 2 ) : 두 도시를 직접적으로 잇는 선로의 수
    3. start ( 1 ≤ start ≤ n ) : 출발지
    4. end ( 1 ≤ end ≤ n, start ≠ end ) : 도착지
    5. s ( 1 ≤ s ≤ 1000 ) : 티켓의 기본 요금 s
    6. p ( 1 ≤ p ≤ 1000 ) : 티켓 혹은 벌금에 추가되는 1 킬로미터당 추가 요금
    7. y ( s < y ≤ 1000 ) : 무임승차 벌금의 기본 요금 y
  • 다음 m개의 줄에는 각각 i 번째 선로에 대한 정보가 네 개의 공백으로 구분된 정수로 주어진다. 모든 선로는 양방향이다. i 번째 줄의 데이터는 다음과 같다.
    1. ai ( 1 ≤ ai ≤ n ) : i 번째 선로의 한쪽 끝 도시 번호
    2. bi ( ai < bi ≤ n ) : i 번째 선로의 다른 쪽 끝 도시
    3. ci ( 0 ≤ ci ≤ 100 ) : i 번째 선로를 운행하는 기차에서 티켓을 확인받을 확률( % 값으로 주어진다. )
    4. di ( 1 ≤ di ≤ 1000 ) : i 번째 선로의 길이(단위는 킬로미터)

i ≠ j 인 모든 (ai, bi) 와 (aj, bj) 쌍에 대하여, (ai, bi) ≠ (aj, bj) 이다. (중복되어 입력되는 선로는 없다)

출력

각 테스트 케이스에 대해 다음을 출력한다.

  • start에서 end까지 가는 데 드는 요금의 기댓값의 최솟값을 소수 두 번째 자리까지 반올림한 값

모든 테스트 케이스에서, 계산 과정에서의 절대 오차가 10^-6 이하인 경우 결과값은 달라지지 않는다.

예제 입력 1

3
2 1 1 2 10 1 100
1 2 20 50
2 1 1 2 10 1 100
1 2 60 50
4 4 1 4 10 1 100
1 4 50 90
1 2 90 10
2 3 10 120
3 4 90 10

예제 출력 1

30.00
60.00
62.00

힌트

벌금의 기댓값은 확인받을 확률 * 벌금 으로 계산할 수 있으며, 각 운행 경로는 독립적이므로 경로 X에서의 기댓값을 E[X], 경로 Y에서의 기댓값을 E[Y]라 할 때, X-Y 경로의 기댓값은 E[X]+E[Y]가 된다.

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10), 2\uc5d0\uc11c 3\uae4c\uc9c0 \uac00\ub294 \uacbd\ub85c\uc5d0\uc120 \ubb34\uc784\uc2b9\ucc28\ub97c \ud558\uba70(\uc774\ub54c\uc758 \ubc8c\uae08\uc758&nbsp;\uae30\ub313\uac12\uc740 0.1&nbsp;&times; (100 + 1&nbsp;&times; 120) = 22), \uadf8\ub9ac\uace0 3\uc5d0\uc11c 4\uae4c\uc9c0\ub294 20\uc758 \uc694\uae08\uc744 \uc9c0\ubd88\ud558\uace0 \ud2f0\ucf13\uc744 \uad6c\uc785\ud55c\ub2e4(10 + 1&nbsp;&times; 10). \uc774\uc640 \uac19\uc774 \ucd9c\uadfc\ud560 \uacbd\uc6b0, \ucd1d&nbsp;\uc694\uae08\uc758 \uae30\ub313\uac12\uc740 62\uc774\uba70(20 + 22 + 20)&nbsp;\uc774 \uacbd\uc6b0\uac00 \ucd5c\uc18c\uc758 \uae30\ub313\uac12\uc73c\ub85c \ucd9c\uadfc\ud558\ub294 \ubc29\ubc95\uc774\ub2e4.<\/p>\r\n","input":"<p>\uccab \uc904\uc5d0 \ud14c\uc2a4\ud2b8 \ucf00\uc774\uc2a4\uc758 \uc218 T\uac00 \uc8fc\uc5b4\uc9c4\ub2e4. (T&le;100)<\/p>\r\n\r\n<p>\uac01 \ud14c\uc2a4\ud2b8 \ucf00\uc774\uc2a4\ub294 \uc544\ub798\uc640 \uac19\uc774 \uad6c\uc131\ub418\uc5b4 \uc788\ub2e4<\/p>\r\n\r\n<ul>\r\n\t<li>\uccab \uc904\uc5d0 \uacf5\ubc31\uc73c\ub85c \uad6c\ubd84\ub41c 7\uac1c\uc758 \uc815\uc218\uac00 \uc8fc\uc5b4\uc9c4\ub2e4.\r\n\t<ol>\r\n\t\t<li>n ( 2&nbsp;&le;&nbsp;n&nbsp;&le;&nbsp;200&nbsp;) : \uae30\ucc28\uc758 \uc6b4\ud589 \uacbd\ub85c\uc5d0 \ub193\uc778&nbsp;\ub3c4\uc2dc\uc758 \uc218<\/li>\r\n\t\t<li>m ( 1&nbsp;&le;&nbsp;m&nbsp;&le;&nbsp;n(n-1)&nbsp;\/&nbsp;2&nbsp;) : \ub450 \ub3c4\uc2dc\ub97c \uc9c1\uc811\uc801\uc73c\ub85c \uc787\ub294 \uc120\ub85c\uc758 \uc218<\/li>\r\n\t\t<li>start&nbsp;( 1&nbsp;&le; start&nbsp;&le; n ) : \ucd9c\ubc1c\uc9c0<\/li>\r\n\t\t<li>end (&nbsp;1&nbsp;&le; end&nbsp;&le; n, start &ne; end ) : \ub3c4\ucc29\uc9c0<\/li>\r\n\t\t<li>s ( 1&nbsp;&le; s&nbsp;&le; 1000 ) : \ud2f0\ucf13\uc758 \uae30\ubcf8 \uc694\uae08 s<\/li>\r\n\t\t<li>p ( 1&nbsp;&le; p&nbsp;&le; 1000 ) : \ud2f0\ucf13 \ud639\uc740 \ubc8c\uae08\uc5d0 \ucd94\uac00\ub418\ub294 1 \ud0ac\ub85c\ubbf8\ud130\ub2f9 \ucd94\uac00 \uc694\uae08<\/li>\r\n\t\t<li>y ( s &lt; y&nbsp;&le; 1000 ) : \ubb34\uc784\uc2b9\ucc28 \ubc8c\uae08\uc758&nbsp;\uae30\ubcf8 \uc694\uae08 y<\/li>\r\n\t<\/ol>\r\n\t<\/li>\r\n\t<li>\ub2e4\uc74c m\uac1c\uc758 \uc904\uc5d0\ub294 \uac01\uac01 i \ubc88\uc9f8 \uc120\ub85c\uc5d0 \ub300\ud55c \uc815\ubcf4\uac00&nbsp;\ub124 \uac1c\uc758 \uacf5\ubc31\uc73c\ub85c \uad6c\ubd84\ub41c \uc815\uc218\ub85c \uc8fc\uc5b4\uc9c4\ub2e4. \ubaa8\ub4e0 \uc120\ub85c\ub294 \uc591\ubc29\ud5a5\uc774\ub2e4. i&nbsp;\ubc88\uc9f8 \uc904\uc758 \ub370\uc774\ud130\ub294 \ub2e4\uc74c\uacfc \uac19\ub2e4.\r\n\t<ol>\r\n\t\t<li>ai ( 1&nbsp;&le; ai&nbsp;&le; n ) : i \ubc88\uc9f8 \uc120\ub85c\uc758 \ud55c\ucabd \ub05d \ub3c4\uc2dc \ubc88\ud638<\/li>\r\n\t\t<li>bi ( ai &lt; bi&nbsp;&le; n ) : i&nbsp;\ubc88\uc9f8 \uc120\ub85c\uc758 \ub2e4\ub978 \ucabd \ub05d \ub3c4\uc2dc<\/li>\r\n\t\t<li>ci ( 0&nbsp;&le; ci&nbsp;&le; 100 ) : i&nbsp;\ubc88\uc9f8 \uc120\ub85c\ub97c \uc6b4\ud589\ud558\ub294 \uae30\ucc28\uc5d0\uc11c \ud2f0\ucf13\uc744 \ud655\uc778\ubc1b\uc744 \ud655\ub960( % \uac12\uc73c\ub85c \uc8fc\uc5b4\uc9c4\ub2e4. )<\/li>\r\n\t\t<li>di ( 1&nbsp;&le; di&nbsp;&le; 1000 ) : i \ubc88\uc9f8 \uc120\ub85c\uc758 \uae38\uc774(\ub2e8\uc704\ub294 \ud0ac\ub85c\ubbf8\ud130)<\/li>\r\n\t<\/ol>\r\n\t<\/li>\r\n<\/ul>\r\n\r\n<p>i&nbsp;&ne; j \uc778&nbsp;\ubaa8\ub4e0 (ai, bi) \uc640 (aj, bj) \uc30d\uc5d0 \ub300\ud558\uc5ec,&nbsp;(ai, bi)&nbsp;&ne;&nbsp;(aj, bj) \uc774\ub2e4. (\uc911\ubcf5\ub418\uc5b4 \uc785\ub825\ub418\ub294 \uc120\ub85c\ub294 \uc5c6\ub2e4)<\/p>\r\n","output":"<p>\uac01 \ud14c\uc2a4\ud2b8 \ucf00\uc774\uc2a4\uc5d0 \ub300\ud574 \ub2e4\uc74c\uc744 \ucd9c\ub825\ud55c\ub2e4.<\/p>\r\n\r\n<ul>\r\n\t<li>start\uc5d0\uc11c end\uae4c\uc9c0 \uac00\ub294 \ub370 \ub4dc\ub294 \uc694\uae08\uc758 \uae30\ub313\uac12\uc758 \ucd5c\uc19f\uac12\uc744 \uc18c\uc218 \ub450 \ubc88\uc9f8 \uc790\ub9ac\uae4c\uc9c0 \ubc18\uc62c\ub9bc\ud55c \uac12<\/li>\r\n<\/ul>\r\n\r\n<p>\ubaa8\ub4e0 \ud14c\uc2a4\ud2b8 \ucf00\uc774\uc2a4\uc5d0\uc11c, \uacc4\uc0b0 \uacfc\uc815\uc5d0\uc11c\uc758 \uc808\ub300 \uc624\ucc28\uac00 10^-6 \uc774\ud558\uc778 \uacbd\uc6b0 \uacb0\uacfc\uac12\uc740 \ub2ec\ub77c\uc9c0\uc9c0 \uc54a\ub294\ub2e4.<\/p>\r\n","hint":"<p>\ubc8c\uae08\uc758 \uae30\ub313\uac12\uc740 \ud655\uc778\ubc1b\uc744 \ud655\ub960 * \ubc8c\uae08 \uc73c\ub85c \uacc4\uc0b0\ud560 \uc218 \uc788\uc73c\uba70, \uac01 \uc6b4\ud589 \uacbd\ub85c\ub294 \ub3c5\ub9bd\uc801\uc774\ubbc0\ub85c&nbsp;\uacbd\ub85c X\uc5d0\uc11c\uc758 \uae30\ub313\uac12\uc744 E[X], \uacbd\ub85c Y\uc5d0\uc11c\uc758 \uae30\ub313\uac12\uc744 E[Y]\ub77c \ud560 \ub54c, X-Y \uacbd\ub85c\uc758 \uae30\ub313\uac12\uc740 E[X]+E[Y]\uac00 \ub41c\ub2e4.<\/p>\r\n","original":"0","html_title":"0","problem_lang_tcode":"Korean"},{"problem_id":"9323","problem_lang":"1","title":"Fare Dodging","description":"<p>Virgil is a loyal employee at a mysterious national security agency. He has a long commute to work by train every day, during which his mind is free to ponder his situation and the facts of life. Always the mathematician, he notices that, although he pays the fares for his journeys without fail, the conductors rarely check his tickets. In fact, through his years of experience he has a pretty good idea of what the chances are of being checked per section between each pair of cities.<\/p>\r\n\r\n<p>Maybe he could just dodge the fares? Of course, if he gets caught in the train without a ticket he has to pay a \ufb01ne that will be much higher than the cost of a ticket would have been. But still... Perhaps it&rsquo;s cheaper to not pay for (part of) the journey and pay the \ufb01nes if and when he does get caught.<\/p>\r\n\r\n<p>A ticket from city A to city B costs a certain start up cost s, plus a small amount p per kilometer on a shortest route between city A and B (the ticket is only valid on a shortest route). If you get caught without a ticket, you have to pay a \ufb01ne consisting of a constant y plus the small amount p per kilometer on the section of railway from the last city the train visited to the next city. The regulations dictate that you have to get off at the next stop if you get caught fare dodging, but being an employee of such a mysterious agency, Virgil is a master of disguise: the conductors will never recognise him and he can continue his journey as if he was never busted.<\/p>\r\n\r\n<p>Help Virgil write a computer program that &ndash; based on his experience &ndash; calculates the route and which tickets to buy so that his expected daily costs are as small as possible.<\/p>\r\n\r\n<p><img alt=\"\" src=\"\/upload\/images\/dogging.png\" style=\"height:171px; width:196px\" \/><\/p>\r\n\r\n<p>A visual representation of the third sample. The best route from 1 to 4 is to buy a ticket for section 1 to 2 for a cost of 20 (10 + 1 &times; 10), then dodge the fares for section 2 to 3 for an expected cost of 22 (0.1 &times; (100 + 1 &times; 120)), and \ufb01nally buy a ticket for section 3 to 4 for a cost of 20 (10 + 1 &times; 10). This leads to a total expected cost of 62 (20 + 22 + 20) which is the best possible for this rail network.<\/p>\r\n","input":"<p>On the \ufb01rst line one positive number: the number of test cases, at most 100. After that per test case:<\/p>\r\n\r\n<ul>\r\n\t<li>one line with seven space-separated integers:\r\n\t<ul>\r\n\t\t<li>n (2 &le; n &le; 200): the number of cities in the rail network;<\/li>\r\n\t\t<li>m (1 &le; m &le; n(n-1) \/ 2): the number of direct connections in the rail network;<\/li>\r\n\t\t<li>start (1 &le; start &le; n): the index of the city that Virgil lives in;<\/li>\r\n\t\t<li>end (1 &le; end &le; n, start &ne; end): the index of the city that Virgil works at;<\/li>\r\n\t\t<li>s (1 &le; s &le; 1 000): the start up cost of a ticket;<\/li>\r\n\t\t<li>p (1 &le; p &le; 1 000): the cost per kilometer of a ticket or \ufb01ne;<\/li>\r\n\t\t<li>y (s &lt; y &le; 1 000): the constant part of a \ufb01ne.<\/li>\r\n\t<\/ul>\r\n\t<\/li>\r\n\t<li>m lines with four space-separated integers, describing the bidirectional sections between cities, with on line i:\r\n\t<ul>\r\n\t\t<li>a<sub>i<\/sub> (1 &le; a<sub>i<\/sub> &le; n): the index of one city on the end of section i;<\/li>\r\n\t\t<li>b<sub>i<\/sub> (a<sub>i<\/sub> &lt; b<sub>i<\/sub> &le; n): the index of the other city of section i;<\/li>\r\n\t\t<li>c<sub>i<\/sub> (0 &le; c<sub>i<\/sub> &le; 100): the probability that a conductor will check your ticket on section i, expressed as a percentage;<\/li>\r\n\t\t<li>d<sub>i<\/sub> (1 &le; d<sub>i<\/sub> &le; 1 000): the distance traveled along section i between city a<sub>i<\/sub> and b<sub>i<\/sub> in kilometers;<\/li>\r\n\t<\/ul>\r\n\t<\/li>\r\n<\/ul>\r\n\r\n<p>For any two pairs (a<sub>i<\/sub>, b<sub>i<\/sub>) and (a<sub>j<\/sub>, b<sub>j<\/sub>) with i &ne; j, we have that (a<sub>i<\/sub>, b<sub>i<\/sub>) &ne; (a<sub>j<\/sub>, b<sub>j<\/sub>).<\/p>\r\n","output":"<p>Per test case:<\/p>\r\n\r\n<ul>\r\n\t<li>the lowest possible expected costs for a single trip between city start and end, rounded to exactly two decimals.<\/li>\r\n<\/ul>\r\n\r\n<p>The test cases are such that an absolute error of at most 10<sup>-6<\/sup> in the \ufb01nal answer does not in\ufb02uence the result of the rounding.<\/p>\r\n","hint":"<p>The expected daily costs can be calculated as \\(E\\left[ C \\right] &nbsp; = \\sum_{k} {P_kC_k}\\), where \\(P_k\\) is the probability that the costs are \\(C_k\\), and sum is over all possible costs. Expectation values are additive, i.e. \\(E\\left[ X+Y \\right] &nbsp;= E\\left[ &nbsp;X \\right] &nbsp;+ E\\left[ &nbsp;Y \\right] &nbsp;\\).<\/p>\r\n","original":"1","html_title":"0","problem_lang_tcode":"English"}]