Kyoya and Train

题意翻译

- 给定一张 $n$ 个点 $m$ 条边的无重边无自环的有向图，你要从 $1$ 号点到 $n$ 号点去。 - 如果你在 $t$ 时刻之后到达 $n$ 号点，你要交 $x$ 元的罚款。 - 每条边从 $a_i$ 到 $b_i$，走过它需要花费 $c_i$ 元，多次走过同一条边需要多次花费。 - 走过每条边所需的时间是随机的，对于 $k \in [1,t]$，$\frac{p_{i,k}}{10^5}$ 表示走过第 $i$ 条边需要时间 $k$ 的概率。因此如果多次走过同一条边，所需的时间也可能不同。 - 你希望花费尽可能少的钱（花费与罚款之和）到达 $n$ 号点，因此每到达一个点，你可能会更改原有的计划。 - 求在最优决策下，你期望花费的钱数。 - $n \le 50$，$m \le 100$，$t \le 2 \times 10^4$，$x,c_i \le 10^6$，$\sum_{k=1}^t p_{i,k} = 10^5$，答案精度误差 $\le 10^{-6}$。

题目描述

Kyoya Ootori wants to take the train to get to school. There are $n$ train stations and $m$ one-way train lines going between various stations. Kyoya is currently at train station $1$ , and the school is at station $n$ . To take a train, he must pay for a ticket, and the train also takes a certain amount of time. However, the trains are not perfect and take random amounts of time to arrive at their destination. If Kyoya arrives at school strictly after $t$ time units, he will have to pay a fine of $x$ . Each train line is described by a ticket price, and a probability distribution on the time the train takes. More formally, train line $i$ has ticket cost $c_{i}$ , and a probability distribution $p_{i,k}$ which denotes the probability that this train will take $k$ time units for all $1<=k<=t$ . Amounts of time that each of the trains used by Kyouya takes are mutually independent random values (moreover, if Kyoya travels along the same train more than once, it is possible for the train to take different amounts of time and those amounts are also independent one from another). Kyoya wants to get to school by spending the least amount of money in expectation (for the ticket price plus possible fine for being late). Of course, Kyoya has an optimal plan for how to get to school, and every time he arrives at a train station, he may recalculate his plan based on how much time he has remaining. What is the expected cost that Kyoya will pay to get to school if he moves optimally?

输入输出格式

输入格式

The first line of input contains four integers $n,m,t,x$ ( $2<=n<=50$ , $1<=m<=100$ , $1<=t<=20000$ , $0<=x<=10^{6}$ ). The next $2m$ lines contain the description of the trains. The $2i$ -th line will have $3$ integers $a_{i},b_{i},c_{i}$ , representing a one way train from station $a_{i}$ to $b_{i}$ with ticket cost $c_{i}$ ( $1<=a_{i},b_{i}<=n$ , $a_{i}≠b_{i}$ , $0<=c_{i}<=10^{6}$ ). There will always be at least one path from any station to the school. The $(2i+1)$ -th line will contain $t$ integers, $p_{i,1},p_{i,2},...,p_{i,t}$ where $p_{i,k}/100000$ is the probability that this train will take $k$ units of time to traverse ( $0<=p_{i,k}<=100000$ for $1<=k<=t$ , ![](https://cdn.luogu.com.cn/upload/vjudge_pic/CF553E/c76a22ee61018b4af95225f4e365759ccc0e9dc9.png)). It is guaranteed that there is no more than one train between each pair of platforms in each of the directions.

输出格式

Print a single real number that is equal to an optimal expected cost of getting to school. The answer will be considered correct if its relative or absolute error doesn't exceed $10^{-6}$ .

输入输出样例

输入样例 #1

4 4 5 1
1 2 0
50000 0 50000 0 0
2 3 0
10000 0 0 0 90000
3 4 0
100000 0 0 0 0
2 4 0
0 0 0 50000 50000


输出样例 #1

0.7000000000


输入样例 #2

4 4 5 1
1 2 100
50000 0 50000 0 0
2 3 100
10000 0 0 0 90000
3 4 100
100000 0 0 0 0
2 4 100
0 0 0 50000 50000


输出样例 #2

200.7500000000


说明

The optimal strategy in the first case is as follows: First, travel along first train line. With probability $1/2$ Kyoya will take $1$ time unit. Otherwise, Kyoya will take $3$ time units. If the train takes $1$ time unit, travel along the 4th train line. Kyoya will make it to school in time with probability $1/2$ . Otherwise, if the train takes 3 time units, travel along the 2nd train line. Kyoya will make it to school in time with probability $1/10$ . Since the cost of all train lines are zero, we can just look at the probability that Kyoya will incur the penalty. The probability that Kyoya will have to pay the penalty is $1/2×1/2+1/2×9/10=7/10$ . We can show that no other strategy is strictly better. The optimal strategy in the second case is to travel along $1→2→4$ no matter what. Kyoya will incur the penalty with probability $3/4$ , and the cost of the trains is $200$ , thus the expected cost is $200.75$ .