102224: [AtCoder]ABC222 E - Red and Blue Tree

Problem Statement

Given are a tree with $N$ vertices, a sequence of $M$ numbers $A=(A_1,\ldots,A_M)$, and an integer $K$.
The vertices are numbered $1$ through $N$, and the $i$-th edge connects Vertices $U_i$ and $V_i$.

We will paint each of the $N-1$ edges of this tree red or blue. Among the $2^{N-1}$ such ways, find the number of ones that satisfies the following condition, modulo $998244353$.

Condition:
Let us put a piece on Vertex $A_1$, and for each $i=1,\ldots,M-1$ in this order, move it from Vertex $A_i$ to Vertex $A_{i+1}$ along the edges in the shortest path. After all of these movements, $R-B=K$ holds, where $R$ and $B$ are the numbers of times the piece traverses a red edge and a blue edge, respectively.

Constraints

• $2 \leq N \leq 1000$
• $2 \leq M \leq 100$
• $|K| \leq 10^5$
• $1 \leq A_i \leq N$
• $1\leq U_i,V_i\leq N$
• The given graph is a tree.
• All values in input are integers.

Input

Input is given from Standard Input in the following format:

$N$ $M$ $K$
$A_1$ $A_2$ $\ldots$ $A_M$
$U_1$ $V_1$
$\vdots$
$U_{N-1}$ $V_{N-1}$

Output

Print the answer.

Sample Input 1

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

Sample Output 1

2

If we paint the $1$-st and $3$-rd edges red and the $2$-nd edge blue, the piece will traverse the following numbers of red and blue edges:

• $0$ red edges and $1$ blue edge when moving from Vertex $2$ to $3$,
• $0$ red edges and $1$ blue edge when moving from Vertex $3$ to $2$,
• $1$ red edge and $0$ blue edges when moving from Vertex $2$ to $1$,
• $2$ red edges and $1$ blue edge when moving from Vertex $1$ to $4$,

for a total of $3$ red edges and $3$ blue edges, satisfying the condition.

Another way to satisfy the condition is to paint the $1$-st and $3$-rd edges blue and the $2$-nd edge red. There is no other way to satisfy it, so the answer is $2$.

Sample Input 2

3 10 10000
1 2 1 2 1 2 2 1 1 2
1 2
1 3

Sample Output 2

0

There may be no way to paint the tree to satisfy the condition.

Sample Input 3

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

Sample Output 3

126

Sample Input 4

5 8 -1
1 4 1 4 2 1 3 5
1 2
4 1
3 1
1 5

Sample Output 4

2

问题描述

给定一个有N个顶点的树，一个长度为M的整数序列A=(A_1,...,A_M)，以及一个整数K。

顶点编号为1到N，第i条边连接顶点U_i和V_i。

我们要在这棵树的N-1条边中，将每一条染成红色或蓝色。在2^{N-1}种这样的方式中，找出满足以下条件的方式的数量，对998244353取模。

条件：
在顶点A_1上放置一个棋子，然后按照顺序，从顶点A_i移动到顶点A_{i+1}，沿着最短路径中的边。经过所有的移动后，R-B=K成立，其中R和B分别是棋子经过红色边和蓝色边的次数。

约束条件

• 2 \leq N \leq 1000
• 2 \leq M \leq 100
• |K| \leq 10^5
• 1 \leq A_i \leq N
• 1\leq U_i,V_i\leq N
• 给定的图是一棵树。
• 输入中的所有值都是整数。

输入

输入通过标准输入给出以下格式：

$N$ $M$ $K$
$A_1$ $A_2$ $\ldots$ $A_M$
$U_1$ $V_1$
$\vdots$
$U_{N-1}$ $V_{N-1}$

输出

输出答案。

样例输入1

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

样例输出1

2

如果我们把第1条和第3条边涂成红色，第2条边涂成蓝色，棋子将沿着以下红色和蓝色边的次数移动：

• 当从顶点2移动到顶点3时，没有红色边和1条蓝色边，
• 当从顶点3移动到顶点2时，没有红色边和1条蓝色边，
• 当从顶点2移动到顶点1时，有1条红色边和0条蓝色边，
• 当从顶点1移动到顶点4时，有2条红色边和1条蓝色边，

总共3条红色边和3条蓝色边，满足条件。

另一种满足条件的方式是把第1条和第3条边涂成蓝色，第2条边涂成红色。没有其他方式可以满足它，所以答案是2。

样例输入2

3 10 10000
1 2 1 2 1 2 2 1 1 2
1 2
1 3

样例输出2

0

可能没有满足条件的涂色方式。

样例输入3

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

样例输出3

126

样例输入4

5 8 -1
1 4 1 4 2 1 3 5
1 2
4 1
3 1
1 5

样例输出4

2