Score : $500$ points

问题描述

给定一棵具有 $N$ 个顶点的树，一个包含 $M$ 个数的序列 $A=(A_1,\ldots,A_M)$，以及一个整数 $K$。

顶点按 $1$ 到 $N$ 编号，第 $i$ 条边连接顶点 $U_i$ 和 $V_i$。

我们将这棵树的 $N-1$ 条边分别涂成红色或蓝色。在所有 $2^{N-1}$ 种涂色方式中，找出满足以下条件的方式的数量，并对 $998244353$ 取模的结果。

条件：

• 首先，在顶点 $A_1$ 上放置一个棋子，按照顺序对于每个 $i=1,\ldots,M-1$，沿着最短路径从顶点 $A_i$ 移动到顶点 $A_{i+1}$。
• 在完成所有这些移动后，令 $R$ 表示棋子经过红色边的次数，$B$ 表示经过蓝色边的次数，则要求满足 $R-B=K$。

以上为通义千问 qwen-max 翻译，仅供参考。

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

update @ 2024/3/10 09:47:32