Red and Blue Tree

AtCoder

IDabc222_e

Time2000ms

Memory256MB

Difficulty—

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

Samples

Input #1

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.
![image](https://img.atcoder.jp/ghi/f9b2b199fb6eedaca02e15ff556b72b1.png)
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 $2$.

Input #2

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

Output #2

0

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

Input #3

Output #3

Input #4

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

Output #4

API Response (JSON)

{
  "problem": {
    "name": "Red and Blue Tree",
    "description": {
      "content": "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",
      "description_type": "Markdown"
    },
    "platform": "AtCoder",
    "limit": {
      "time_limit": 2000,
      "memory_limit": 262144
    },
    "difficulty": "None",
    "is_remote": true,
    "is_sync": true,
    "sync_url": null,
    "sign": "abc222_e"
  },
  "statements": [
    {
      "statement_type": "Markdown",
      "content": "Given are a tree with $N$ vertices, a sequence of $M$ numbers $A=(A_1,\\ldots,A_M)$, and an integer $K$.  \nThe vertices are numbered $1$ through $N$, and the $i$\\-th edge connects Vertices $U_i$ and $V...",
      "is_translate": false,
      "language": "English"
    }
  ]
}

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