310002: CF1770E. Koxia and Tree
Memory Limit:256 MB
Time Limit:3 S
Judge Style:Text Compare
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Description
Koxia and Tree
题意翻译
有一棵 $n$ 个结点的无根树,边从 $1$ 到 $n-1$ 标号,第 $i$ 条边连接结点 $u_i$ 和 $v_i$。 有 $k$ 个结点带有标记,分别是 $a_1,a_2,\cdots,a_k$($\forall i\not= j,a_i\not = a_j$)。接下来,进行如下操作: 1. 给每条边随机定向。 2. **按照编号顺序**,对于每一条边 $u\to v$,如果结点 $u$ 有标记而 $v$ 没有,那么就把 $u$ 的标记转移到 $v$ 上。 3. 随机选取两个不同的标记点,计算它们的距离。 你需要给出第三步中距离的期望值。对 998244353 取模。题目描述
Imi has an undirected tree with $ n $ vertices where edges are numbered from $ 1 $ to $ n-1 $ . The $ i $ -th edge connects vertices $ u_i $ and $ v_i $ . There are also $ k $ butterflies on the tree. Initially, the $ i $ -th butterfly is on vertex $ a_i $ . All values of $ a $ are pairwise distinct. Koxia plays a game as follows: - For $ i = 1, 2, \dots, n - 1 $ , Koxia set the direction of the $ i $ -th edge as $ u_i \rightarrow v_i $ or $ v_i \rightarrow u_i $ with equal probability. - For $ i = 1, 2, \dots, n - 1 $ , if a butterfly is on the initial vertex of $ i $ -th edge and there is no butterfly on the terminal vertex, then this butterfly flies to the terminal vertex. Note that operations are sequentially in order of $ 1, 2, \dots, n - 1 $ instead of simultaneously. - Koxia chooses two butterflies from the $ k $ butterflies with equal probability from all possible $ \frac{k(k-1)}{2} $ ways to select two butterflies, then she takes the distance $ ^\dagger $ between the two chosen vertices as her score. Now, Koxia wants you to find the expected value of her score, modulo $ 998\,244\,353^\ddagger $ . $ ^\dagger $ The distance between two vertices on a tree is the number of edges on the (unique) simple path between them. $ ^\ddagger $ Formally, let $ M = 998\,244\,353 $ . It can be shown that the answer can be expressed as an irreducible fraction $ \frac{p}{q} $ , where $ p $ and $ q $ are integers and $ q \not \equiv 0 \pmod{M} $ . Output the integer equal to $ p \cdot q^{-1} \bmod M $ . In other words, output such an integer $ x $ that $ 0 \le x < M $ and $ x \cdot q \equiv p \pmod{M} $ .输入输出格式
输入格式
The first line contains two integers $ n $ , $ k $ ( $ 2 \leq k \leq n \leq 3 \cdot {10}^5 $ ) — the size of the tree and the number of butterflies. The second line contains $ k $ integers $ a_1, a_2, \dots, a_k $ ( $ 1 \leq a_i \leq n $ ) — the initial position of butterflies. It's guaranteed that all positions are distinct. The $ i $ -th line in following $ n − 1 $ lines contains two integers $ u_i $ , $ v_i $ ( $ 1 \leq u_i, v_i \leq n $ , $ u_i \neq v_i $ ) — the vertices the $ i $ -th edge connects. It is guaranteed that the given edges form a tree.
输出格式
Output a single integer — the expected value of Koxia's score, modulo $ 998\,244\,353 $ .
输入输出样例
输入样例 #1
3 2
1 3
1 2
2 3
输出样例 #1
748683266
输入样例 #2
5 3
3 4 5
1 2
1 3
2 4
2 5
输出样例 #2
831870296