**Part 1: Problem Statement**

A tree $$$T=(V, E)$$$ has $$$n$$$ vertices and $$$n-1$$$ edges, the weight of each vertex $$$i$$$ is $$$a_i$$$.

For each edge $$$e$$$, you can determine its direction, i.e., for two vertices $$$u, v$$$, there are two states: $$$u \rightarrow v$$$ and $$$v \rightarrow u$$$. There are $$$2^{n-1}$$$ states in total.

For each state $$$S$$$, we define $$$f(S)$$$ as

$$$f(S) := \sum\limits_{(u, v) \in V \times V, v\,\text{is reachable from}\,u} |a_u - a_v|$$$.

Compute the sum of $$$f(S)$$$ over all $$$2^{n-1}$$$ states $$$S$$$, modulo $$$998244353$$$.

Example:

Suppose $$$n=3$$$, and two edges connect $$$(1, 2), (2, 3)$$$, respectively. $$$a_1 = 3, a_2 = 1, a_3 = 2$$$, the answer is $$$14$$$.

Constraints: $$$2 \leq n \leq 2 \cdot 10^5$$$, $$$1 \leq a_i \leq 10^9$$$.

**Part 2: My idea**

My only attempt is using the counting twice trick:

$$$ans = 2\sum\limits_{(u, v) \in V \times V, u < v}|a_u - a_v|2^{n-1-d(u, v)} = \sum\limits_{(u, v) \in V \times V, u < v}|a_u - a_v|2^{n-d(u, v)}$$$, but I don't know how to solve it in $$$O(n)$$$ or $$$O(nlogn)$$$.

Maybe we could use $$$d(u, v) = d(u) + d(v) - 2d(lca(u, v))$$$?

I think you can use Centroid Decomposition + Heuristic Merge to solve it.

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I upvoted. Would you please elaborate on this please?

Start with Centroid Decomposition (you can think of it as cdq on tree).

Subsequently, at Centroid Decomposition, the points of the two subtrees are sorted by ai and multiplied by 2^(size of tree-dep of node) to make a prefix sum respectively, so that |ax-ay| becomes ∑ two ax-ay. I'm not sure how to analyze the complexity, maybe nlog^2n .

I think you are in the right direction. Please give me some times to figure out! Thanks for your idea.

https://ac.nowcoder.com/discuss/1128009

I found editorial, it is the same as my idea.

Thank you ShaoNian, I think you are super clever. I am not familiar with centroid decomposition at all, so it takes time for me to understand it. Hope you perform well in your coming CSP tests!

PS. Are you the real ShaoNian?

i'm Karry5307 and i'm good at ssh

## I will back in NOI2077