F - Graph Smoothing

Score : $600$ points

Problem Statement

We have a simple undirected graph with $N$ vertices and $M$ edges. The vertices are numbered $1$ through $N$ and the edges are numbered $1$ through $M$.
Edge $i$ connects Vertex $X_i$ and Vertex $Y_i$. Initially, Vertex $i$ has an integer $A_i$ written on it.
You will do the following operation $K$ times:

• Choose one from the $M$ edges uniformly at random and independently from other choices. Let $x$ be the arithmetic mean of the numbers written on the two vertices connected by that edge. Replace each number written on those vertices with $x$.

For each vertex $i$, find the expected value of the number written on that vertex after $K$ operations, and print it modulo $(10^9 + 7)$ as described in Notes.

Notes

When printing a rational number, first, represent it as a fraction $\frac{y}{x}$, where $x$ and $y$ are integers and $x$ is not divisible by $(10^9+7)$ (under the Constraints of this problem, such a representation is always possible).
Then, print the only integer $z$ between $0$ and $(10^9+6)$ (inclusive) such that $xz \equiv y \pmod {10^9+7}$.

Constraints

• $2 \le N \le 100$
• $1 \le M \le \frac{N(N - 1)}{2}$
• $0 \le K \le 10^9$
• $0 \le A_i \le 10^9$
• $1 \le X_i \le N$
• $1 \le Y_i \le N$
• The given graph is simple.
• All values in input are integers.

Input

Input is given from Standard Input in the following format:

$N$ $M$ $K$

$A_1$ $A_2$ $A_3$ $\dots$ $A_N$

$X_1$ $Y_1$

$X_2$ $Y_2$

$X_3$ $Y_3$

$\hspace{15pt} \vdots$

$X_M$ $Y_M$

Output

Print $N$ lines.
The $i$-th line should contain the expected value of the number written on that vertex after $K$ operations, modulo $(10^9 + 7)$, as described in Notes.

3 2 1
3 1 5
1 2
1 3

3
500000005
500000008

• If Edge $1$ is chosen in the only operation: the vertices written on Vertices $1, 2, 3$ will be $2, 2, 5$, respectively.
• If Edge $2$ is chosen in the only operation: the vertices written on Vertices $1, 2, 3$ will be $4, 1, 4$, respectively.

Thus, the expected values of the numbers written on Vertices $1, 2, 3$ are $3, \frac{3}{2}, \frac{9}{2}$, respectively.
If we express them modulo $(10^9 + 7)$ as described in Notes, they will be $3, 500000005, 500000008$, respectively.

3 2 2
12 48 36
1 2
1 3

750000036
36
250000031

• If Edge $1$ is chosen in the $1$-st operation: The numbers written on Vertices $1, 2, 3$ will be $30, 30, 36$, respectively.
  • If Edge $1$ is chosen in the $2$-nd operation: The numbers written on Vertices $1, 2, 3$ will be $30, 30, 36$, respectively.
  • If Edge $2$ is chosen in the $2$-nd operation: The numbers written on Vertices $1, 2, 3$ will be $33, 30, 33$, respectively.
• If Edge $2$ is chosen in the $1$-st operation: The numbers written on Vertices $1, 2, 3$ will be $24, 48, 24$, respectively.
  • If Edge $1$ is chosen in the $2$-nd operation: The numbers written on Vertices $1, 2, 3$ will be $36, 36, 24$, respectively.
  • If Edge $2$ is chosen in the $2$-nd operation: The numbers written on Vertices $1, 2, 3$ will be $24, 48, 24$, respectively.

Each of these four scenarios happen with probability $\frac{1}{4}$, so the expected values of the numbers written on Vertices $1, 2, 3$ are $\frac{123}{4}, \frac{144}{4} (=36), \frac{117}{4}$, respectively.

4 5 1000
578 173 489 910
1 2
2 3
3 4
4 1
1 3

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