A14087. BFS Trees

We define a spanning tree of a graph to be a BFS tree rooted at vertex $s$ if and only if for every node $t$ the shortest distance between $s$ and $t$ in the graph is equal to the shortest distance between $s$ and $t$ in the spanning tree.

Given a graph, we define $f(x,y)$ to be the number of spanning trees of that graph that are BFS trees rooted at vertices $x$ and $y$ at the same time.

You are given an undirected connected graph with $n$ vertices and $m$ edges. Calculate $f(i,j)$ for all $i$ , $j$ by modulo $998\,244\,353$ .

The first line contains two integers $n$ , $m$ ( $1 \le n \le 400$ , $0 \le m \le 600$ ) — the number of vertices and the number of edges in the graph.

The $i$ -th of the next $m$ lines contains two integers $a_i$ , $b_i$ ( $1 \leq a_i, b_i \leq n$ , $a_i < b_i$ ), representing an edge connecting $a_i$ and $b_i$ .

It is guaranteed that all edges are distinct and the graph is connected.

Print $n$ lines, each consisting of $n$ integers.

The integer printed in the row $i$ and the column $j$ should be $f(i,j) \bmod 998\,244\,353$ .

The following picture describes the first example.


The tree with red edges is a BFS tree rooted at both $1$ and $2$ .


Similarly, the BFS tree for other adjacent pairs of vertices can be generated in this way.