CF1707C.DFS Trees

You are given a connected undirected graph consisting of $n$ vertices and $m$ edges. The weight of the $i$ -th edge is $i$ .

Here is a wrong algorithm of finding a minimum spanning tree (MST) of a graph:

<pre class="verbatim"><br></br>vis := an array of length n<br></br>s := a set of edges<br></br><br></br>function dfs(u):<br></br>    vis[u] := true<br></br>    iterate through each edge (u, v) in the order from smallest to largest edge weight<br></br>        if vis[v] = false<br></br>            add edge (u, v) into the set (s)<br></br>            dfs(v)<br></br><br></br>function findMST(u):<br></br>    reset all elements of (vis) to false<br></br>    reset the edge set (s) to empty<br></br>    dfs(u)<br></br>    return the edge set (s)<br></br>

Each of the calls findMST(1), findMST(2), ..., findMST(n) gives you a spanning tree of the graph. Determine which of these trees are minimum spanning trees.

The first line of the input contains two integers $n$ , $m$ ( $2\le n\le 10^5$ , $n-1\le m\le 2\cdot 10^5$ ) — the number of vertices and the number of edges in the graph.

Each of the following $m$ lines contains two integers $u_i$ and $v_i$ ( $1\le u_i, v_i\le n$ , $u_i\ne v_i$ ), describing an undirected edge $(u_i,v_i)$ in the graph. The $i$ -th edge in the input has weight $i$ .

It is guaranteed that the graph is connected and there is at most one edge between any pair of vertices.

You need to output a binary string $s$ , where $s_i=1$ if findMST(i) creates an MST, and $s_i = 0$ otherwise.

Here is the graph given in the first example.

There is only one minimum spanning tree in this graph. A minimum spanning tree is $(1,2),(3,5),(1,3),(2,4)$ which has weight $1+2+3+5=11$ .

Here is a part of the process of calling findMST(1):

• reset the array vis and the edge set s;
• calling dfs(1);
• vis[1] := true;
• iterate through each edge $(1,2),(1,3)$ ;
• add edge $(1,2)$ into the edge set s, calling dfs(2):
  • vis[2] := true
  • iterate through each edge $(2,1),(2,3),(2,4)$ ;
  • because vis[1] = true, ignore the edge $(2,1)$ ;
  • add edge $(2,3)$ into the edge set s, calling dfs(3):
    • ...

In the end, it will select edges $(1,2),(2,3),(3,5),(2,4)$ with total weight $1+4+2+5=12>11$ , so findMST(1) does not find a minimum spanning tree.

It can be shown that the other trees are all MSTs, so the answer is 01111.