C. DFS Trees
time limit per test
1 second
memory limit per test
256 megabytes
standard input
standard output

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:

vis := an array of length n
s := a set of edges

function dfs(u):
vis[u] := true
iterate through each edge (u, v) in the order from smallest to largest edge weight
if vis[v] = false
add edge (u, v) into the set (s)

function findMST(u):
reset all elements of (vis) to false
reset the edge set (s) to empty
return the edge set (s)

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.

5 5
1 2
3 5
1 3
3 2
4 2
10 11
1 2
2 5
3 4
4 2
8 1
4 5
10 5
9 5
8 2
5 7
4 6

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.