All Tracks Algorithms Graphs Breadth First Search Problem

Delete and Cut Game

Algorithms, Breadth-first search, Graphs, Probablity


You are given a connected graph with \(N\) nodes and \(M\) edges. Two players \(A\) and \(B\) are playing a game with this graph. A person \(X\) chooses an edge uniformly, randomly, and removes it. If the size (number of nodes in component) of two non-empty connected component created are EVEN, then \(A\) wins otherwise player \(B\) wins.

Find the probability of winning the game for \(A\) and \(B\). The probability is of the \(\frac P Q\) form where \(P\) and \(Q\) are both coprimes. Print \(PQ^{-1}\ modulo\ 1000000007\).
Note: \(X\) can only select the edges which divide the graph into two non-empty connected components after they are removed. If no such edge is present in the graph, then the probability to win can be 0 for both \(A\) and \(B\).

Input format

  • The first line contains two space-separated integers \(N\ M\) denoting the number of nodes and edges.
  • The next \(M\) lines contain two space-separated integers \(u\ v\) denoting an edge between node \(u\) and node \(v\).
    Note: The graph does not have multiple edges between two vertices

Output format
Print two space-separated integers denoting the probability of winning for \(A\) and \(B\) respectively.

\(1 \le  N,\ M \le 1e5\)

6 7
1 2
2 3
3 1
4 5
5 6
6 4
1 4
0 1

X can choose only edge 1-4 and B wins in that case.

Time Limit: 1.0 sec(s) for each input file.
Memory Limit: 256 MB
Source Limit: 1024 KB

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