The Door Problem
Contest: ICM Technex 2017 and Codeforces Round 400 (Div. 1 + Div. 2, combined)
Difficulty rating: 2000
Tags: 2-sat, dfs and similar, dsu, graphs

Moriarty has trapped n people in n distinct rooms in a hotel. Some rooms are locked, others are unlocked. But, there is a condition that the people in the hotel can only escape when all the doors are unlocked at the same time. There are m switches. Each switch control doors of some rooms, but each door is controlled by exactly two switches.

You are given the initial configuration of the doors. Toggling any switch, that is, turning it ON when it is OFF, or turning it OFF when it is ON, toggles the condition of the doors that this switch controls. Say, we toggled switch 1, which was connected to room 1, 2 and 3 which were respectively locked, unlocked and unlocked. Then, after toggling the switch, they become unlocked, locked and locked.

You need to tell Sherlock, if there exists a way to unlock all doors at the same time.

Input format:
First line of input contains two integers n and m (2 ≤ n ≤ 105, 2 ≤ m ≤ 105) — the number of rooms and the number of switches.

Next line contains n space-separated integers r1, r2, ..., rn (0 ≤ ri ≤ 1) which tell the status of room doors. The i-th room is locked if ri = 0, otherwise it is unlocked.

The i-th of next m lines contains an integer xi (0 ≤ xi ≤ n) followed by xi distinct integers separated by space, denoting the number of rooms controlled by the i-th switch followed by the room numbers that this switch controls. It is guaranteed that the room numbers are in the range from 1 to n. It is guaranteed that each door is controlled by exactly two switches.

Output format:
Output "YES" without quotes, if it is possible to open all doors at the same time, otherwise output "NO" without quotes.

Constraints:
Time limit: 2.0 s
Memory limit: 256.0 MB

Note:
In the second example input, the initial statuses of the doors are [1, 0, 1] (0 means locked, 1 — unlocked).

After toggling switch 3, we get [0, 0, 0] that means all doors are locked.

Then, after toggling switch 1, we get [1, 1, 1] that means all doors are unlocked.

It can be seen that for the first and for the third example inputs it is not possible to make all doors unlocked.

Example input:
3 3
1 0 1
2 1 3
2 1 2
2 2 3

Example output:
NO

Example input:
3 3
1 0 1
3 1 2 3
1 2
2 1 3

Example output:
YES

Editorial:
Hint : Try to model the situation as a graph with rooms as edges and switches as nodes.
Editorial : All rooms are represented as edges. Mark the edges as 1 if the room is open else mark the edge as closed. The answer will be "YES" if you can color the graph in such a manner that the edges having value 0 have both nodes under different color (if the door is locked then one of the switches should be selected) and the edges having 1 have both nodes under same color (if the door is unlocked you should either select both switches or neither of them). For checking the same start a bfs from switch 1 and toggle it and proceed, if you are able to color all of the switches then the answer is "YES" else it is not possible.
Complexity: O(N + M)
See the setter's solution for implementation details.