Problem:
Score : 300 points

Problem Statement

We have N switches with "on" and "off" state, and M bulbs. The switches are numbered 1 to N, and the bulbs are numbered 1 to M.

Bulb i is connected to k_i switches: Switch s_{i1}, s_{i2}, ..., and s_{ik_i}. It is lighted when the number of switches that are "on" among these switches is congruent to p_i modulo 2.

How many combinations of "on" and "off" states of the switches light all the bulbs?

Constraints

1 \leq N, M \leq 10

1 \leq k_i \leq N

1 \leq s_{ij} \leq N

s_{ia} \neq s_{ib} (a \neq b)

p_i is 0 or 1.

All values in input are integers.

Input

Input is given from Standard Input in the following format:

N M
k_1 s_{11} s_{12} ... s_{1k_1}
:
k_M s_{M1} s_{M2} ... s_{Mk_M}
p_1 p_2 ... p_M

Output

Print the number of combinations of "on" and "off" states of the switches that light all the bulbs.

Sample Input 1

2 2
2 1 2
1 2
0 1

Sample Output 1

1

Bulb 1 is lighted when there is an even number of switches that are "on" among the following: Switch 1 and 2.

Bulb 2 is lighted when there is an odd number of switches that are "on" among the following: Switch 2.

There are four possible combinations of states of (Switch 1, Switch 2): (on, on), (on, off), (off, on) and (off, off). Among them, only (on, on) lights all the bulbs, so we should print 1.

Sample Input 2

2 3
2 1 2
1 1
1 2
0 0 1

Sample Output 2

0

Bulb 1 is lighted when there is an even number of switches that are "on" among the following: Switch 1 and 2.

Bulb 2 is lighted when there is an even number of switches that are "on" among the following: Switch 1.

Bulb 3 is lighted when there is an odd number of switches that are "on" among the following: Switch 2.

Switch 1 has to be "off" to light Bulb 2 and Switch 2 has to be "on" to light Bulb 3, but then Bulb 1 will not be lighted. Thus, there are no combinations of states of the switches that light all the bulbs, so we should print 0.

Sample Input 3

5 2
3 1 2 5
2 2 3
1 0

Sample Output 3

8

Solution in Scala (2.11.7) (judge verdict: Accepted)

import java.util.Scanner

import scala.annotation.tailrec
import scala.collection.mutable.ArrayBuffer
import scala.math._

object Main {
  def main(args: Array[String]): Unit = {
    val sc = new Scanner(System.in)
    val writer = new java.io.PrintWriter(System.out)
    writer.println(solve(sc))
    writer.flush()
  }

  case class LightBulb(switches: Seq[Int], condition: Int)

  def solve(sc: Scanner): String = {
    val (n, m, lightbulbs) = parseInput(sc)
    debug(n, m, lightbulbs)

    var countOfAllOn = 0
    getSwitchPatterns(n).foreach { case (_, sp) =>
      val countOfOn = lightbulbs.foldLeft(0) { (acc, lightbulb) =>
        if (sp.zipWithIndex.filter { case (_, i) => lightbulb.switches.contains(i) }.map(_._1).sum % 2 == lightbulb.condition) {
          acc + 1
        } else {
          acc
        }
      }
      if (countOfOn == lightbulbs.size) countOfAllOn += 1
    }
    countOfAllOn.toString
  }

  def getSwitchPatterns(n: Int): Vector[(Int, ArrayBuffer[Int])] = {
    @tailrec
    def dfs(sp: Vector[(Int, ArrayBuffer[Int])], n: Int): Vector[(Int, ArrayBuffer[Int])] = {
      if (n > 0) {
        sp.foreach { case (i, a) =>
          a += floor(i / (pow(2, n) / 2) % 2).toInt
        }
        dfs(sp, n - 1)
      } else sp
    }

    val list = (0 until pow(2, n).toInt).map(i => (i, ArrayBuffer.empty[Int])).toVector
    dfs(list, n)
  }

  def parseInput(sc: Scanner) = {
    val (n, m) = (sc.nextInt(), sc.nextInt())
    val switchesList = (0 until m).map { _ =>
      val k = sc.nextInt()
      IndexedSeq.fill(k)(sc.nextInt() - 1)
    }
    val conditionList = IndexedSeq.fill(m)(sc.nextInt())
    val lightbulbs = (0 until m).foldLeft(ArrayBuffer.empty[LightBulb]) { (acc, i) =>
      acc += LightBulb(switchesList(i), conditionList(i))
      acc
    }.toIndexedSeq
    (n, m, lightbulbs)
  }

  private def debug(x: Any): Unit = {
    if (System.getenv("LOCAL_DEBUG") != null) println(x)
  }
}