IFCHAIN - If Chain
Consider the following code:
where a,b and c are boolean variables. If we run this code in C++, the function foo() is called if and only if all three variables are true. However, recently a new language - C-- - is being developed. In this language, when an if() evaluates to false, only the statement directly following it is not executed; for example, if a was false, the program would jump from if(a) to if(c).
Using this convention, there are 5 different possible assignments of truth values to the variables a,b,c which end up calling foo(). Considering a,b,c as three bits in that order, they are 111, 101, 100, 011 and 001.
Given n boolean variables within a chain of m if()'s, where the variables within one if() are separated using logical or, count the number of ways to assign truth values to them which end up calling the function foo() (the call to foo() is after the last if()).
The first line of the input is the number of test cases 1 ≤ T ≤ 30. T test cases follow.
The first line of each test case contains two nonnegative integers n ≤ 105 - the number of boolean variables (they are numbered 1 through n) - and m ≤ 105 - the number of if()'s. m lines follow, the i-th line describing the i-th if(). The first integer in each line is a positive integer ki - the number of variables in the i-th if() (implicitly separated by the logical or operator) - followed by ki positive integers in the range [1,n]: the variables in the i-th if(). Not all variables need necessarily appear within the if() chain, and the variables within one if() need not be distinct.
The sum of ki within a test case will not exceed 5*105. Additionally, the sum of n,m and ki within a single input file will not exceed 2*106.
The input is quite large - make sure to read it efficiently.
For each case, output the string "Case x: y" in a single line, where x is the number of the test case, starting from 1, and y is the number of ways of assigning truth values to the n boolean variables (out of 2n), which when run in C-- end up calling foo(), modulo 109+9.
Input: 2 3 3 1 1 1 2 1 3 5 3 2 1 2 3 1 3 5 2 2 4 Output: Case 1: 5 Case 2: 24