- Time limit: 6.00 s
- Memory limit: 512 MB
SAT solvers find satisfying assignments to Boolean formulas. The formulas are given in the conjunctive normal form:
- the formula is a conjunction of m clauses: c_1 \land c_2 \land \ldots
- each clause c_j is a disjunction of literals: \ell_1 \lor \ell_2 \lor \ldots
- each literal \ell_k is either a Boolean variable x_i or is negation \lnot x_i.
Here \land is the "AND" operation, \lor is the "OR" operation, and \lnot is the "NOT" operation. We will always assume that the variables are called x_1, x_2, \ldots, x_n.
For example, here is a Boolean formula in the conjunctive normal form; we have m = 3 clauses and n = 3 variables: (x_1 \lor x_2 \lor x_3) \land (\lnot x_1 \lor \lnot x_2 \lor \lnot x_3) \land (\lnot x_1 \lor x_2 \lor x_3). This formula is satisfiable; we can find the following variable assignment that makes the formula true: x_1 = \mathrm{false},\ x_2 = \mathrm{true},\ x_3 = \mathrm{true}. Such formulas are commonly represented in the standard DIMACS format: the first line contains the magic words "p" ("problem") and "cnf" ("conjunctive normal form"), and then it is followed by the number of variables n and the number of clauses m. This is followed by m lines, each describing one clause. The line contains just one number per literal: a positive literal x_i is written as i, and a negated literal \lnot x_i is written as -i. The clause is terminated with a 0.
For example, the above formula would be written as follows in the DIMACS format:
p cnf 3 3 1 2 3 0 -1 -2 -3 0 -1 2 3 0
Now we could take a commonly used SAT solver, for example "picosat", and ask it to solve this. We will get e.g. the following output:
s SATISFIABLE v -1 2 3 0
Here the first line of output that starts with an "s" indicates that this formula was indeed satisfiable, and the second line of output that starts with a "v" gives the variable assingment. For each i = 1, \dotsc, n, we write i if x_i is true and -i if x_i is false. Again, the list is terminated with a 0.
Your task here is to write a SAT solver that solves formulas in certain special cases.
Input
The input is a Boolean formula in the conjunctive normal form, given in the DIMACS format.
We promise the following:
- Each clause contains exactly 3 literals.
- Each variable occurs at most 3 times in the entire formula (either negated or unnegated).
- The same variable does not occur multiple times in a one clause.
- The formula will be satisfiable; there will be a way to solve it.
Output
Output a satisfying assignment in a format similar to the one used in the "picosat" example above. There should be two lines of output: first one that says "s SATISFIABLE", and then one that starts with "v", then contains n values that indicate the satisfying assignment, and finally a "0".
Limits
- Number of variables: 3 \le n \le 10^5
- Number of clauses: 3 \le m \le 10^4
Example
Input:
p cnf 8 5 4 -5 3 0 -5 -2 -4 0 8 -1 2 0 -3 5 6 0 2 1 8 0
Output:
s SATISFIABLE v -1 2 -3 -4 -5 6 -7 8 0
Note that all variables occur at most 3 times in the entire formula: e.g. variable x_5 occurs 3 times and x_8 occurs 2 times.