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 clause: $3 \le n \le 10^4$
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.