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Antti H S Laaksonen 2017-02-18 14:00:23 +02:00
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luku17.tex
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@ -188,14 +188,13 @@ in the following order:
\end{center} \end{center}
The notation $x/y$ means that The notation $x/y$ means that
processing the node started at moment $x$ processing the node started
and ended at moment $y$. at time $x$ and finished at time $y$.
The following list contains the nodes Thus, the corresponding list is as follows:
sorted according to their ending times:
\begin{tabular}{ll} \begin{tabular}{ll}
\\ \\
node & ending time \\ node & processing time \\
\hline \hline
4 & 5 \\ 4 & 5 \\
5 & 6 \\ 5 & 6 \\
@ -218,7 +217,7 @@ forms the strongly connected components
of the graph. of the graph.
First, the algorithm reverses every First, the algorithm reverses every
edge in the graph. edge in the graph.
This makes sure that during the second search, This guarantees that during the second search,
we will always find strongly connected we will always find strongly connected
components that do not have extra nodes. components that do not have extra nodes.
@ -287,7 +286,7 @@ begins at node 3:
\end{tikzpicture} \end{tikzpicture}
\end{center} \end{center}
Note that since all edges were reversed, Note that since all edges are reversed,
the component does not ''leak'' to other parts in the graph. the component does not ''leak'' to other parts in the graph.
\begin{samepage} \begin{samepage}
@ -359,8 +358,7 @@ that create the remaining strongy connected components:
The time complexity of the algorithm is $O(n+m)$, The time complexity of the algorithm is $O(n+m)$,
because the algorithm because the algorithm
performs two depth-first searches and performs two depth-first searches.
both searches take $O(n+m)$ time.
\section{2SAT problem} \section{2SAT problem}
@ -390,8 +388,17 @@ L_1 = (x_2 \lor \lnot x_1) \land
(\lnot x_2 \lor \lnot x_3) \land (\lnot x_2 \lor \lnot x_3) \land
(x_1 \lor x_4) (x_1 \lor x_4)
\] \]
is true when $x_1$ and $x_2$ are false is true when the variables are assigned as follows:
and $x_3$ and $x_4$ are true.
\[
\begin{cases}
x_1 = \textrm{false} \\
x_2 = \textrm{false} \\
x_3 = \textrm{true} \\
x_4 = \textrm{true} \\
\end{cases}
\]
However, the formula However, the formula
\[ \[
L_2 = (x_1 \lor x_2) \land L_2 = (x_1 \lor x_2) \land
@ -400,14 +407,14 @@ L_2 = (x_1 \lor x_2) \land
(\lnot x_1 \lor \lnot x_3) (\lnot x_1 \lor \lnot x_3)
\] \]
is always false, regardless of how we is always false, regardless of how we
choose the values of the variables. assign the values.
The reason for this is that we cannot The reason for this is that we cannot
choose a value for variable $x_1$ choose a value for $x_1$
without creating a contradiction. without creating a contradiction.
If $x_1$ is false, both $x_2$ and $\lnot x_2$ If $x_1$ is false, both $x_2$ and $\lnot x_2$
should hold which is impossible, should be true which is impossible,
and if $x_1$ is true, both $x_3$ and $\lnot x_3$ and if $x_1$ is true, both $x_3$ and $\lnot x_3$
should hold which is also impossible. should be true which is also impossible.
The 2SAT problem can be represented as a graph The 2SAT problem can be represented as a graph
whose nodes correspond to whose nodes correspond to
@ -466,19 +473,21 @@ And the graph for the formula $L_2$ is:
\end{tikzpicture} \end{tikzpicture}
\end{center} \end{center}
The structure of the graph indicates whether The structure of the graph tells us whether
the corresponding 2SAT problem can be solved. it is possible to assign the values
If there is a variable $x_i$ such that of the variables so
both $x_i$ and $\lnot x_i$ belong to the that the formula is true.
same strongly connected component, It turns out that this can be done
then there are no solutions. exactly when there are no nodes
In this case, the graph contains $x_i$ and $\lnot x_i$ such that
a path from $x_i$ to $\lnot x_i$, both nodes belong to the
same strongly connected component.
If there are such nodes,
the graph contains
a path from $x_i$ to $\lnot x_i$
and also a path from $\lnot x_i$ to $x_i$, and also a path from $\lnot x_i$ to $x_i$,
so both $x_i$ and $\lnot x_i$ should hold so both $x_i$ and $\lnot x_i$ should be true
which is not possible. which is not possible.
However, if the graph does not contain
such a variable $x_i$, then there is always a solution.
In the graph of the formula $L_1$ In the graph of the formula $L_1$
there are no nodes $x_i$ and $\lnot x_i$ there are no nodes $x_i$ and $\lnot x_i$
@ -490,7 +499,7 @@ all nodes belong to the same strongly connected component,
so there are no solutions. so there are no solutions.
If a solution exists, the values for the variables If a solution exists, the values for the variables
can be found by processing the nodes of the can be found by going through the nodes of the
component graph in a reverse topological sort order. component graph in a reverse topological sort order.
At each step, we process a component At each step, we process a component
that does not contain edges that lead to an that does not contain edges that lead to an
@ -502,7 +511,7 @@ according to the values in the component,
and if they already have values, and if they already have values,
they remain unchanged. they remain unchanged.
The process continues until all variables The process continues until all variables
have been assigned a value. have been assigned values.
The component graph for the formula $L_1$ is as follows: The component graph for the formula $L_1$ is as follows:
\begin{center} \begin{center}
@ -533,8 +542,8 @@ All variables have been assigned a value,
so the remaining components $A$ and $B$ so the remaining components $A$ and $B$
do not change the variables. do not change the variables.
Note that this method, works because the Note that this method works, because the
structure of the graph is special. graph has a special structure.
If there are paths from node $x_i$ to node $x_j$ If there are paths from node $x_i$ to node $x_j$
and from node $x_j$ to node $\lnot x_j$, and from node $x_j$ to node $\lnot x_j$,
then node $x_i$ never becomes true. then node $x_i$ never becomes true.