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\newtheorem{definition}{Definition}[section] 

\newtheorem{lemma}{Lemma}[section] 

\newtheorem{corollary}{Corollary}[section] 

\newtheorem{theorem}{Theorem}[section] 

\newtheorem{example}{Example}[section] 

\newtheorem{proposition}{Proposition}[section] 
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\jairheading{24}{2005}{1-15}{04/05}{08/05}


\ShortHeadings{Binary Encodings of Non-binary CSPs: Algorithms \&
Experimental Results} {Samaras, \& Stergiou}








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$(1,*,*)$%
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{\newpage\clearpage
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$(2,*,*)$%
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{\newpage\clearpage
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$(*)$%
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{\newpage\clearpage
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$v_3$%
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{\newpage\clearpage
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$(x_6,x_7,x_3)$%
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{\newpage\clearpage
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$D(v_1)$%
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline2042}%
$(*,*,0)$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline2044}%
$(*,*,1)$%
\lthtmlinlinemathZ
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline2046}%
$(*,*,2)$%
\lthtmlinlinemathZ
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline2048}%
$O(ed)$%
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{\newpage\clearpage
\lthtmlfigureA{figure377}%
\begin{figure}
\begin{center}
\begin{tabbing}
16 16\= 16 \= 16 \= 16 \= 16 \= 16 \= 16 \= 16 \= 16 \= \kill
{\bf function} $PW-AC$\\
1:\>$Q\leftarrow \emptyset$\\
2:\>initialize all group counters to 0\\
3:\>{\bf for} each variable $v_i$\\
4:\>\>{\bf for} each variable $v_j$\  constrained with $v_i$\\
5:\>\>\>{\bf for} each tuple $\tau\in D(v_i)$\\
6:\>\>\>\>\>$counter(GroupOf(S(v_i,v_j),\tau))\leftarrow counter(GroupOf(S(v_i,v_j),\tau))+1$\\
7:\>{\bf for} each variable $v_i$\\
8:\>\>{\bf for} each variable $v_j$\  constrained with $v_i$\\
9:\>\>\>{\bf for} each group $s_{l}(v_i,v_j)$\\
10:\>\>\>\>{\bf if} $counter(s_{l}(v_i,v_j))=0$\\
11:\>\>\>\>\>put $s_{l}(v_i,v_j)$\  in $Q$\\
12:\>{\bf return} $Propagation$\\
\\
{\bf function} $Propagation$\\
13:\>{\bf while} $Q$\  is not empty\\
14:\>\>pop group $s_{l}(v_i,v_j)$\  from $Q$\\
15:\>\>\>$\delta\leftarrow \emptyset$\\
16:\>\>\>$\delta\leftarrow Revise(v_i,v_j,s_{l}(v_i,v_j))$\\
17:\>\>\>\>{\bf if} $D(v_j)$\  is empty {\bf return} INCONSISTENCY\\
18:\>\>\>\>{\bf for} each group $s_{l'}(v_j,v_k)$\  in $\delta$\  put $s_{l'}(v_j,v_k)$\  in $Q$\\
19:\>{\bf return} CONSISTENCY\\
\\
{\bf function} $Revise(v_i,v_j,s_{l}(v_i,v_j))$\\
20:\>{\bf for} each tuple $\tau \in D(v_j)$\  where $\tau\in sup(s_{l}(v_i,v_j))$\  \\
21:\>\>remove $\tau$\  from $D(v_j)$\\
22:\>\>{\bf for} each group $s_{l'}(v_j,v_k)$\  that includes $\tau$\\
23:\>\>\>$counter(s_{l'}(v_j,v_k))\leftarrow counter(s_{l'}(v_j,v_k))-1$\\
24:\>\>\>{\bf if} $counter(s_{l'}(v_j,v_k))=0$\\
25:\>\>\>\>add $s_{l'}(v_j,v_k)$\  to $\delta$\\
26:\>{\bf return} $\delta$
\par
\end{tabbing}
\vspace{-1mm} 
\end{center}
\end{figure}%
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$prob5$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

\stepcounter{subsection}
\stepcounter{paragraph}
\stepcounter{paragraph}
\stepcounter{section}
\stepcounter{section}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline2826}%
$|T|$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline2830}%
$|T|-currentSupport_{x_i,a,c}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline2836}%
$\tau>currentSupport_{x_i,a,c}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline2838}%
$\tau-currentSupport_{x_i,a,c}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline2842}%
$|T|-currentSupport_{x_i,a,v_c}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline2844}%
$\tau-currentSupport_{x_i,a,v_c}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline2846}%
$currentSupport_{x_i,a,c}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline2848}%
$currentSupport_{x_i,a,v_c}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline2874}%
$j$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline2910}%
$\{0,\ldots,9\}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline2918}%
$vars(c_1)=\{x_1,x_2,x_3\}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline2920}%
$vars(c_2)=\{x_1,x_2,x_4\}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline2926}%
$(x_1,1)$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline2932}%
$(x_2,0)$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline2938}%
$0,\ldots,9$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline2946}%
$(x_2,1)$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline2950}%
$D(x_1)$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}


\end{document}
