\def\thesection{\Roman {section}}
\def\thesubsection{\it \Alph{subsection}.}
\def\thesubsubsection{\thesubsection.\arabic{subsubsection}}

	
\newtheorem{corollary}{Corollary}
\newtheorem{lemma}{Lemma}
\newtheorem{theorem}{Theorem}
\newtheorem{claim}{Claim}
\newtheorem{proposition}{Proposition}
\newcommand{\proof}{\vspace*{-2.6ex}\noindent {\bf Proof. }}
\newcommand{\proof}{\noindent {\bf Proof. }}
\newcommand{\proofsketch}{\vspace*{-2.6ex}\noindent {\bf Proof sketch. }}
\newcommand{\QED}{\hfill\rule[0.2ex]{1.5ex}{1.5ex}}    

\newcommand{\bgeq}{\begin{equation}}
\newcommand{\eneq}{\end{equation}}


\newcommand{\myheader}[1]{\noindent {\bf {#1}}}
\newcommand{\myrunninghead}[1]{\markright{\protect\small\protect\sl{#1}}}


\newcommand{\correspondence}{
\hspace*{-0.3in}\begin{minipage}[t]{7in}{\small \lineskip 0.1ex
\rule{\textwidth}{1pt}\\
Computer Science \& Electrical Engineering Department,
University of Maryland Baltimore County,\\ 
1000 Hilltop Circle,
Baltimore, MD 21250, U.S.A.; Email: dasgupta@cs.umbc.edu\\
\rule{\textwidth}{1pt}
}
\end{minipage}
}

\newcommand{\cell}[1]{c_{#1}}

\newcommand{\sensor}[1]{s_{#1}}

\newcommand{\costs}[1]{A_{#1}}
\newcommand{\costr}[1]{R\textsc{x}_{#1}}
\newcommand{\costt}[2]{T\textsc{x}_{#1,#2}}

\newcommand{\minrate}[1]{\rho_{#1}^{\textsc{min}}}
\newcommand{\maxrate}[1]{\rho_{#1}^{\textsc{max}}}


\newcommand{\objectSpace}{S}
\newcommand{\replicaSpace}{R}
\newcommand{\diskSpace}{\Delta}
\newcommand{\emptySpace}{A}

\newcommand{\fact}{\noindent{\bf Fact:}}

%% Complement of a set
\newcommand{\comp}[1]{\overline{#1}}
%% cutset (S,V-S)
\newcommand{\cutInSet}[1]{(\comp{#1}, #1)}
\newcommand{\cutOutSet}[1]{(#1,\comp{#1})}
%% cut set capacity
\newcommand{\cutCap}[2]{\omega_{#1}(#2)}
%%
\newcommand{\multiGraph}[1]{\hat{#1}}
%%
\newcommand{\transposeGraph}[1]{#1^{T}}
%%
\newcommand{\partition}[1]{\Pi(\phi_{#1})}
\newcommand{\superIn}[1]{\phi_{#1}}
\newcommand{\superOut}[1]{\comp{\phi_{#1}}}
\newcommand{\energy}[1]{{\cal E}_{#1}}
\newcommand{\renergy}[1]{{\cal E}^r[#1]}
\newcommand{\rev}{{\cal E}^r}
\newcommand{\ienergy}{{\cal E}}
\newcommand{\schedule}{{\cal S}}
\newcommand{\superi}[1]{\phi_{#1}}
\newcommand{\super}{\phi}

\newcommand{\atree}[1]{{\cal T}_{#1}}
\newcommand{\as}{{\cal T}}

\newcommand{\treecount}[1]{{\lambda}_{#1}}
\newcommand{\select}{{\cal A}}
\newcommand{\selecti}[1]{A_{#1}}
\newcommand{\arlrs}{$\select$--\textsc{r}--\textsc{lrs}}
\newcommand{\rlrs}{$\select$--\textsc{Randomized}--\textsc{lrs}}
\newcommand{\alrs}{$\select$--\textsc{lrs}}
\newcommand{\lrs}{\textsc{lrs}}
\newcommand{\mlda}{\textsc{mlda}}
\newcommand{\gcmlda}{\textscm{G--cmlda}}
\newcommand{\icmlda}{\textsc{I--cmlda}}
