423 lines
14 KiB
TeX
423 lines
14 KiB
TeX
% -*- root: main.tex -*-
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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% NOTATION
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%Circuits
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\newcommand{\caseheading}[1]{\smallskip \noindent \textbf{#1}.~}
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%%%%%
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\newcommand{\wElem}{w} %an element of \vct{w}
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\newcommand{\suchthat}{\;|\;} %such that
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\newcommand{\kElem}{k}%the kth element
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%RA-to-Poly Notation
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\newcommand{\polyinput}[2]{\left(#1,\ldots, #2\right)}
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\newcommand{\numvar}{n}
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\newcommand{\rel}{R}
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\newcommand{\prel}{\mathcal{\rel}}
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\newcommand{\reli}{S}
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\newcommand{\relii}{T}
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\newcommand{\db}{D}
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\newcommand{\idb}{\Omega}
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\newcommand{\pdb}{\mathcal{D}}
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\newcommand{\pxdb}{\mathbf{D}}
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\newcommand{\nxdb}{D(\vct{X})}%\mathbb{N}[\vct{X}] db
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\newcommand{\tset}{\mathcal{T}}%the set of tuples in a database
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\newcommand{\pd}{\vct{P}}%pd for probability distribution
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\newcommand{\eval}[1]{\llbracket #1 \rrbracket}%evaluation double brackets
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\newcommand{\evald}[2]{\eval{{#1}}_{#2}}
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\newcommand{\query}{Q}
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\newcommand{\join}{\Join}
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\newcommand{\select}{\sigma}
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\newcommand{\project}{\pi}
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\newcommand{\union}{\cup}
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\newcommand{\sch}{sch}
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\newcommand{\attr}[1]{attr\left(#1\right)}
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\newcommand{\rw}{\textbf{W}}%\rw for random world
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\newcommand{\graph}[1]{G^{(#1)}}
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\newcommand{\eset}[1]{S^{(#1)}} %edge set for arbitrary subgraph
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\newcommand{\linsys}[1]{LS(\graph{#1})}
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\newcommand{\lintime}[1]{LT^{\graph{#1}}}
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\newcommand{\aug}[1]{AUG^{\graph{#1}}}
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\newcommand{\mtrix}[1]{M_{#1}}
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\newcommand{\dtrm}[1]{Det\left(#1\right)}
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\newcommand{\tuple}[1]{\left<#1\right>}
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\newcommand{\indicator}[1]{\onesymbol\inparen{#1}}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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% Query Classes
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\newcommand{\qClass}{\mathcal{Q}}
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\newcommand{\raPlus}{\ensuremath{\mathcal{RA}^{+}}\xspace}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%Approx Alg
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\newcommand{\randvar}{Y}
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\newcommand{\coeffset}{S}
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\newcommand{\distinctvars}{d}
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\newcommand{\coeffitem}[1]{c_{#1}\cdot\prob^{\distinctvars_{#1}}}
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\newcommand{\unidist}[1]{Uniform\left(#1\right)}
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\newcommand{\samplesize}{N}
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\newcommand{\setsize}{m}
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\newcommand{\empmean}{\overline{\vct{\randvar}}}
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\newcommand{\setsum}{SUM}
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\newcommand{\ave}[1]{AVG(#1)}
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\newcommand{\hoeffestsum}{EST_{\setsum}}
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\newcommand{\error}{\epsilon}
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\newcommand{\conf}{\delta}
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%Pseudo Code Notation
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\newcommand{\plus}{\texttt{+}}
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\newcommand{\mult}{\texttt{\times}}
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\newcommand{\algname}[1]{\textsc{#1}\xspace}
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\newcommand{\approxq}{\algname{Approximate$\rpoly$}}
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\newcommand{\onepass}{\algname{OnePass}}
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\newcommand{\sampmon}{\algname{SampleMonomial}}
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\newcommand{\reduce}{\algname{Reduce}}
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\newcommand{\ceil}[1]{\left\lceil #1 \right\rceil}
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\newcommand{\vari}[1]{\texttt{#1}\xspace}
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\newcommand{\accum}{\vari{acc}}
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\newcommand{\numsamp}{\vari{N}}
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\newcommand{\numedge}{m}
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\newcommand{\bivec}{\vari{b}_{\vari{vec}}}
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%expression tree T
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\newcommand{\etree}{\vari{T}}
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\newcommand{\stree}{\vari{S}}
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\newcommand{\lchild}{\vari{L}}
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\newcommand{\rchild}{\vari{R}}
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%members of T
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\newcommand{\val}{\vari{val}\xspace}
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\newcommand{\wght}{\vari{weight}\xspace}
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\newcommand{\vpartial}{\vari{partial}\xspace}
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%types of T
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\newcommand{\var}{\textsc{var}\xspace}
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\newcommand{\tnum}{\textsc{num}\xspace}
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%%%%%%%
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\renewcommand{\algorithmicrequire}{\textbf{Input:}}
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\renewcommand{\algorithmicensure}{\textbf{Output:}}
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\newcommand{\smb}{\poly\left(\vct{X}\right)}%smb for standard monomial basis
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\newcommand{\smbOf}[1]{\textsc{SMB}(#1)}
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\newcommand{\etreeset}[1]{\vari{ET}\left(#1\right)}
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%\expandtree is a placeholder until I change other files with the new macro name \expansion
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\newcommand{\expandtree}[1]{\vari{E}(#1)}
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\newcommand{\expansion}[1]{\vari{E}(#1)}
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\newcommand{\elist}[1]{\vari{List}\pbox{#1}}
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\newcommand{\equivtree}{\vari{EET}}
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%expandtree tuple elements:
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\newcommand{\monom}{\vari{v}}
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\newcommand{\coef}{\vari{c}}
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%----------------------------------
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\newcommand{\abs}[1]{\left|#1\right|}
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\newcommand{\func}[1]{\textsc{#1}\xspace}
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\newcommand{\polyf}{\func{poly}}
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\newcommand{\evalmp}{\func{eval}}
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\newcommand{\degree}{\func{deg}}
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\newcommand{\size}{\func{size}}
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\newcommand{\depth}{\func{depth}}
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\newcommand{\topord}{\func{TopOrd}}
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%saving \treesize for now to keep latex from breaking
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\newcommand{\treesize}{\func{size}}
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\newcommand{\sign}{\func{sgn}}
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%PDBs
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\newcommand{\pdbx}{X_{DB}}
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\newcommand{\prob}{p}
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\newcommand{\probOf}{P}
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\newcommand{\probDist}{\vct{\probOf}}
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\newcommand{\probAllTup}{\vct{\prob}}
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\newcommand{\wSet}{\Omega}
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\newcommand{\ti}{TIDB\xspace}
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\newcommand{\tis}{TIDBs\xspace}
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\newcommand{\bi}{BIDB\xspace}
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\newcommand{\bis}{BIDBs\xspace}
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\newcommand{\tiabb}{ti}
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\newcommand{\biabb}{bi}
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\newcommand{\biwset}{\idb_{\biabb}}
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\newcommand{\block}{b}
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\newcommand{\biord}{\leq_{x_\block}}
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\newcommand{\tiwset}{\idb_{\tiabb}}
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\newcommand{\bipd}{\pd_{\biabb}}
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\newcommand{\tipd}{\pd_{\tiabb}}
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\newcommand{\bipdb}{\pdb_{\biabb}}
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\newcommand{\bivar}{x_{\block, i}}
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\newcommand{\tipdb}{\pdb_{\tiabb}}
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% REPRESENTATIONS
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\newcommand{\rmod}{Mod}
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\newcommand{\reprs}{\mathcal{M}}
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\newcommand{\repr}{M}
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%Polynomial Reformulation
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\newcommand{\wbit}{w}
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\newcommand{\expct}{\mathop{\mathbb{E}}}
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\newcommand{\pbox}[1]{\left[#1\right]}
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\newcommand{\pbrace}[1]{\left\{#1\right\}}
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\newcommand{\vct}[1]{{\bf #1}}
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%using \wVec for world bit vector notation
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\newcommand{\poly}{Q}
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\newcommand{\rpoly}{\widetilde{Q}}%r for reduced as in reduced 'Q'
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\newcommand{\polyForTuple}{\poly_{\tup}}
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\newcommand{\out}{output}%output aggregation over the output vector
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\newcommand{\numocc}[2]{\#\left(#1, #2\right)}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%Graph Symbols
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%Shift macro
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\newcommand{\patternshift}[1]{\hspace*{-0.5mm}\raisebox{-0.35mm}{#1}\hspace*{-0.5mm} }
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%Global styles
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\tikzset{
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default_node/.style={align=center, inner sep=0pt},
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pattern_node/.style={fill=gray!50, draw=black, semithick, inner sep=0pt, minimum size = 2pt, circle},
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tree_node/.style={default_node, draw=black, black, circle, text width=0.3cm, font=\bfseries, minimum size=0.65cm},
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highlight_color/.style={black}, wght_color/.style={black},
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highlight_treenode/.style={tree_node, draw=black, black},
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edge from parent path={(\tikzparentnode) -- (\tikzchildnode)}
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}
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%Subgraph patterns
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\newcommand{\ed}{\patternshift{
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\begin{tikzpicture}[every path/.style={thick, draw}]%[baseline=0.00005cm]
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%\begin{scope}[yshift=-5cm]
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\node at (0, 0)[pattern_node](bottom){};
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\node [above=0.07cm of bottom, pattern_node] (top){};
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\draw (top) -- (bottom);
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% \node at (0, -2)[pattern_node, blue](b2){};
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% \node [above=0.07cm of b2, pattern_node, blue] (t2){};
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% \draw (t2) -- (b2);
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%\end{scope}
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\end{tikzpicture}
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}
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}
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\newcommand{\kmatch}{\ed\cdots\ed^\kElem}
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\newcommand{\twodis}{\patternshift{
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\begin{tikzpicture}[every path/.style={thick, draw}]
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\node at (0, 0) [pattern_node] (bottom1) {};
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\node[above=0.07cm of bottom1, pattern_node] (top1) {} edge (bottom1);
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\node at (0.14, 0) [pattern_node] (bottom2) {};
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\node [above=0.07cm of bottom2, pattern_node] (top2) {} edge (bottom2);
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\end{tikzpicture}
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}
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}
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\newcommand{\twopath}{\patternshift{
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\begin{tikzpicture}[every path/.style={thick, draw}]
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\node at (0, 0.08) [pattern_node] (top){};
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\node [below left=0.095cm and 0.05cm of top, pattern_node](left){};
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\node[below right=0.095cm and 0.05cm of top, pattern_node](right){};
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\draw (top) -- (left);
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\draw (top) -- (right);
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\end{tikzpicture}
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}
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}
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\newcommand{\threedis}{\patternshift{
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\begin{tikzpicture}[every path/.style={thick, draw}]
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\node at (0, 0) [pattern_node] (bottom1) {};
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\node[above=0.07cm of bottom1, pattern_node] (top1) {} edge (bottom1);
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\node at (0.14, 0) [pattern_node] (bottom2) {};
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\node [above=0.07cm of bottom2, pattern_node] (top2) {} edge (bottom2);
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\node at (0.28, 0) [pattern_node] (bottom3) {};
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\node [above=0.07cm of bottom3, pattern_node] (top3) {} edge (bottom3);
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\end{tikzpicture}
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}
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}
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\newcommand{\tri}{\patternshift{
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\begin{tikzpicture}[every path/.style={ thick, draw}]
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\node at (0, 0.08) [pattern_node] (top){};
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\node [below left=0.08cm and 0.01cm of top, pattern_node](left){} edge (top);
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\node[below right=0.08cm and 0.01cm of top, pattern_node](right){} edge (top) edge (left);
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\end{tikzpicture}
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}
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}
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\newcommand{\twopathdis}{\ed~\twopath}
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\newcommand{\threepath}{\patternshift{
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\begin{tikzpicture}[every path/.style={thick, draw}]
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\node at (0, 0) [pattern_node] (node1a) {};
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\node [above=0.07cm of node1a, pattern_node] (node1b) {} edge (node1a);
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\node [right=0.099cm of node1a, pattern_node] (node2b) {}; %edge [semithick] (node1b);
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\node [above=0.07cm of node2b, pattern_node] (node3b) {} edge (node2b);
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\draw (node1b) -- (node3b);
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\end{tikzpicture}
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}
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}
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\newcommand{\oneint}{\patternshift{
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\begin{tikzpicture}[level/.style={sibling distance=0.14cm, level distance=0.15cm}, every path/.style={thick, draw}]
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\node at (0, 0) [pattern_node] {} [grow=down]
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child{node [pattern_node]{}}
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child {node [pattern_node] {}}
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child{node [pattern_node] {}};
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\end{tikzpicture}
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}
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}
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\newcommand{\bsym}[1]{\boldsymbol{#1}}%b for bold; sym for symbol
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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% COMMENTS
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\newcommand{\BG}[1]{\todo[inline]{\textbf{Boris says:$\,$} #1}}
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\newcommand{\SF}[1]{\todo[inline]{\textbf{Su says:$\,$} #1}}
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\newcommand{\OK}[1]{\todo[inline]{\textbf{Oliver says:$\,$} #1}}
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\newcommand{\AH}[1]{\todo[inline, backgroundcolor=cyan]{\textbf{Aaron says:$\,$} #1}}
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\newcommand{\SR}[1]{\todo[inline, backgroundcolor=white]{\textbf{Note to self:$\,$} #1}}
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\newcommand{\AR}[1]{\todo[inline, color=green]{\textbf{Atri says:$\,$} #1}}
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%\newcommand{\AR}[1]{}
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%\newcommand{\AH}[1]{}
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%\newcommand{\SF}[1]{}
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%\newcommand{\BG}[1]{}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%Revision Parameters
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\newcommand{\revision}[1]{\color{blue}{#1}\color{black}\xspace}
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\newcommand{\oldstuff}[1]{\color{gray}{#1}\color{black}\xspace}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%Circuit Notation
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\newcommand{\circuit}{\vari{C}}
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\newcommand{\circuitset}[1]{\vari{CSet}(#1)}
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\newcommand{\circmult}{\times}
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\newcommand{\circplus}{+}
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%formally \rchild and \lchild
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\newcommand{\rinput}{\vari{R}}
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\newcommand{\linput}{\vari{L}}
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\newcommand{\inp}{\vari{input}}
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\newcommand{\inputs}{\vari{inputs}}
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\newcommand{\subcircuit}{\vari{S}}
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\newcommand{\gate}{\vari{g}}
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\newcommand{\lwght}{\vari{Lweight}}
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\newcommand{\rwght}{\vari{Rweight}}
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\newcommand{\prt}{\vari{partial}}
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\newcommand{\degval}{\vari{degree}}
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\newcommand{\type}{\vari{type}}
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\newcommand{\subgraph}{\vari{S}_{\equivtree(\circuit)}}
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\newcommand{\cost}{\func{Cost}}
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\newcommand{\nullval}{NULL}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%Sets
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\newcommand{\reals}{\mathbb{R}}
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%
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%borrowed from Su and Boris; Ihave them here for reference purposes.
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%needs to be cleaned up
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%
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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% THEOREM LIKE ENVIRONMENTS
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\DeclareMathAlphabet{\mathbbold}{U}{bbold}{m}{n}
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\newtheorem{Theorem}{Theorem}[section]
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\newtheorem{Definition}[Theorem]{Definition}
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\newtheorem{Lemma}[Theorem]{Lemma}
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\newtheorem{Proposition}[Theorem]{Proposition}
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\newtheorem{Property}[Theorem]{Property}
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\newtheorem{Corollary}[Theorem]{Corollary}
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\newtheorem{Claim}[Theorem]{Claim}
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\newtheorem{Example}[Theorem]{Example}
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\newtheorem{Axiom}[Theorem]{Axiom}
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\newtheorem{Question}[Theorem]{Question}
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\newtheorem{Assumption}[Theorem]{Assumption}
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\newtheorem{hypo}[Theorem]{Conjecture}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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% Rel model
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\newcommand{\tup}{t}
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\newcommand{\comprehension}[2]{\left\{\;#1\;|\;#2\;\right\}}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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% SEMIRINGS
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\newcommand{\udom}{\mathcal{U}}
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\newcommand{\semN}{\mathbb{N}}
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\newcommand{\domN}{\mathbb{N}}
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\newcommand{\semB}{\mathbb{B}}
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\newcommand{\semNX}{\mathbb{N}[\vct{X}]}
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\newcommand{\domK}{K}
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\newcommand{\semK}{\mathcal{K}}
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\newcommand{\support}[1]{supp({#1})}
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\newcommand{\onesymbol}{\mathbbold{1}}
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\newcommand{\zerosymbol}{\mathbbold{0}}
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\newcommand{\multsymb}{\otimes}
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\newcommand{\addsymbol}{\oplus}
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\newcommand{\addormultsymbol}{\bigcdot}
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\newcommand{\monsymbol}{\ominus}
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\newcommand{\addK}{\addsymbol_{\semK}}
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\newcommand{\multK}{\multsymb_{\semK}}
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\newcommand{\genopK}{\odot_{\semK}}
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\newcommand{\monK}{\monsymbol_{\semK}}
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\newcommand{\oneK}{\onesymbol_{\semK}}
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\newcommand{\zeroK}{\zerosymbol_{\semK}}
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\newcommand{\addOf}[1]{\addsymbol_{#1}}
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\newcommand{\multOf}[1]{\multsymb_{#1}}
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\newcommand{\monOf}[1]{\monsymbol_{#1}}
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\newcommand{\oneOf}[1]{\onesymbol_{#1}}
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\newcommand{\zeroOf}[1]{\zerosymbol_{#1}}
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\newcommand{\dbDomK}[1]{\mathcal{DB}_{#1}}
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\newcommand{\assign}{\varphi}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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% COMPLEXITY
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\newcommand{\sharpphard}{\#P-hard\xspace}
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\newcommand{\sharpwonehard}{\#W[1]-hard\xspace}
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\newcommand{\ptime}{PTIME\xspace}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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% TERMINOLOGY AND ABBREVIATIONS
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\newcommand{\expectProblem}{\textsc{Expected Result Multiplicity Problem}\xspace}
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\newcommand{\termSMB}{standard monomial basis\xspace}
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\newcommand{\abbrSMB}{SMB\xspace}
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\newcommand{\termSOP}{sum of products\xspace}
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\newcommand{\abbrSOP}{SOP}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\newcommand{\eps}{\epsilon}
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\newcommand{\inparen}[1]{\left({#1}\right)}
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\newcommand{\inset}[1]{\left\{{#1}\right\}}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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% Forcing Layouts
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\newcommand{\trimfigurespacing}{\vspace*{-5mm}}
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%%%Adding stuff below so that long chain of display equatoons can be split across pages
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\allowdisplaybreaks
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%%% Local Variables:
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%%% mode: latex
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%%% TeX-master: "main"
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%%% End:
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