From b85044e7b35c3a6944776cbca9c6217413e0d356 Mon Sep 17 00:00:00 2001 From: Aaron Huber Date: Wed, 16 Mar 2022 08:28:18 -0400 Subject: [PATCH] Finished layout rework for the main part of the paper in acmart format. --- approx_alg.tex | 8 +++++--- main.pdf | Bin 1017433 -> 1017505 bytes main.synctex.gz | Bin 830749 -> 831008 bytes 3 files changed, 5 insertions(+), 3 deletions(-) diff --git a/approx_alg.tex b/approx_alg.tex index 469280e..7eba36e 100644 --- a/approx_alg.tex +++ b/approx_alg.tex @@ -2,7 +2,7 @@ %!TEX root=./main.tex \section{$1 \pm \epsilon$ Approximation Algorithm}\label{sec:algo} -In \Cref{sec:hard}, we showed that \Cref{prob:bag-pdb-poly-expected} cannot be solved in $\bigO{\qruntime{\optquery{\query},\tupset,\bound}}$ runtime. In light of this, we desire to produce an approximation algorithm that runs in time $\bigO{\qruntime{\optquery{\query},\tupset,\bound}}$. We do this by showing the result via circuits, +We showed in~\Cref{sec:hard} that a runtime of $\bigO{\qruntime{\optquery{\query},\tupset,\bound}}$ cannot be acheived for~\Cref{prob:bag-pdb-poly-expected}. In light of this, we desire to produce an approximation algorithm that runs in time $\bigO{\qruntime{\optquery{\query},\tupset,\bound}}$. We do this by showing the result via circuits, such that our approximation algorithm for this problem runs in $\bigO{\abs{\circuit}}$ for a very broad class of circuits, (thus affirming~\Cref{prob:intro-stmt}); see the discussion after \Cref{lem:val-ub} for more. The following approximation algorithm applies to bag query semantics over both \abbrCTIDB lineage polynomials and general \abbrBIDB lineage polynomials in practice, where for the latter we note that a $1$-\abbrTIDB is equivalently a \abbrBIDB (blocks are size $1$). Our experimental results (see~\Cref{app:subsec:experiment}) which use queries from the PDBench benchmark~\cite{pdbench} show a low $\gamma$ (see~\Cref{def:param-gamma}) supporting the notion that our bounds hold for general \abbrBIDB in practice. @@ -21,7 +21,7 @@ $\expansion{\circuit}$ has the following recursive definition ($\circ$ is list c $\expansion{\circuit} = \begin{cases} \expansion{\circuit_\linput} \circ \expansion{\circuit_\rinput} &\textbf{ if }\circuit.\type = \circplus\\ - \left\{(\monom_\linput \cup \monom_\rinput, \coef_\linput \cdot \coef_\rinput) ~|~(\monom_\linput, \coef_\linput) \in \expansion{\circuit_\linput}, (\monom_\rinput, \coef_\rinput) \in \expansion{\circuit_\rinput}\right\} &\textbf{ if }\circuit.\type = \circmult\\ + \left\{(\monom_\linput \cup \monom_\rinput, \coef_\linput \cdot \coef_\rinput) \right.\\\left.~|~(\monom_\linput, \coef_\linput) \in \expansion{\circuit_\linput}, (\monom_\rinput, \coef_\rinput) \in \expansion{\circuit_\rinput}\right\} &\textbf{ if }\circuit.\type = \circmult\\ \elist{(\emptyset, \circuit.\val)} &\textbf{ if }\circuit.\type = \tnum\\ \elist{(\{\circuit.\val\}, 1)} &\textbf{ if }\circuit.\type = \var.\\ \end{cases} @@ -102,11 +102,13 @@ satisfying \probOf\left(\left|\mathcal{E} - \rpoly(\prob_1,\dots,\prob_\numvar)\right|> \error' \cdot \rpoly(\prob_1,\dots,\prob_\numvar)\right) \leq \conf \end{equation} can be computed in time +\begin{footnotesize} \begin{equation} \label{eq:approx-algo-runtime} O\left(\left(\size(\circuit) + \frac{\log{\frac{1}{\conf}}\cdot k\cdot \log{k} \cdot \depth(\circuit))}{\inparen{\error'}^2\cdot(1-\gamma)^2\cdot \prob_0^{2k}}\right)\cdot\multc{\log\left(\abs{\circuit}(1,\ldots, 1)\right)}{\log\left(\size(\circuit)\right)}\right). \end{equation} -In particular, if $\prob_0>0$ and $\gamma<1$ are absolute constants then the above runtime simplifies to $O_k\left(\left(\frac 1{\inparen{\error'}^2}\cdot\size(\circuit)\cdot \log{\frac{1}{\conf}}\right)\cdot\multc{\log\left(\abs{\circuit}(1,\ldots, 1)\right)}{\log\left(\size(\circuit)\right)}\right)$. +\end{footnotesize} +In particular, if $\prob_0>0$ and $\gamma<1$ are absolute constants then the above runtime simplifies to $$O_k\left(\left(\frac 1{\inparen{\error'}^2}\cdot\size(\circuit)\cdot \log{\frac{1}{\conf}}\right)\cdot\multc{\log\left(\abs{\circuit}(1,\ldots, 1)\right)}{\log\left(\size(\circuit)\right)}\right).$$ \end{Theorem} The restriction on $\gamma$ is satisfied by any diff --git a/main.pdf b/main.pdf index 75807dc1977b164e146844a64f458654ce9e83f8..5542b2316cded4c8becb8761a63d8dfa04417b86 100644 GIT binary patch delta 69621 zcmV(%K;pmIj6R`_K7h0V2S@@lGnc`x11x_V$93QNE4C^w3mX&Tz7NNaN+Km&j&*M$ zr7Twk#DcpcVF7vpkfM|P_q^BFbT@itmNKQXlT>XjW_qTt*RNl{d&tGj1-ba`qxY{K z{qjR@?1k#A=H}w+`Gw@2w&p@J(J9FT44n#E?f-drx3B#YG=@ArSZ z*LVK@ov*)s?5}=*+J$!r|AxS?3=bHgCR~1XX~<%KxzzsO_2KA#*WZQbpTLh;{QKpf zu6}<^;<`~Al?XcxX 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