From 70b254979173ccf6970433f873328a8c2a6319e7 Mon Sep 17 00:00:00 2001 From: Arnau Abella Date: Thu, 31 Mar 2022 11:35:50 +0200 Subject: [PATCH 1/2] Improve Algorithm 2: Enumeration - Simplify some of the conditionals. - Improve explanation. --- ..._Complex_Event_Recognition-Arnau_Abela.pdf | Bin 1009275 -> 0 bytes docs/thesis/chapters/chapter_5.tex | 18 ++++++++---------- docs/thesis/preamble.tex | 2 +- docs/thesis/thesis.cls | 1 + 4 files changed, 10 insertions(+), 11 deletions(-) delete mode 100644 Distributed_Complex_Event_Recognition-Arnau_Abela.pdf diff --git a/Distributed_Complex_Event_Recognition-Arnau_Abela.pdf b/Distributed_Complex_Event_Recognition-Arnau_Abela.pdf deleted file mode 100644 index f665bd7da561ffc82f3c8f6044b0773640aa7ad2..0000000000000000000000000000000000000000 GIT binary patch literal 0 HcmV?d00001 literal 1009275 zcma%iWlUwwwk_`N?kE%!`Qe!4%^fV=tVlSySxElx z1IHw3ZRck0Lc%0zXY6J!Zf@#mW)3GP2-N}&@1p8&0fl&%%t*H 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\sigma' - paths(left(n'))$\; push(st, $\langle$right($n'$), $P$, $\sigma''$, $\delta''\rangle$)\; } } - \eIf{$paths(left(n')) > \sigma'$}{\label{line:enumeration:if:2} + \eIf{$paths(left(n')) > \sigma'$}{\label{line:enumeration:if:3} $n' \leftarrow left(n')$\; }{ \textbf{break}\; @@ -219,9 +215,11 @@ \section{The Enumeration procedure}\label{sec:enumeration} \end{figure} -\aref{algo:enumeration} receives as an input a \acrshort{tecs}, a node $n$, a time-window $\epsilon$, a position $j$, and a process $p$ and traverses a fraction of $G_{\mathcal{E}}$ in a DFS-way left-to-right order. First, computes the parameters $\sigma, \delta$ corresponding to the starting and ending path to enumerate, respectively. The value of these parameters can be computed statically i.e. without message interchanging. Each iteration of the \code{while} of line~\ref{line:enumeration:while:1} traverses a new path starting from the point it branches from the previous path (except for the first iteration). For this, the stack $st$ is used to store the node and partial complex event of that branching point. Then, the \code{while} of line~\ref{line:enumeration:while:2} traverses through the nodes of the next path, following the left direction whenever a union node is reached and adding the right node to the stack whenever need. The \code{if}s of line~\ref{line:enumeration:if:1}~and~line~\ref{line:enumeration:if:2} make sure that enumeration starts on path $\pi_{\sigma}$ so only $paths_{\ge j - \epsilon, \sigma, \delta}$ are traversed. Moreover, by checking for every node $n'$ its value $max(n')$ before adding it to the stack, it makes sure of only going through paths in $paths_{\ge j - \epsilon}$. +\aref{algo:enumeration} receives as an input a \acrshort{tecs}, a node $n$, a time-window $\epsilon$, a position $j$, and a process $p$ and traverses a fraction of $G_{\mathcal{E}}$ in a DFS-way left-to-right order. First, computes the parameters $\sigma, \delta$ corresponding to the starting and ending path to enumerate, respectively. The value of these parameters can be computed statically i.e. without message passing. Notice, the last process may have less paths to enumerate due to the rounding on $\delta$. The enumeration starts from the root node $n$. Each iteration of the \code{while} of line~\ref{line:enumeration:while:1} traverses a new path starting from the point it branches from the previous path. The stack $st$ is used to store the node, the partial complex event, and the parameters $\sigma, \delta$ of that branching point. The \code{while} of line~\ref{line:enumeration:while:2} traverses through the nodes of the next path, following the left direction whenever a union node is reached and adding the right node to the stack whenever need. The \code{if} of line~\ref{line:enumeration:if:1} guarantees that traversed paths $\in paths_{\ge \tau}$ i.e. paths outside the time window are skipped. The \code{if}s of line~\ref{line:enumeration:if:2}~and~\ref{line:enumeration:if:3} assert that we enumerate at most $\delta$ paths and the enumeration starts on path $\pi_{\sigma}$, respectively. All three conditionals guarantee that only paths $\in paths_{\ge \tau, \sigma, \delta}$ are enumerated on process $p$. +\begin{remark*} A simpler recursive algorithm could have been used, however, the constant-delay output might not be guaranteed because the number of backtracking steps after branching might be as long as the longest path of $G_{\mathcal{E}}$. To guarantee constant steps after branching and assure constant-delay output, \aref{algo:enumeration} uses a stack which allows to jump immediately to the next branch. We assume that storing $P$ in the stack takes constant time. We materialize this assumption by modelling $P$ as a linked list of positions, where the list is ordered by the last element added. To update $P$ with position $i$, we only need to create a node $i$ that points to the previous last element of $P$. Then, storing $P$ on the stack is just storing the pointer of the last element added. +\end{remark*} This concludes the presentation of Algorithm~\ref{algo:enumeration}. In the reminding of this section, we prove properties (1), (2) and (3). diff --git a/docs/thesis/preamble.tex b/docs/thesis/preamble.tex index 0e13e27..c5214d1 100644 --- a/docs/thesis/preamble.tex +++ b/docs/thesis/preamble.tex @@ -103,8 +103,8 @@ %%%% Extras \usepackage{multirow} -\usepackage{todonotes} \usepackage{adjustbox} +\usepackage{soul} %%%% Glossaries & Acronyms diff --git a/docs/thesis/thesis.cls b/docs/thesis/thesis.cls index 99ea52f..1d176af 100644 --- a/docs/thesis/thesis.cls +++ b/docs/thesis/thesis.cls @@ -149,6 +149,7 @@ \newtheorem{definition}[theorem]{Definition} \theoremstyle{remark} \newtheorem{remark}[theorem]{Remark} +\newtheorem*{remark*}{Remark} \usepackage[centerlast,small,sc]{caption} \setlength{\captionmargin}{20pt} \newcommand{\fref}[1]{Figure~\ref{#1}} From a162f25a16d335a76f653711f44249b7b8c9b56e Mon Sep 17 00:00:00 2001 From: Arnau Abella Date: Tue, 5 Apr 2022 11:03:26 +0200 Subject: [PATCH 2/2] Time windows into consideration during enumeration This is a draft. Green color means added. --- docs/thesis/chapters/chapter_3.tex | 44 ++++++++++++++++++++++++++++-- docs/thesis/chapters/chapter_5.tex | 30 +++++++++++--------- docs/thesis/preamble.tex | 2 ++ 3 files changed, 60 insertions(+), 16 deletions(-) diff --git a/docs/thesis/chapters/chapter_3.tex b/docs/thesis/chapters/chapter_3.tex index e2369d0..2b85d50 100644 --- a/docs/thesis/chapters/chapter_3.tex +++ b/docs/thesis/chapters/chapter_3.tex @@ -144,9 +144,9 @@ \section{Computational model}\label{sec:cea} \section{Timed Enumerable Compact Set}\label{sec:data_structure} -The data structure that lazily represents the set of partial matches in CORE is called \acrfull{tecs}. A tECS is a \emph{directed acyclic graph (DAG)} $\mathcal{E}$ with two types of nodes: \emph{union nodes} and \emph{non-union nodes}. Union nodes have two children: \code{left} and \code{right}. Non-union nodes are labelled by a stream position and are divided in \emph{output nodes} and \emph{bottom nodes}. The former have exactly one child and the latter have none. To simplify presentation in what follows, we write nodes of any kind as \textrm{n}, bottom, output and union nodes as \textrm{b, o, u}, respectively, and we denote the sets of bottom, output and union nodes by $N_{B}$, $N_{O}$ and $N_{U}$, respectively. - -For a node \textrm{n}, define its \emph{descending-paths}, denoted \code{paths(\textrm{n})}, recursively as follows: if \textrm{n} is a bottom node, then \code{paths(\textrm{n})} = 1; if \textrm{n} is an output node, then \code{paths(\textrm{n}) = paths(next(\textrm{n}))}; if \textrm{n} is a union node, \code{paths(\textrm{n}) = paths(left(\textrm{n})) + paths(right(\textrm{n}))}. Every node \textrm{n} carries paths(\textrm{n}) as an extra label; thus the descending-paths can be retrieved in constant time. The descending-paths attribute is going to be used during the enumeration phase of the distributed evaluation algorithm to balance the workload of each processing unit. +The data structure that lazily represents the set of partial matches in CORE is called \acrfull{tecs}. A tECS is a \emph{directed acyclic graph (DAG)} $\mathcal{E}$ with two types of nodes: \emph{union nodes} and \emph{non-union nodes}. Union nodes have two children: \code{left} and \code{right}. Non-union nodes are labelled by a stream position and are divided in \emph{output nodes} and \emph{bottom nodes}. The former have exactly one child and the latter have none. +{\color{Comment} We should integrate descending-paths here (or at least mention it)}. +To simplify presentation in what follows, we write nodes of any kind as \textrm{n}, bottom, output and union nodes as \textrm{b, o, u}, respectively, and we denote the sets of bottom, output and union nodes by $N_{B}$, $N_{O}$ and $N_{U}$, respectively. {\color{Comment} missing definition of $pos$, $next$, $right$ and $left$}. An \emph{open complex event}, denoted $(i, D)$, is a complex event $([i, j], D)$, where the closing event $j$ hasn't been reached yet. A \acrshort{tecs} represents sets of open complex events. Let $\bar{p} = n_{1}, n_{2}, \ldots, n_{k}$ be a \emph{full-path} in $\mathcal{E}$ such that $n_{k}$ is a bottom node. Then $\bar{p}$ represents the open complex event ${\llbracket \bar{p} \rrbracket}_{\mathcal{E}} = (i, D)$, where $i = pos(n_{k})$ is the label of bottom node $n_{k}$ and $D$ are the labels of the other non-union nodes in $\bar{p}$. Given a node \textrm{n} in $\mathcal{E}$, ${\llbracket \textrm{n} \rrbracket}_{\mathcal{E}}$ is the set of open complex events represented by \textrm{n} and consists of all open complex events ${\llbracket \bar{p} \rrbracket}_{\mathcal{E}}$ with $\bar{p}$ a full-path in $\mathcal{E}$ starting at \textrm{n}. @@ -240,6 +240,44 @@ \section{Auxiliary data structures}\label{sec:auxiliary_data_structure} We assume that hash table lookups and insertions take constant time, and iterating over has constant delay. +{\color{Added} +\section{Descending-paths}\label{sec:desc-path} + +{\color{Comment} I don't think we need an exclusive section for this, but when I tried to integrate this into the description of the tECS it was too much information interrupting the flow of the description} + +For a node \textrm{n}, define its \emph{descending-paths}, denoted \code{paths(\textrm{n})}, as a binary relation $R$ between event position and the number of paths starting from that position that reaches \textrm{n}. This binary relation is defined recursively as follows: + +\[ +paths(\textrm{n}) = +\begin{cases} + \{(pos(\textrm{n}), 1)\} &\quad \textrm{n} \in N_{B} \\ + paths(next(\textrm{n})) &\quad \textrm{n} \in N_{O} \\ + paths(left(\textrm{n})) \cup paths(right(\textrm{n})) &\quad \textrm{n} \in N_{U} \\ +\end{cases} +\] + +We define the union of two binary relations $R := R_{1} \cup R_{2}$ as follows: + +\begin{align*} + R := & \{ \ (x_{1}, y_{1} + y_{2}) \ | \ \forall (x_{1}, y_{1}) \in R_{1}, \exists (x_{2}, y_{2}) \in R_{2}, x_{1} = x_{2} \ \} \\ + & \cup \{ \ (x_{1}, y_{1}) \ | \ \forall (x_{1}, y_{1}) \in R_{1}, \nexists (x_{2}, y_{2}) \in R_{2} \ \} \\ + & \cup \{ \ (x_{2}, y_{2}) \ | \ \forall (x_{2}, y_{2}) \in R_{2}, \nexists (x_{1}, y_{1}) \in R_{1} \ \} +\end{align*} + +For example, suppose node \textrm{n} has two paths: one starting at position $0$ and the other starting at position $2$, then $paths(\textrm{n}) = {\{0 \to 1, 2 \to 1\}}$. The descending-paths attribute is used during the enumeration phase of the distributed evaluation algorithm to balance the workload of each processing unit. + +Every node \textrm{n} carries paths(\textrm{n}) as an extra attribute; thus the descending-paths can be retrieved in constant time. This attribute will contain a pointer to a hash table that encodes the descending-path binary relation efficiently. Additionally, we define the method \code{pathsc(n, $\tau$)} that counts the number of paths starting after position $\tau$. This method will be useful for the enumeration phase of the evaluation algorithm \ref{sec:enumeration}. + +Notice, the size of this hash table may grow linearly with respect to the length of the stream. During the evaluation algorithm \ref{sec:evaluation}, to keep the size constant, we will only preserve paths that are inside the time window. The hash table can be efficiently generated as follows: + +\begin{itemize} + \item Bottom nodes will create a new hash table with a single entry corresponding to the current position. + \item Output nodes will point to the hash table of their child node. + \item Union nodes will create a fresh hash table and initialize it with all binary relations from \code{left(\textrm{u})} and \code{right(\textrm{u})} such that $\{(x, y) \ | \ x \ge \tau\}$, where $\tau = j - \epsilon$, $j$ is the current position and $\epsilon$ is the time window. +\end{itemize} + +All three operations can be computed in constant time $\mathcal{O}(\tau)$. +} \section{Chapter summary} diff --git a/docs/thesis/chapters/chapter_5.tex b/docs/thesis/chapters/chapter_5.tex index 8a2b722..f2cd1e5 100644 --- a/docs/thesis/chapters/chapter_5.tex +++ b/docs/thesis/chapters/chapter_5.tex @@ -151,7 +151,7 @@ \section{The Enumeration procedure}\label{sec:enumeration} \SetKwProg{Procedure}{procedure}{}{} \SetKwFunction{Enumerate}{\textsc{Enumerate}} \Procedure{\Enumerate{$\mathcal{E}, n, \epsilon, j, p$}}{ - $\delta \leftarrow \lceil \text{paths(n)} \ / \ {|\mathcal{P}|} \rceil$\; + {\color{Added}$\delta \leftarrow \lceil \text{pathsc(n, }\tau) \ / \ {|\mathcal{P}|} \rceil$}\; $\sigma \leftarrow \text{index}(p) \cdot \delta$\; st $\leftarrow$ new-stack()\; $\tau \leftarrow j - \epsilon $\; @@ -168,18 +168,20 @@ \section{The Enumeration procedure}\label{sec:enumeration} $P \leftarrow P \ \cup $ {pos($n'$)}\; $n' \leftarrow $ next($n'$)\; } - \ElseIf{$n' \in N_{U}$}{ - \If{$max(right(n')) \ge \tau$}{\label{line:enumeration:if:1} - $\delta'' \leftarrow \delta' - max(0, paths(left(n')) - \sigma')$\; - \If{$\delta'' > 0$}{\label{line:enumeration:if:2} - $\sigma'' \leftarrow \sigma' - paths(left(n'))$\; - push(st, $\langle$right($n'$), $P$, $\sigma''$, $\delta''\rangle$)\; + {\color{Added} + \ElseIf{$n' \in N_{U}$}{ + \If{$max(right(n')) \ge \tau$}{\label{line:enumeration:if:1} + $\delta'' \leftarrow \delta' - max(0, pathsc(left(n'), \tau) - \sigma')$\; + \If{$\delta'' > 0$}{\label{line:enumeration:if:2} + $\sigma'' \leftarrow \sigma' - pathsc(left(n'), \tau)$\; + push(st, $\langle$right($n'$), $P$, $\sigma''$, $\delta''\rangle$)\; + } + } + \eIf{$pathsc(left(n'), \tau) > \sigma'$}{\label{line:enumeration:if:3} + $n' \leftarrow left(n')$\; + }{ + \textbf{break}\; } - } - \eIf{$paths(left(n')) > \sigma'$}{\label{line:enumeration:if:3} - $n' \leftarrow left(n')$\; - }{ - \textbf{break}\; } } } @@ -215,7 +217,9 @@ \section{The Enumeration procedure}\label{sec:enumeration} \end{figure} -\aref{algo:enumeration} receives as an input a \acrshort{tecs}, a node $n$, a time-window $\epsilon$, a position $j$, and a process $p$ and traverses a fraction of $G_{\mathcal{E}}$ in a DFS-way left-to-right order. First, computes the parameters $\sigma, \delta$ corresponding to the starting and ending path to enumerate, respectively. The value of these parameters can be computed statically i.e. without message passing. Notice, the last process may have less paths to enumerate due to the rounding on $\delta$. The enumeration starts from the root node $n$. Each iteration of the \code{while} of line~\ref{line:enumeration:while:1} traverses a new path starting from the point it branches from the previous path. The stack $st$ is used to store the node, the partial complex event, and the parameters $\sigma, \delta$ of that branching point. The \code{while} of line~\ref{line:enumeration:while:2} traverses through the nodes of the next path, following the left direction whenever a union node is reached and adding the right node to the stack whenever need. The \code{if} of line~\ref{line:enumeration:if:1} guarantees that traversed paths $\in paths_{\ge \tau}$ i.e. paths outside the time window are skipped. The \code{if}s of line~\ref{line:enumeration:if:2}~and~\ref{line:enumeration:if:3} assert that we enumerate at most $\delta$ paths and the enumeration starts on path $\pi_{\sigma}$, respectively. All three conditionals guarantee that only paths $\in paths_{\ge \tau, \sigma, \delta}$ are enumerated on process $p$. +{\color{Added} \aref{algo:enumeration} receives as an input a \acrshort{tecs}, a node $n$, a time-window $\epsilon$, a position $j$, and a process $p$ and traverses a fraction of $G_{\mathcal{E}}$ in a DFS-way left-to-right order. First, it computes the parameters $\sigma$ and $\delta$ corresponding to the starting and ending path to enumerate using the method \code{pathsc(n, $j - \epsilon$)}. This method can be executed in constant time. We remark that the value of parameters $\sigma$ and $\delta$ can be computed statically and locally before the execution of the algorithm on each node.} + +The enumeration starts from the root node $n$. Each iteration of the \code{while} of line~\ref{line:enumeration:while:1} traverses a new path starting from the point it branches from the previous path. The stack $st$ is used to store the node, the partial complex event, and the parameters $\sigma, \delta$ of that branching point. {\color{Added} The \code{while} of line~\ref{line:enumeration:while:2} traverses through the nodes of the next path, following the left direction whenever a union node is reached and adding the right node to the stack whenever need. The \code{if} of line~\ref{line:enumeration:if:1} guarantees that traversed paths $\in paths_{\ge \tau}$ i.e. paths outside the time window are skipped. The \code{if}s of line~\ref{line:enumeration:if:2}~and~\ref{line:enumeration:if:3} assert that we enumerate at most $\delta$ paths and the enumeration starts on path $\pi_{\sigma}$, respectively. All three conditionals guarantee that only paths $\in paths_{\ge \tau, \sigma, \delta}$ are enumerated on process $p$.} \begin{remark*} A simpler recursive algorithm could have been used, however, the constant-delay output might not be guaranteed because the number of backtracking steps after branching might be as long as the longest path of $G_{\mathcal{E}}$. To guarantee constant steps after branching and assure constant-delay output, \aref{algo:enumeration} uses a stack which allows to jump immediately to the next branch. We assume that storing $P$ in the stack takes constant time. We materialize this assumption by modelling $P$ as a linked list of positions, where the list is ordered by the last element added. To update $P$ with position $i$, we only need to create a node $i$ that points to the previous last element of $P$. Then, storing $P$ on the stack is just storing the pointer of the last element added. diff --git a/docs/thesis/preamble.tex b/docs/thesis/preamble.tex index c5214d1..9fcf267 100644 --- a/docs/thesis/preamble.tex +++ b/docs/thesis/preamble.tex @@ -93,6 +93,8 @@ \usepackage{xcolor} % \definecolor, \color{codegray} \definecolor{codegray}{rgb}{0.9, 0.9, 0.9} +\colorlet{Added}{green!70!black} +\colorlet{Comment}{red!50!yellow} % \color{codegray} ... ... % \textcolor{red}{easily}

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