99-figures.tex

\chapter{First images}
\label{chapter:dummy-layout-text}

% Explanation of the details of transactions on multiversion data.
% The transaction model.
% No implementation (index structure) details.

The theory of general transaction processing in traditional 
databases is well-defined and mature, and the basic principles are
well presented in many
textbooks~\cite{bernstein:1987:cc-n-r,gray:1993:transactionprocessing,papadimitriou:1986:cc-theory}.
In this chapter, we concentrate on the theory of multiversion
transaction processing, and highlight the differences to the classical
transaction theory.
The traditional read-write model assumes that transactions are sequences
of reads and writes on data items, without distinguishing item deletions and
insertions from
updates~\cite{bernstein:1987:cc-n-r,papadimitriou:1986:cc-theory}.
The theory of transaction processing in this dissertation is based on
the recoverable transaction model presented by Sippu and
Soisalon-Soininen~\cite{sippu:2001:theory}, which in turn is based on the
model proposed by C.~Mohan~\cite{mohan:1990:aries-kvl,mohan:1992:aries-im}.
In this model, data-item insertions and deletions are made explicit,
and structure-modification operations are included in the model.
We assume the partially persistent transaction-time model, as
described in the introduction.

\begin{figure}[htb]
\centering
\input{images/tx-snapshot.tex}
\figcaption{Transaction in a snapshot database}{The transaction has
executed the schedule $b w[a,\alpha,\alpha'] d[b] w[e,\epsilon] c$.}
\label{fig:tx-snapshot}
\end{figure}

\begin{figure}[htb]
\centering
\input{images/tx-valid.tex}
\figcaption{Transaction in a valid-time database}{The transaction has
executed the schedule $b w[a,\alpha,\alpha'] d[b] w[e,\epsilon] c$.}
\label{fig:tx-valid}
\end{figure}

\begin{figure}[htb]
\begin{center}
\subfigure[A single-version \Btree\ index]{
\input{images/btree-example}}
\subfigure[A common design in multiversion indexes]{
\label{fig:sv-mv-index-comparison:mv}
\input{images/mv-example}}
\figcaption{Comparison of single-version and multiversion indexes}{}
\label{fig:sv-mv-index-comparison}
\end{center}
\end{figure}

\begin{figure}[htb]
\begin{center}
  \input{images/mv-height-decrease}
  \figcaption{Search trees of different heights}
  {Search trees of different versions in a multiversion index can
  have different heights.}
  \label{fig:mv-height-decrease}
\end{center}
\end{figure}

\begin{figure}[htb]
\begin{center}
\input{images/mv-index-example}
\figcaption{Structure of a balanced multiversion index}{}
\label{fig:mv-index-example}
\end{center}
\end{figure}

\chapter{Second images}

\begin{figure}[htb]
\begin{center}
\subfigure[Level 1]{\input{images/mvbt-space-level-1.tex}
\label{fig:mvbt-space-partition:level-1}}
\subfigure[Level 2]{\input{images/mvbt-space-level-2.tex}
\label{fig:mvbt-space-partition:level-2}}
\figcaption{Partitioning of the key-version space in the MVBT}%
{In the image, $\comver = v_6$, which is indicated by the open ends of the
version ranges of pages $p_7$, $p_8$, and $p_{10}$.
The level \num{2} image shows routers to page $p_5$ in pages $p_9$ and
$p_{10}$.}
\label{fig:mvbt-space-partition}
\end{center}
\end{figure}

\begin{figure}[htb]
\begin{center}
  \input{images/mvbt-problem}
  \figcaption{\abbr{MVBT} problem scenario}{Insertion of key~4 causes
  an invalid split.}
  \label{fig:mvbt-invalid-split}
\end{center}
\end{figure}

\begin{figure}[htb]
\begin{center}
  \input{images/mvbt-problem-solved}
  \figcaption{Key split without version split in the TMVBT}%
  {A key-split is triggered by the insertion of key~\num{4}.}
  \label{fig:mvbt-invalid-split-solved}
\end{center}
\end{figure}

\begin{figure}[!htb]
\begin{center}
\subfigure[Active index entries]{\input{images/tmvbt-active-index-entries.tex}
\label{fig:tmvbt-active-entries:index}}
\subfigure[Active leaf entries]{\input{images/tmvbt-active-leaf-entries.tex}
\label{fig:tmvbt-active-entries:leaf}}
\figcaption{Active entries in the TMVBT index}%
{The index page contains three index entries with routers to pages $p_1$,
$p_2$, and $p_3$. 
The leaf page contains three entries, namely $e_1$, $e_2$, and $e_3$.}
\label{fig:tmvbt-active-entries}
\end{center}
\end{figure}

\chapter{Third images}

\begin{figure}[!hbt]
\begin{center}
  \input{images/tmvbt-oper-1}
  \figcaption{Example of a TMVBT index after insertions}
  {
  The page header shows the page identifier followed by (key range, version
  range); 
  the format of index-page entries is (key range, life span, page
  identifier); and 
  the format of leaf-page entries is (key, life span, data), but
  the associated data has been left out for clarity. 
  This TMVBT has been created by transaction~$T_1$
  inserting keys~\range{1}{6} and transaction~$T_2$ inserting
  keys~\num{7} and~\num{8}.
  }
  \label{fig:example-oper-1}
\end{center}
\end{figure}

\begin{figure}[!htb]
\begin{center}
  \input{images/tmvbt-oper-2}
  \figcaption{\abbr{TMVBT} after inserting entry with key~\num{9}}
  {
  The format of the figure is the same as in \figref{fig:example-oper-1}.
  White rectangles denote live pages, and gray rectangles denote dead pages.
  Transaction~$T_2$ has caused a version-split on~$p_5$ by inserting
  key~\num{9}.}
  \label{fig:example-oper-2}
\end{center}
\end{figure}

\begin{figure}[!htb]
\begin{center}
  \input{images/tmvbt-oper-3}
  \figcaption{\abbr{TMVBT} after deleting most of the entries}
  {Transaction~$T_2$ deleted keys~\range{4}{9}, thus shrinking
  the current-version search tree to a single page.}
  \label{fig:example-oper-3}
\end{center}
\end{figure}


\begin{figure*}[htb]
\begin{center}
  \input{images/tmvbt-example-1}
  \figcaption{Example of a TMVBT index after insertions}{In this
  figure, transaction $T_1$ has inserted keys \range{1}{9}.}
  \label{fig:tmvbt-example:1}
\end{center}
\end{figure*}

\begin{figure*}[htb]
\begin{center}
  \input{images/tmvbt-example-2}
  \figcaption{Example of a TMVBT index after some deletions}{In this
  figure, transaction $T_2$ has deleted keys \range{7}{9}.}
  \label{fig:tmvbt-example:2}
\end{center}
\end{figure*}

\begin{figure*}[htb]
\begin{center}
  \input{images/tmvbt-example-3}
  \figcaption{Example of a TMVBT index after more insertions}{In this
  figure, transaction $T_2$ has inserted keys \range{10}{15}.}
  \label{fig:tmvbt-example:3}
\end{center}
\end{figure*}

\begin{figure}[!htb]
\begin{center}
  \input{images/tmvbt-split-a.tex}
  \figcaption{Key-splitting an active page~$p$}%
  {The horizontal axis represents version ranges, and the vertical axis key
  ranges.}
 % [$p$ active, $\liveentries{p} = \capacity$]
  \label{fig:split-active} 
\end{center}
\end{figure}

\chapter{Fourth images}

\begin{figure}[!htb]
\begin{center}
  \subfigure[$\minsplit \leq \liveentries{p} \leq
  \maxsplit$]{\label{fig:split:is1}
    \input{images/tmvbt-split-is1.tex}}
  \subfigure[$\liveentries{p} > \maxsplit$]{\label{fig:split:is2} 
    \input{images/tmvbt-split-is2.tex}}
  \\
  \subfigure[$s$ inactive, $\liveentries{p} < \minsplit$ and 
    $\liveentries{p} + \liveentries{s} \leq
  \maxsplit$]{\label{fig:split:imi1}
    \input{images/tmvbt-split-imi1.tex}}
  \subfigure[$s$ inactive, $\liveentries{p} < \minsplit$ and  
    $\liveentries{p} + \liveentries{s} >
  \maxsplit$]{\label{fig:split:imi2}
    \input{images/tmvbt-split-imi2.tex}}
  \\
  \subfigure[$s$ active, $\liveentries{p} < \minsplit$ and  
    $\liveentries{p} + \liveentries{s} \leq
  \maxsplit$]{\label{fig:split:ima1}
    \input{images/tmvbt-split-ima1.tex}}
  \subfigure[$s$ active, $\liveentries{p} < \minsplit$ and  
    $\liveentries{p} + \liveentries{s} >
  \maxsplit$]{\label{fig:split:ima2}
    \input{images/tmvbt-split-ima2.tex}}
  \figcaption{Version-splitting an inactive page~$p$}
  {The horizontal axis represents version ranges, and the vertical
  axis key ranges. 
  Case 
  (a)~represents a version split, 
  (b)~a version split followed by a key split,
  (c)~a version split followed by a merge with an inactive sibling, 
  (d)~a version split followed by a redistribution of live entries
  with an inactive sibling, 
  (e)~a version split followed by a merge with an active
  sibling, and 
  (f)~a version split followed by a redistribution of live entries
  with an active sibling.}
  \label{fig:split}
\end{center}
\end{figure}

\begin{figure}[!htb]
\begin{center}
  \subfigure[$p$ inactive, $s$ active, $\liveentries{p} + \liveentries{s}
  \leq \maxsplit$]{\label{fig:merge:ima1}
    \input{images/tmvbt-merge-ima1.tex}}
  \subfigure[$p$ inactive, $s$ active, $\liveentries{p} + \liveentries{s} >
  \maxsplit$]{\label{fig:merge:ima2}
    \input{images/tmvbt-merge-ima2.tex}}
  \subfigure[$p$ inactive, $s$ inactive, $\liveentries{p} + \liveentries{s} \leq
  \maxsplit$]{\label{fig:merge:imi1} 
    \input{images/tmvbt-merge-imi1.tex}}
  \subfigure[$p$ inactive, $s$ inactive, $\liveentries{p} + \liveentries{s} >
  \maxsplit$]{\label{fig:merge:imi2}
    \input{images/tmvbt-merge-imi2.tex}}
  \figcaption{Merging an inactive page~$p$}
  {The horizontal axis represents version ranges, and the vertical
  axis key ranges.
  Case
  (a)~a represents a version split followed by a merge,
  (b)~a version split followed by a redistribution of live entries,
  (c)~a version split followed by a merge with an inactive
  sibling, and
  (d)~a version split followed by a redistribution of live entries
  with an inactive sibling.}
  \label{fig:merge-inactive}
\end{center}
\end{figure}


The theory of general transaction processing in traditional 
databases is well-defined and mature, and the basic principles are
well presented in many
textbooks~\cite{bernstein:1987:cc-n-r,gray:1993:transactionprocessing,papadimitriou:1986:cc-theory}.
In this chapter, we concentrate on the theory of multiversion
transaction processing, and highlight the differences to the classical
transaction theory.
The traditional read-write model assumes that transactions are sequences
of reads and writes on data items, without distinguishing item deletions and
insertions from
updates~\cite{bernstein:1987:cc-n-r,papadimitriou:1986:cc-theory}.
The theory of transaction processing in this dissertation is based on
the recoverable transaction model presented by Sippu and
Soisalon-Soininen~\cite{sippu:2001:theory}, which in turn is based on the
model proposed by C.~Mohan~\cite{mohan:1990:aries-kvl,mohan:1992:aries-im}.
In this model, data-item insertions and deletions are made explicit,
and structure-modification operations are included in the model.
We assume the partially persistent transaction-time model, as
described in the introduction.

\begin{figure}[htb]
\begin{center}
  \input{images/cmvbt-setup}\\
  \figcaption{The CMVBT index structure organization}
  {User transactions issue queries both to the VBT and the
  TMVBT indexes, but updates are performed only on the VBT\@.
  A system maintenance transaction is run periodically to apply the
  pending updates of committed transactions into the TMVBT index and to
  delete them from the VBT\@.}
  \label{fig:cmvbt-setup}
\end{center}
\end{figure}

\begin{figure}[htb]
\begin{center}
%  \includegraphics[width=0.4\textwidth]{images/cmvbt-example}\\
  \input{images/cmvbt-example}\\
  \figcaption{Example of the logical contents of a CMVBT index}
  {The database contains the updates of five committed versions, versions
  \range{1}{5}. 
  The TMVBT index contains all the updates of stable versions
  \range{1}{3}, and the updates of committed transient versions \range{4}{5}
  are still located in the VBT index, identified with transaction
  identifiers \num{103} and~\num{101}.
  The VBT additionally contains updates by two active updating transactions
  that have transaction identifiers \num{102} and~\num{104}.}
  \label{fig:cmvbt-example}
\end{center}
\end{figure}

\begin{figure}[htb]
\begin{center}
  \input{images/cmvbt-example-contents}
  \figcaption{Example of data entries stored in a CMVBT index}
  {This example represents the same situation as
  \figref{fig:cmvbt-example}.
  The format of entries in the TMVBT is (key, life span, data), and the
  format of entries in the VBT is (key, transaction identifier, update).
  In addition to the committed transactions, the VBT also contains the
  updates of two active transactions with transaction identifiers \num{102}
  and \num{104}.}
  \label{fig:cmvbt-contents}
\end{center}
\end{figure}

The theory of general transaction processing in traditional 
databases is well-defined and mature, and the basic principles are
well presented in many
textbooks~\cite{bernstein:1987:cc-n-r,gray:1993:transactionprocessing,papadimitriou:1986:cc-theory}.
In this chapter, we concentrate on the theory of multiversion
transaction processing, and highlight the differences to the classical
transaction theory.
The traditional read-write model assumes that transactions are sequences
of reads and writes on data items, without distinguishing item deletions and
insertions from
updates~\cite{bernstein:1987:cc-n-r,papadimitriou:1986:cc-theory}.
The theory of transaction processing in this dissertation is based on
the recoverable transaction model presented by Sippu and
Soisalon-Soininen~\cite{sippu:2001:theory}, which in turn is based on the
model proposed by C.~Mohan~\cite{mohan:1990:aries-kvl,mohan:1992:aries-im}.
In this model, data-item insertions and deletions are made explicit,
and structure-modification operations are included in the model.
We assume the partially persistent transaction-time model, as
described in the introduction.