\chapter{Background}
\label{chap:background}

This chapter will introduce important background information and
existing work in the area of data structure dynamization. We will
first discuss the concept of a search problem, which is central to
dynamization techniques.  While one might imagine that restrictions on
dynamization would be functions of the data structure to be dynamized,
in practice the requirements placed on the data structure are quite mild,
and it is the necessary properties of the search problem that the data
structure is used to address that provide the central difficulty to
applying dynamization techniques in a given area. After this, database
indices will be discussed briefly. Indices are the primary use of data
structures within the database context that is of interest to our work.
Following this, existing theoretical results in the area of data structure
dynamization will be discussed, which will serve as the building blocks
for our techniques in subsequent chapters. The chapter will conclude with
a discussion of some of the limitations of these existing techniques.

\section{Queries and Search Problems}
\label{sec:dsp}

Data access lies at the core of most database systems. We want to ask
questions of the data, and ideally get the answer efficiently. We
will refer to the different types of question that can be asked as
\emph{search problems}. We will be using this term in a similar way as
the word \emph{query} \footnote{
    The term query is often abused and used to
    refer to several related, but slightly different things. In the
    vernacular, a query can refer to either a) a general type of search
    problem (as in "range query"), b) a specific instance of a search
    problem, or c) a program written in a query language.
}
is often used within the database systems literature: to refer to a
general class of questions. For example, we could consider range scans,
point-lookups, nearest neighbor searches, predicate filtering, random
sampling, etc., to each be a general search problem.  Formally, for the
purposes of this work, a search problem is defined as follows,

\begin{definition}[Search Problem] 
    Given three multi-sets, $\mathcal{D}$, $\mathcal{R}$, and $\mathcal{Q}$, a search problem is a function
    $F: (\mathcal{D}, \mathcal{Q}) \to \mathcal{R}$, where $\mathcal{D}$ represents the domain of data to be searched,
    $\mathcal{Q}$ represents the domain of query parameters, and $\mathcal{R}$ represents the
answer domain.\footnote{
    It is important to note that it is not required for $\mathcal{R} \subseteq \mathcal{D}$. As an
example, a \texttt{COUNT} aggregation might map a set of strings onto
    an integer. Most common queries do satisfy $\mathcal{R} \subseteq \mathcal{D}$, but this need
not be a universal constraint.
}
\end{definition}

We will use the term \emph{query} to mean a specific instance of a search
problem,

\begin{definition}[Query]
    Given three multi-sets, $\mathcal{D}$, $\mathcal{R}$, and $\mathcal{Q}$, a search problem $F$ and
    a specific set of query parameters $q \in \mathcal{Q}$, a query is a specific
    instance of the search problem, $F(\mathcal{D}, q)$.
\end{definition}

As an example of using these definitions, a \emph{membership test}
or \emph{range scan} would be considered search problems, and a range
scan over the interval $[10, 99]$ would be a query.  We've drawn this
distinction because, as we'll see as we enter into the discussion of
our work in later chapters, it is useful to have separate, unambiguous
terms for these two concepts.

\subsection{Decomposable Search Problems}

Dynamization techniques require the partitioning of one data structure
into several, smaller ones. As a result, these techniques can only
be applied in situations where the search problem to be answered can
be answered from this set of smaller data structures, with the same
answer as would have been obtained had all of the data been used to
construct a single, large structure. This requirement is formalized in
the definition of a class of problems called \emph{decomposable search
problems (DSP)}.  This class was first defined by Bentley and Saxe in
their work on dynamization, and we will adopt their definition,

\begin{definition}[Decomposable Search Problem~\cite{saxe79}]
	\label{def:dsp}
    A search problem $F: (\mathcal{D}, \mathcal{Q}) \to \mathcal{R}$ is decomposable if and
    only if there exists a constant-time computable, associative, and
    commutative binary operator $\mergeop$ such that,
    \begin{equation*}
    F(A \cup B, q) = F(A, q)~ \mergeop ~F(B, q)
    \end{equation*}
\end{definition}

The requirement for $\mergeop$ to be constant-time was used by Bentley and
Saxe to prove specific performance bounds for answering queries from a
decomposed data structure. However, it is not strictly \emph{necessary},
and later work by Overmars lifted this constraint and considered a more
general class of search problems called \emph{$C(n)$-decomposable search
problems},

\begin{definition}[$C(n)$-decomposable Search Problem~\cite{overmars83}]
    A search problem $F: (\mathcal{D}, \mathcal{Q}) \to \mathcal{R}$ is $C(n)$-decomposable
    if and only if there exists an $O(C(n))$-time computable, associative,
    and commutative binary operator $\mergeop$ such that,
    \begin{equation*}
    F(A \cup B, q) = F(A, q)~ \mergeop ~F(B, q)
    \end{equation*}
\end{definition}

To demonstrate that a search problem is decomposable, it is necessary to
show the existence of the merge operator, $\mergeop$, with the necessary
properties, and to show that $F(A \cup B, q) = F(A, q)~ \mergeop ~F(B,
q)$. With these two results, induction demonstrates that the problem is
decomposable even in cases with more than two partial results.

As an example, consider  range scans,
\begin{definition}[Range Count]
    Let $d$ be a set of $n$ points in $\mathbb{R}$. Given an interval,
    $ q = [x, y],\quad x,y \in \mathbb{R}$, a range count returns
    the cardinality, $|d \cap q|$.
\end{definition}

\begin{theorem}
Range Count is a decomposable search problem.
\end{theorem}

\begin{proof}
Let $\mergeop$ be addition ($+$). Applying this to
Definition~\ref{def:dsp}, gives
\begin{align*}
	|(A \cup B) \cap q| = |(A \cap q)| + |(B \cap q)|
\end{align*}
which is true by the distributive property of union and
intersection. Addition is an associative and commutative
operator that can be calculated in $O(1)$ time. Therefore, range counts
are DSPs.
\end{proof}

Because the codomain of a DSP is not restricted, more complex output
structures can be used to allow for problems that are not directly
decomposable to be converted to DSPs, possibly with some minor
post-processing. For example, calculating the arithmetic mean of a set
of numbers can be formulated as a DSP,
\begin{theorem}
The calculation of the arithmetic mean of a set of numbers is a DSP.
\end{theorem}
\begin{proof}
    Consider the search problem $A:\mathcal{D} \to (\mathbb{R}, \mathbb{Z})$,
    where $\mathcal{D}\subset\mathbb{R}$ and is a multi-set. The output tuple
contains the sum of the values within the input set, and the
cardinality of the input set. For two disjoint partitions of the data,
$D_1$ and $D_2$, let  $A(D_1) = (s_1, c_1)$ and $A(D_2) = (s_2, c_2)$. Let
$A(D_1) \mergeop A(D_2) = (s_1 + s_2, c_1 + c_2)$.

Applying Definition~\ref{def:dsp}, gives
\begin{align*}
	A(D_1 \cup D_2) &= A(D_1)\mergeop A(D_2)  \\
	(s_1 + s_2, c_1 + c_2) &= (s_1 + s_2, c_1 + c_2) = (s, c)
\end{align*}
From this result, the average can be determined in constant time by
taking $\nicefrac{s}{c}$. Therefore, calculating the average of a set
of numbers is a DSP.
\end{proof}



\section{Database Indexes}
\label{sec:indexes}

Within a database system, search problems are expressed using
some high level language (or mapped directly to commands, for
simpler systems like key-value stores), which is processed by
the database system to produce a result. Within many database
systems, the most basic access primitive is a table scan, which
sequentially examines each record within the data set. There are many
situations in which the same query could be answered in less time using
a more sophisticated data access scheme, however, and databases support
a limited number of such schemes through the use of specialized data
structures called \emph{indices} (or indexes). Indices can be built over
a set of attributes in a table and provide faster access for particular
search problems.

The term \emph{index} is often abused within the database community
to refer to a range of closely related, but distinct, conceptual
categories.\footnote{
The word index can be used to refer to a structure mapping record
information to the set of records matching that information, as a
general synonym for ``data structure'', to data structures used
specifically in query processing, etc.
} 
This ambiguity is rarely problematic, as the subtle differences between
these categories are not often significant, and context clarifies the
intended meaning in situations where they are.  However, this work
explicitly operates at the interface of two of these categories, and so
it is important to disambiguate between them.

\subsection{The Classical Index}

A database index is a specialized data structure that provides a means
to efficiently locate records that satisfy specific criteria. This
enables more efficient query processing for supported search problems. A
classical index can be modeled as a function, mapping a set of attribute
values, called a key, $\mathcal{K}$, to a set of record identifiers,
$\mathcal{R}$. The codomain of an index can be either the set of
record identifiers, a set containing sets of record identifiers, or
the set of physical records, depending upon the configuration of the
index.~\cite{cowbook} For our purposes here, we'll focus on the first of
these, but the use of other codmains wouldn't have any material effect
on our discussion.

We will use the following definition of a "classical" database index,

\begin{definition}[Classical Index~\cite{cowbook}]
Consider a set of database records, $\mathcal{D}$. An index over
these records, $\mathcal{I}_\mathcal{D}$ is a map of the form
    $\mathcal{I}_\mathcal{D}:(\mathcal{K}, \mathcal{D}) \to \mathcal{R}$, where
$\mathcal{K}$ is a set of attributes of the records in $\mathcal{D}$,
called a \emph{key}.
\end{definition}

In order to facilitate this mapping, indexes are built using data
structures.  The selection of data structure has implications on the
performance of the index, and the types of search problem it can be
used to accelerate. Broadly speaking, classical indices can be divided
into two categories: ordered and unordered. Ordered indices allow for
the iteration over a set of record identifiers in a particular sorted
order of keys, and the efficient location of a specific key value in
that order. These indices can be used to accelerate range scans and
point-lookups. Unordered indices are specialized for point-lookups on a
particular key value, and do not support iterating over records in some
order.~\cite{cowbook, mysql-btree-hash}

There is a very small set of data structures that are usually used for
creating classical indexes. For ordered indices, the most commonly used
data structure is the B-tree~\cite{ubiq-btree},\footnote{
    By \emph{B-tree} here, we are referring not to the B-tree data
    structure, but to a wide range of related structures derived from
    the B-tree.  Examples include the B$^+$-tree, B$^\epsilon$-tree, etc.
}
and the log-structured merge (LSM) tree~\cite{oneil96} is also often
used within the context of key-value stores~\cite{rocksdb}. Some databases
implement unordered indices using hash tables~\cite{mysql-btree-hash}.


\subsection{The Generalized Index}

The previous section discussed the classical definition of index
as might be found in a database systems textbook. However, this
definition is limited by its association specifically with mapping
key fields to records. For the purposes of this work, a broader
definition of index will be considered,

\begin{definition}[Generalized Index]
Consider a set of database records, $\mathcal{D}$, and search
problem, $\mathcal{Q}$.
A generalized index, $\mathcal{I}_\mathcal{D}$
is a map of the form $\mathcal{I}_\mathcal{D}:(\mathcal{Q}, \mathcal{D}) \to
\mathcal{R})$.
\end{definition}

A classical index is a special case of a generalized index, with $\mathcal{Q}$
being a point-lookup or range scan based on a set of record attributes.

There are a number of generalized indexes that appear in some database systems.
For example, some specialized databases or database extensions have support for
indexes based the R-tree\footnote{ Like the B-tree, R-tree here is used as a
signifier for a general class of related data structures.} for spatial
databases~\cite{postgis-doc, ubiq-rtree} or hierarchical navigable small world
graphs for similarity search~\cite{pinecone-db}, among others. These systems
are typically either an add-on module, or a specialized standalone database
that has been designed specifically for answering particular types of queries
(such as spatial queries, similarity search, string matching, etc.).

%\subsection{Indexes in Query Processing} 

%A database management system utilizes indexes to accelerate certain
%types of query. Queries are expressed to the system in some high
%level language, such as SQL or Datalog. These are generalized
%languages capable of expressing a wide range of possible queries.
%The DBMS is then responsible for converting these queries into a
%set of primitive data access procedures that are supported by the
%underlying storage engine. There are a variety of techniques for
%this, including mapping directly to a tree of relational algebra
%operators and interpreting that tree, query compilation, etc. But,
%ultimately, this internal query representation is limited by the routines 
%supported by the storage engine.~\cite{cowbook}

%As an example, consider the following SQL query (representing a
%2-dimensional k-nearest neighbor problem)\footnote{There are more efficient
%ways of answering this query, but I'm aiming for simplicity here
%to demonstrate my point},
%
%\begin{verbatim}
%SELECT dist(A.x, A.y, Qx, Qy) as d, A.key FROM A
%       WHERE A.property = filtering_criterion
%       ORDER BY d
%       LIMIT 5;
%\end{verbatim}
%
%This query will be translated into a logical query plan (a sequence
%of relational algebra operators) by the query planner, which could
%result in a plan like this,
%
%\begin{verbatim}
%query plan here
%\end{verbatim}
%
%With this logical query plan, the DBMS will next need to determine
%which supported operations it can use to most efficiently answer
%this query. For example, the selection operation (A) could be
%physically manifested as a table scan, or could be answered using
%an index scan if there is an ordered index over \texttt{A.property}.
%The query optimizer will make this decision based on its estimate
%of the selectivity of the predicate. This may result in one of the
%following physical query plans
%
%\begin{verbatim}
%physical query plan
%\end{verbatim}
%
%In either case, however, the space of possible physical plans is
%limited by the available access methods: either a sorted scan on
%an attribute (index) or an unsorted scan (table scan). The database
%must filter for all elements matching the filtering criterion,
%calculate the distances between all of these points and the query,
%and then sort the results to get the final answer. Additionally,
%note that the sort operation in the plan is a pipeline-breaker. If
%this plan were to appear as a sub-tree in a larger query plan, the
%overall plan would need to wait for the full evaluation of this
%sub-query before it could proceed, as sorting requires the full
%result set.
%
%Imagine a world where a new index was available to the DBMS: a
%nearest neighbor index. This index would allow the iteration over
%records in sorted order, relative to some predefined metric and a
%query point. If such an index existed over \texttt{(A.x, A.y)} using
%\texttt{dist}, then a third physical plan would be available to the DBMS,
%
%\begin{verbatim}
%\end{verbatim}
%
%This plan pulls records in order of their distance to \texttt{Q}
%directly, using an index, and then filters them, avoiding the
%pipeline breaking sort operation. While it's not obvious in this
%case that this new plan is superior (this would depend upon the
%selectivity of the predicate), it is a third option. It becomes
%increasingly superior as the selectivity of the predicate grows,
%and is clearly superior in the case where the predicate has unit
%selectivity (requiring only the consideration of $5$ records total).
%
%This use of query-specific indexing schemes presents a query
%optimization challenge: how does the database know when a particular
%specialized index can be used for a given query, and how can
%specialized indexes broadcast their capabilities to the query optimizer
%in a general fashion? This work is focused on the problem of enabling
%the existence of such indexes, rather than facilitating their use;
%however these are important questions that must be considered in
%future work for this solution to be viable. There has been work
%done surrounding the use of arbitrary indexes in queries in the past,
%such as~\cite{byods-datalog}. This problem is considered out-of-scope
%for the proposed work, but will be considered in the future.

\section{Classical Dynamization Techniques}

Because data in a database is regularly updated, data structures
intended to be used as an index must support updates (inserts, in-place
modification, and deletes). Not all potentially useful data structures
support updates, and so a general strategy for adding update support
would increase the number of data structures that could be used as
database indices. We refer to a data structure with update support as
\emph{dynamic}, and one without update support as \emph{static}.\footnote{
    The term static is distinct from immutable. Static refers to the
    layout of records within the data structure, whereas immutable
    refers to the data stored within those records. This distinction
    will become relevant when we discuss different techniques for adding
    delete support to data structures.  The data structures used are
    always static, but not necessarily immutable, because the records may
    contain header information (like visibility) that is updated in place.
}

This section discusses \emph{dynamization}, the construction of a dynamic
data structure based on an existing static one. When certain conditions
are satisfied by the data structure and its associated search problem,
this process can be done automatically, and with provable asymptotic
bounds on amortized insertion performance, as well as worst case query
performance. We will first discuss the necessary data structure
requirements, and then examine several classical dynamization techniques.
The section will conclude with a discussion of delete support within the
context of these techniques.

It is worth noting that there are a variety of techniques
discussed in the literature for dynamizing structures with specific
properties, or under very specific sets of circumstances. Examples
include frameworks for adding update support succinct data
structures~\cite{dynamize-succinct} or taking advantage of batching
of insert and query operations~\cite{batched-decomposable}. This
section discusses techniques that are more general, and don't require
workload-specific assumptions.


\subsection{Global Reconstruction}

The most fundamental dynamization technique is that of \emph{global
reconstruction}. While not particularly useful on its own, global
reconstruction serves as the basis for the techniques to follow, and so
we will begin our discussion of dynamization with it.

Consider a class of data structure, $\mathcal{I}$, capable of answering a
search problem, $\mathcal{Q}$. Insertion via global reconstruction is
possible if $\mathcal{I}$ supports the following two operations,
\begin{align*}
\mathtt{build} : \mathcal{PS}(\mathcal{D})& \to \mathcal{I} \\
\mathtt{unbuild} : \mathcal{I}& \to \mathcal{PS}(\mathcal{D})
\end{align*}
where $\mathtt{build}$ constructs an instance $\mathscr{I}\in\mathcal{I}$
over the data structure over a set of records $d \subseteq \mathcal{D}$
in $C(|d|)$ time, and $\mathtt{unbuild}$ returns the set of records $d
\subseteq \mathcal{D}$ used to construct $\mathscr{I} \in \mathcal{I}$ in
$\Theta(1)$ time,\footnote{
    There isn't any practical reason why $\mathtt{unbuild}$ must run
    in constant time, but this is the assumption made in \cite{saxe79}
    and in subsequent work based on it, and so we will follow the same
    definition here.
} such that $\mathscr{I} = \mathtt{build}(\mathtt{unbuild}(\mathscr{I}))$.

Given this structure, an insert of record $r \in \mathcal{D}$ into a
data structure $\mathscr{I} \in \mathcal{I}$ can be defined by,
\begin{align*}
\mathscr{I}_{i}^\prime = \text{build}(\text{unbuild}(\mathscr{I}_i) \cup \{r\})
\end{align*}

It goes without saying that this operation is sub-optimal, as the
insertion cost is $\Theta(C(n))$, and $C(n) \in \Omega(n)$ at best for
most data structures. However, this global reconstruction strategy can
be used as a primitive for more sophisticated techniques that can provide
reasonable performance.

\subsection{Amortized Global Reconstruction}
\label{ssec:agr}

The problem with global reconstruction is that each insert must rebuild
the entire data structure, involving all of its records. This results
in a worst-case insert cost of $\Theta(C(n))$. However, opportunities
for improving this scheme can present themselves when considering the
\emph{amortized} insertion cost.

Consider the cost accrued by the dynamized structure under global
reconstruction over the lifetime of the structure. Each insert will result
in all of the existing records being rewritten, so at worst each record
will be involved in $\Theta(n)$ reconstructions, each reconstruction
having $\Theta(C(n))$ cost. We can amortize this cost over the $n$ records
inserted to get an amortized insertion cost for global reconstruction of,

\begin{equation*}
I_a(n) = \frac{C(n) \cdot n}{n} = C(n)
\end{equation*}

This doesn't improve things as is, however it does present two
opportunities for improvement. If we could either reduce the size of
the reconstructions, or the number of times a record is reconstructed,
then we could reduce the amortized insertion cost.

The key insight, first discussed by Bentley and Saxe, is that
this goal can be accomplished by \emph{decomposing} the data
structure into multiple, smaller structures, each built from a
disjoint partition of the data. As long as the search problem
being considered is decomposable, queries can be answered from
this structure with bounded worst-case overhead, and the amortized
insertion cost can be improved~\cite{saxe79}. Significant theoretical
work exists in evaluating different strategies for decomposing the
data structure~\cite{saxe79, overmars81, overmars83} and for leveraging
specific efficiencies of the data structures being considered to improve
these reconstructions~\cite{merge-dsp}.

There are two general decomposition techniques that emerged from this
work. The earliest of these is the logarithmic method, often called
the Bentley-Saxe method in modern literature, and is the most commonly
discussed technique today. A later technique, the equal block method,
was also examined. It is generally not as effective as the Bentley-Saxe
method, but it has some useful properties for explanatory purposes and
so will be discussed here as well.

\subsection{Equal Block Method~\cite[pp.~96-100]{overmars83}}
\label{ssec:ebm}

Though chronologically later, the equal block method is theoretically a
bit simpler, and so we will begin our discussion of decomposition-based
technique for dynamization of decomposable search problems with it. The
core concept of the equal block method is to decompose the data structure
into several smaller data structures, called blocks, over partitions
of the data. This decomposition is performed such that each block is of
roughly equal size.

Consider a data structure $\mathscr{I} \in \mathcal{I}$ that solves
some decomposable search problem, $F$ and is built over a set of records
$d \in \mathcal{D}$. This structure can be decomposed into $s$ blocks,
$\mathscr{I}_1, \mathscr{I}_2, \ldots, \mathscr{I}_s$ each built over
partitions of $d$, $d_1, d_2, \ldots, d_s$. Fixing $s$ to a specific value
makes little sense when the number of records changes, and so it is taken
to be governed by a smooth, monotonically increasing function $f(n)$ such
that, at any point, the following two constraints are obeyed.
\begin{align}
    f\left(\frac{n}{2}\right) \leq s \leq f(2n) \label{ebm-c1}\\
    \forall_{1 \leq j \leq s} \quad | \mathscr{I}_j |  \leq \frac{2n}{i} \label{ebm-c2}
\end{align}
where $|\mathscr{I}_j|$ is the number of records in the block,
$|\text{unbuild}(\mathscr{I}_j)|$.

A new record is inserted by finding the smallest block and rebuilding it
using the new record. If $k = \argmin_{1 \leq j \leq s}(|\mathscr{I}_j|)$,
then an insert is done by,
\begin{equation*}
\mathscr{I}_k^\prime = \text{build}(\text{unbuild}(\mathscr{I}_k) \cup \{r\})
\end{equation*}
Following an insert, it is possible that Constraint~\ref{ebm-c1} is violated.\footnote{
    Constraint~\ref{ebm-c2} cannot be violated by inserts, but may be
    violated by deletes. We're omitting deletes from the discussion at
    this point, but will circle back to them in Section~\ref{sec:deletes}.
} In this case, the constraints are enforced by "re-configuring" the
structure. $s$ is updated to be exactly $f(n)$, all of the existing
blocks are unbuilt, and then the records are redistributed evenly into
$s$ blocks.

A query with parameters $q$ is answered by this structure by individually
querying the blocks, and merging the local results together with $\mergeop$,
\begin{equation*}
F(\mathscr{I}, q) = \bigmergeop_{j=1}^{s}F(\mathscr{I}_j, q)
\end{equation*}
where $F(\mathscr{I}, q)$ is a slight abuse of notation, referring to
answering the query over $d$ using the data structure $\mathscr{I}$.

This technique provides better amortized performance bounds than global
reconstruction, at the possible cost of increased query performance for
sub-linear queries. We'll omit the details of the proof of performance
for brevity and streamline some of the original notation (full details
can be found in~\cite{overmars83}), but this technique ultimately
results in a data structure with the following performance characteristics,
\begin{align*}
\text{Amortized Insertion Cost:}&\quad \Theta\left(\frac{C(n)}{n} + C\left(\frac{n}{f(n)}\right)\right) \\
\text{Worst-case Query Cost:}& \quad \Theta\left(f(n) \cdot \mathscr{Q}\left(\frac{n}{f(n)}\right)\right) \\
\end{align*}
where $C(n)$ is the cost of statically building $\mathcal{I}$, and
$\mathscr{Q}(n)$ is the cost of answering $F$ using $\mathcal{I}$.

%TODO: example?


\subsection{The Bentley-Saxe Method~\cite{saxe79}} 
\label{ssec:bsm}

%FIXME: switch this section (and maybe the previous?) over to being
%       indexed at 0 instead of 1

The original, and most frequently used, dynamization technique is the
Bentley-Saxe Method (BSM), also called the logarithmic method in older
literature. Rather than breaking the data structure into equally sized
blocks, BSM decomposes the structure into logarithmically many blocks
of exponentially increasing size. More specifically, the data structure
is decomposed into $h = \lceil \log_2 n \rceil$ blocks, $\mathscr{I}_1,
\mathscr{I}_2, \ldots, \mathscr{I}_h$. A given block $\mathscr{I}_i$
will be either empty, or contain exactly $2^i$ records within it.

The procedure for inserting a record, $r \in \mathcal{D}$, into
a BSM dynamization is as follows. If the block $\mathscr{I}_0$
is empty, then $\mathscr{I}_0 = \text{build}{\{r\}}$. If it is not
empty, then there will exist a maximal sequence of non-empty blocks
$\mathscr{I}_0, \mathscr{I}_1, \ldots, \mathscr{I}_i$ for some $i \geq
0$, terminated by an empty block $\mathscr{I}_{i+1}$. In this case,
$\mathscr{I}_{i+1}$ is set to $\text{build}(\{r\} \cup \bigcup_{l=0}^i
\text{unbuild}(\mathscr{I}_l))$ and blocks $\mathscr{I}_0$ through
$\mathscr{I}_i$ are emptied. New empty blocks can be freely added to the
end of the structure as needed.

%FIXME: switch the x's to r's for consistency
\begin{figure}
\centering
\includegraphics[width=.8\textwidth]{diag/bsm.pdf}
\caption{An illustration of inserts into the Bentley-Saxe Method}
\label{fig:bsm-example}
\end{figure}

Figure~\ref{fig:bsm-example} demonstrates this insertion procedure. The
dynamization is built over a set of records $x_1, x_2, \ldots,
x_{10}$ initially, with eight records in $\mathscr{I}_3$ and two in
$\mathscr{I}_1$. The first new record, $x_{11}$, is inserted directly
into $\mathscr{I}_0$. For the next insert following this, $x_{12}$, the
first empty block is $\mathscr{I}_2$, and so the insert is performed by
doing $\mathscr{I}_2 = \text{build}\left(\{x_{12}\} \cup
\text{unbuild}(\mathscr{I}_1) \cup \text{unbuild}(\mathscr{I}_2)\right)$
and then emptying $\mathscr{I}_1$ and $\mathscr{I}_2$.

This technique is called a \emph{binary decomposition} of the data
structure.  Considering a BSM dynamization of a structure containing $n$
records, labeling each block with a $0$ if it is empty and a $1$ if it
is full will result in the binary representation of $n$. For example,
the final state of the structure in Figure~\ref{fig:bsm-example} contains
$12$ records, and the labeling procedure will result in $0\text{b}1100$,
which is $12$ in binary. Inserts affect this representation of the
structure in the same way that incrementing the binary number by $1$ does.

By applying BSM to a data structure, a dynamized structure can be created
with the following performance characteristics,
\begin{align*}
\text{Amortized Insertion Cost:}&\quad \Theta\left(\left(\frac{C(n)}{n}\cdot \log_2 n\right)\right) \\
\text{Worst Case Insertion Cost:}&\quad \Theta\left(C(n)\right) \\
\text{Worst-case Query Cost:}& \quad \Theta\left(\log_2 n\cdot \mathscr{Q}\left(n\right)\right) \\
\end{align*}
This is a particularly attractive result because, for example, a data
structure having $C(n) \in \Theta(n)$ will have an amortized insertion
cost of $\log_2 (n)$, which is quite reasonable. The cost is an extra
logarithmic multiple attached to the query complexity. It is also worth
noting that the worst-case insertion cost remains the same as global
reconstruction, but this case arises only very rarely. If you consider the
binary decomposition representation, the worst-case behavior is triggered
each time the existing number overflows, and a new digit must be added.

\subsection{Delete Support}

Classical dynamization techniques have also been developed with
support for deleting records. In general, the same technique of global
reconstruction that was used for inserting records can also be used to
delete them. Given a record $r \in \mathcal{D}$ and a data structure
$\mathscr{I} \in \mathcal{I}$ such that $r \in \mathscr{I}$, $r$ can be
deleted from the structure in $C(n)$ time as follows,
\begin{equation*}
\mathscr{I}^\prime = \text{build}(\text{unbuild}(\mathscr{I}) - \{r\})
\end{equation*}
However, supporting deletes within the dynamization schemes discussed
above is more complicated. The core problem is that inserts affect the
dynamized structure in a deterministic way, and as a result certain
partitioning schemes can be leveraged to reason about the
performance. But, deletes do not work like this.

\begin{figure}
\caption{A Bentley-Saxe dynamization for the integers on the
interval $[1, 100]$.}
\label{fig:bsm-delete-example}
\end{figure}

For example, consider a Bentley-Saxe dynamization that contains all
integers on the interval $[1, 100]$, inserted in that order, shown in
Figure~\ref{fig:bsm-delete-example}. We would like to delete all of the
records from this structure, one at a time, using global reconstruction.
This presents several problems,
\begin{itemize}
    \item For each record, we need to identify which block it is in before
          we can delete it. 
    \item The cost of performing a delete is a function of which block the
          record is in, which is a question of distribution and not easily
          controlled.
    \item As records are deleted, the structure will potentially violate
          the invariants of the decomposition scheme used, which will
          require additional work to fix.
\end{itemize}



\section{Limitations of Classical Dynamization Techniques}
\label{sec:bsm-limits}

While fairly general, these dynamization techniques have a number of
limitations that prevent them from being directly usable as a general
solution to the problem of creating database indices. Because of the
requirement that the query being answered be decomposable, many search
problems cannot be addressed--or at least efficiently addressed, by
decomposition-based dynamization. The techniques also do nothing to reduce
the worst-case insertion cost, resulting in extremely poor tail latency
performance relative to hand-built dynamic structures. Finally, these
approaches do not do a good job of exposing the underlying configuration
space to the user, meaning that the user can exert limited control on the
performance of the dynamized data structure. This section will discuss
these limitations, and the rest of the document will be dedicated to
proposing solutions to them.

\subsection{Limits of Decomposability}
\label{ssec:decomp-limits}
Unfortunately, the DSP abstraction used as the basis of classical
dynamization techniques has a few significant limitations that restrict
their applicability,

\begin{itemize}
    \item The query must be broadcast identically to each block and cannot
    be adjusted based on the state of the other blocks.

    \item The query process is done in one pass--it cannot be repeated.

    \item The result merge operation must be $O(1)$ to maintain good query
          performance.

    \item The result merge operation must be commutative and associative, 
          and is called repeatedly to merge pairs of results.
\end{itemize}

These requirements restrict the types of queries that can be supported by
the method efficiently. For example, k-nearest neighbor and independent
range sampling are not decomposable. 

\subsubsection{k-Nearest Neighbor} 
\label{sssec-decomp-limits-knn}
The k-nearest neighbor (k-NN) problem is a generalization of the nearest
neighbor problem, which seeks to return the closest point within the
dataset to a given query point. More formally, this can be defined as,
\begin{definition}[Nearest Neighbor]
    
    Let $D$ be a set of $n>0$ points in $\mathbb{R}^d$ and $f(x, y)$
    be some function $f: D^2 \to \mathbb{R}^+$ representing the distance
    between two points within $D$. The nearest neighbor problem, $NN(D,
    q)$ returns some $d \in D$ having $\min_{d \in D} \{f(d, q)\}$
    for some query point, $q \in \mathbb{R}^d$.

\end{definition}

In practice, it is common to require $f(x, y)$ be a metric,\footnote
{
    Contrary to its vernacular usage as a synonym for ``distance'', a
    metric is more formally defined as a valid distance function over
    a metric space. Metric spaces require their distance functions to
    have the following properties,
    \begin{itemize}
        \item The distance between a point and itself is always 0.
        \item All distances between non-equal points must be positive.
        \item For all points, $x, y \in D$, it is true that 
              $f(x, y) = f(y, x)$.
        \item For any three points $x, y, z \in D$ it is true that 
              $f(x, z) \leq f(x, y) + f(y, z)$.
    \end{itemize}

    These distances also must have the interpretation that $f(x, y) <
    f(x, z)$ means that $y$ is ``closer'' to $x$ than $z$ is to $x$. This
    is the opposite of the definition of similarity, and so some minor
    manipulations are usually required to make similarity measures work
    in metric-based indexes. \cite{intro-analysis}
}
and this will be done in the examples of indices for addressing
this problem in this work, but it is not a fundamental aspect of the problem
formulation. The nearest neighbor problem itself is decomposable,
with a simple merge function that accepts the result with the smallest
value of $f(x, q)$ for any two inputs\cite{saxe79}.

The k-nearest neighbor problem generalizes nearest-neighbor to return
the $k$ nearest elements,
\begin{definition}[k-Nearest Neighbor]

    Let $D$ be a set of $n \geq k$ points in $\mathbb{R}^d$ and $f(x, y)$
    be some function $f: D^2 \to \mathbb{R}^+$ representing the distance
    between two points within $D$. The k-nearest neighbor problem,
    $KNN(D, q, k)$ seeks to identify a set $R\subset D$ with $|R| = k$
    such that $\forall d \in D - R, r \in R, f(d, q) \geq f(r, q)$.

\end{definition}

This can be thought of as solving the nearest-neighbor problem $k$ times,
each time removing the returned result from $D$ prior to solving the
problem again.  Unlike the single nearest-neighbor case (which can be
thought of as k-NN with $k=1$), this problem is \emph{not} decomposable.

\begin{theorem}
    k-NN is not a decomposable search problem.
\end{theorem}

\begin{proof}
To prove this, consider the query $KNN(D, q, k)$ against some partitioned
dataset $D = D_0 \cup D_1 \ldots \cup D_\ell$. If k-NN is decomposable,
then there must exist some constant-time, commutative, and associative
binary operator $\mergeop$, such that $R = \mergeop_{0 \leq i \leq l}
R_i$ where $R_i$ is the result of evaluating the query $KNN(D_i, q,
k)$. Consider the evaluation of the merge operator against two arbitrary
result sets, $R = R_i \mergeop R_j$.  It is clear that $|R| = |R_i| =
|R_j| = k$, and that the contents of $R$ must be the $k$ records from
$R_i \cup R_j$ that are nearest to $q$. Thus, $\mergeop$ must solve the
problem $KNN(R_i \cup R_j, q, k)$. However, k-NN cannot be solved in $O(1)$
time. Therefore, k-NN is not a decomposable search problem.
\end{proof}

With that said, it is clear that there isn't any fundamental restriction
preventing the merging of the result sets; it is only the case that an
arbitrary performance requirement wouldn't be satisfied. It is possible
to merge the result sets in non-constant time, and so it is the case
that k-NN is $C(n)$-decomposable. Unfortunately, this classification
brings with it a reduction in query performance as a result of the way
result merges are performed.

As a concrete example of these costs, consider using the Bentley-Saxe
method to extend the VPTree~\cite{vptree}. The VPTree is a static,
metric index capable of answering k-NN queries in $KNN(D, q, k) \in O(k
\log n)$.  One possible merge algorithm for k-NN would be to push all
of the elements in the two arguments onto a min-heap, and then pop off
the first $k$. In this case, the cost of the merge operation would be
$C(k) = k \log k$. Were $k$ assumed to be constant, then the operation
could be considered to be constant-time.  But given that $k$ is only
bounded in size above by $n$, this isn't a safe assumption to make in
general. Evaluating the total query cost for the extended structure,
this would yield,

\begin{equation} 
    k-NN(D, q, k) \in O\left(k\log n \left(\log n + \log k\right) \right)
\end{equation}

The reason for this large increase in cost is the repeated application
of the merge operator. The Bentley-Saxe method requires applying the
merge operator in a binary fashion to each partial result, multiplying
its cost by a factor of $\log n$. Thus, the constant-time requirement
of standard decomposability is necessary to keep the cost of the merge
operator from appearing within the complexity bound of the entire
operation in the general case.\footnote {
    There is a special case, noted by Overmars, where the total cost is
    $O(Q(n) + C(n))$, without the logarithmic term, when $(Q(n) + C(n))
    \in \Omega(n^\epsilon)$ for some $\epsilon >0$. This accounts for the
    case where the cost of the query and merge operation are sufficiently
    large to consume the logarithmic factor, and so it doesn't represent
    a special case with better performance.
} 
If we could revise the result merging operation to remove this duplicated
cost, we could greatly reduce the cost of supporting $C(n)$-decomposable
queries.

\subsubsection{Independent Range Sampling}

Another problem that is not decomposable is independent sampling. There
are a variety of problems falling under this umbrella, including weighted
set sampling, simple random sampling, and weighted independent range
sampling, but we will focus on independent range sampling here.

\begin{definition}[Independent Range Sampling~\cite{tao22}]
    Let $D$ be a set of $n$ points in $\mathbb{R}$. Given a query
    interval $q = [x, y]$ and an integer $k$, an independent range
    sampling query returns $k$ independent samples from $D \cap q$
    with each point having equal probability of being sampled.
\end{definition}

This problem immediately encounters a category error when considering
whether it is decomposable: the result set is randomized, whereas
the conditions for decomposability are defined in terms of an exact
matching of records in result sets. To work around this, a slight abuse
of definition is in order: assume that the equality conditions within
the DSP definition can be interpreted to mean ``the contents in the two
sets are drawn from the same distribution''. This enables the category
of DSP to apply to this type of problem. More formally,
\begin{definition}[Decomposable Sampling Problem]
    A sampling problem $F: (D, Q) \to R$, $F$ is decomposable if and
    only if there exists a constant-time computable, associative, and
    commutative binary operator $\mergeop$ such that,
    \begin{equation*}
    F(A \cup B, q) \sim F(A, q)~ \mergeop ~F(B, q)
    \end{equation*}
\end{definition}

Even with this abuse, however, IRS cannot generally be considered
decomposable; it is at best $C(n)$-decomposable.  The reason for this is
that matching the distribution requires drawing the appropriate number
of samples from each each partition of the data. Even in the special
case that $|D_0| = |D_1| = \ldots = |D_\ell|$, the number of samples
from each partition that must appear in the result set cannot be known
in advance due to differences in the selectivity of the predicate across
the partitions.

\begin{example}[IRS Sampling Difficulties]

    Consider three partitions of data, $D_0 = \{1, 2, 3, 4, 5\}, D_1 =
    \{1, 1, 1, 1, 3\}, D_2 = \{4, 4, 4, 4, 4\}$ using bag semantics and
    an IRS query over the interval $[3, 4]$ with $k=12$. Because all three
    partitions have the same size, it seems sensible to evenly distribute
    the samples across them ($4$ samples from each partition). Applying
    the query predicate to the partitions results in the following,
    $d_0 = \{3, 4\}, d_1 = \{3 \}, d_2 = \{4, 4, 4, 4\}$.

    In expectation, then, the first result set will contain $R_0 = \{3,
    3, 4, 4\}$ as it has a 50\% chance of sampling a $3$ and the same
    probability of a $4$. The second and third result sets can only
    be ${3, 3, 3, 3}$ and ${4, 4, 4, 4}$ respectively. Merging these
    together, we'd find that the probability distribution of the sample
    would be $p(3) = 0.5$ and $p(4) = 0.5$. However, were were to perform
    the same sampling operation over the full dataset (not partitioned),
    the distribution would be $p(3) = 0.25$ and $p(4) = 0.75$.

\end{example}

The problem is that the number of samples drawn from each partition needs to be
weighted based on the number of elements satisfying the query predicate in that
partition. In the above example, by drawing $4$ samples from $D_1$, more weight
is given to $3$ than exists within the base dataset. This can be worked around
by sampling a full $k$ records from each partition, returning both the sample
and the number of records satisfying the predicate as that partition's query
result, and then performing another pass of IRS as the merge operator, but this
is the same approach as was used for k-NN above. This leaves IRS firmly in the
$C(n)$-decomposable camp. If it were possible to pre-calculate the number of
samples to draw from each partition, then a constant-time merge operation could
be used.

\subsection{Insertion Tail Latency}

\subsection{Configurability}

\section{Conclusion}
This chapter discussed the necessary background information pertaining to
queries and search problems, indexes, and techniques for dynamic extension. It
described the potential for using custom indexes for accelerating particular
kinds of queries, as well as the challenges associated with constructing these
indexes. The remainder of this document will seek to address these challenges
through modification and extension of the Bentley-Saxe method, describing work
that has already been completed, as well as the additional work that must be
done to realize this vision.
