From 9f730256a5526564f7db0d6f802bd0b82178731e Mon Sep 17 00:00:00 2001
From: Douglas Rumbaugh <dbr4@psu.edu>
Date: Sun, 25 May 2025 19:09:29 -0400
Subject: Updates

---
 chapters/design-space.tex | 564 ++++++++++++++++++++++++++++++++++++++++++++--
 1 file changed, 540 insertions(+), 24 deletions(-)

(limited to 'chapters/design-space.tex')

diff --git a/chapters/design-space.tex b/chapters/design-space.tex
index 10278bd..f639999 100644
--- a/chapters/design-space.tex
+++ b/chapters/design-space.tex
@@ -56,6 +56,7 @@ in Chapter~\ref{chap:framework} show that, for other types of problem,
 the technique does not fair quite so well.
 
 \section{Asymptotic Analsyis}
+\label{sec:design-asymp}
 
 Before beginning with derivations for
 the cost functions of dynamized structures within the context of our
@@ -65,7 +66,14 @@ involves adjusting constants, we will leave the design-space related
 constants within our asymptotic expressions. Additionally, we will
 perform the analysis for a simple decomposable search problem. Deletes
 will be entirely neglected, and we won't make any assumptions about
-mergability. These assumptions are to simplify the analysis.
+mergability. We will also neglect the buffer size, $N_B$, during this
+analysis. Buffering isn't fundamental to the techniques we are examining
+in this chapter, and including it would increase the complexity of the
+analysis without contributing any useful insights.\footnote{
+	The contribution of the buffer size is simply to replace each of the
+	individual records considered in the analysis with batches of $N_B$
+	records. The same patterns hold.
+} 
 
 \subsection{Generalized Bentley Saxe Method}
 As a first step, we will derive a modified version of the Bentley-Saxe
@@ -76,7 +84,6 @@ like this before simply out of a lack of interest in constant factors in
 theoretical asymptotic analysis. During our analysis, we'll intentionally
 leave these constant factors in place.
 
-
 When generalizing the Bentley-Saxe method for arbitrary scale factors, we
 decided to maintain the core concept of binary decomposition. One interesting
 mathematical property of a Bentley-Saxe dynamization is that the internal
@@ -87,21 +94,49 @@ with all other levels being empty. If we represent a full level with a 1
 and an empty level with a 0, then we'd have $1100$, which is $20$ in
 base 2.
 
+\begin{algorithm}
+\caption{The Generalized BSM Layout Policy}
+\label{alg:design-bsm}
+
+\KwIn{$r$: set of records to be inserted, $\mathscr{I}$: a dynamized structure, $n$: number of records in $\mathscr{I}$}
+
+\BlankLine
+\Comment{Find the first non-full level}
+$target \gets -1$ \;
+\For{$i=0\ldots \log_s n$} {
+	\If {$|\mathscr{I}_i| < N_B (s - 1)\cdot s^i$} {
+		$target \gets i$ \;
+		break \;
+	}
+}
+
+\BlankLine
+\Comment{If the structure is full, we need to grow it}
+\If {$target = -1$} {
+	$target \gets 1 + (\log_s n)$ \;
+}
+
+\BlankLine
+\Comment{Build the new structure}
+$\mathscr{I}_{target} \gets \text{build}(\text{unbuild}(\mathscr{I}_0) \cup  \ldots \text{unbuild}(\mathscr{I}_{target}) \cup r)$ \;
+\BlankLine
+\Comment{Empty the levels used to build the new shard}
+\For{$i=0\ldots target-1$} {
+	$\mathscr{I}_i \gets \emptyset$ \;
+}
+\end{algorithm}
+
 Our generalization, then, is to represent the data as an $s$-ary
 decomposition, where the scale factor represents the base of the
 representation. To accomplish this, we set of capacity of level $i$ to
-be $N_b (s - 1) \cdot s^i$, where $N_b$ is the size of the buffer. The
-resulting structure will have at most $\log_s n$ shards. Unfortunately,
-the approach used by Bentley and Saxe to calculate the amortized insertion
-cost of the BSM does not generalize to larger bases, and so we will need
-to derive this result using a different approach. Note that, for this
-analysis, we will neglect the buffer size $N_b$ for simplicity. It cancels
-out in the analysis, and so would only serve to increase the complexity
-of the expressions without contributing any additional insights.\footnote{
-	The contribution of the buffer size is simply to replace each of the
-	individual records considered in the analysis with batches of $N_b$
-	records. The same patterns hold.
-} 
+be $N_B (s - 1) \cdot s^i$, where $N_b$ is the size of the buffer. The
+resulting structure will have at most $\log_s n$ shards. The resulting
+policy is described in Algorithm~\ref{alg:design-bsm}.
+
+Unfortunately, the approach used by Bentley and Saxe to calculate the
+amortized insertion cost of the BSM does not generalize to larger bases,
+and so we will need to derive this result using a different approach.
+
 
 \begin{theorem}
 The amortized insertion cost for generalized BSM with a growth factor of
@@ -185,38 +220,519 @@ in the dynamized structure, and will thus have a cost of $I(n) \in
 
 \begin{theorem}
 The worst-case query cost for generalized BSM for a decomposable
-search problem with cost $\mathscr{Q}_s(n)$ is $\Theta(\log_s(n) \cdot
+search problem with cost $\mathscr{Q}_S(n)$ is $O(\log_s(n) \cdot
 \mathscr{Q}_s(n))$.
 \end{theorem}
 \begin{proof}
+The worst-case scenario for queries in BSM occurs when every existing
+level is full. In this case, there will be $\log_s n$ levels that must
+be queried, with the $i$th level containing $(s - 1) \cdot s^i$ records.
+Thus, the total cost of querying the structure will be,
+\begin{equation}
+\mathscr{Q}(n) = \sum_{i=0}^{\log_s n} \mathscr{Q}_S\left((s - 1) \cdot s^i\right)
+\end{equation}
+The number of records per shard will be upper bounded by $O(n)$, so
+\begin{equation}
+\mathscr{Q}(n) \in O\left(\sum_{i=0}^{\log_s n} \mathscr{Q}_S(n)\right)
+	\in O\left(\log_s n \cdot \mathscr{Q}_S(n)\right)
+\end{equation}
 \end{proof}
 
 \begin{theorem}
-The best-case insertion cost for generalized BSM for a decomposable
-search problem is $I_B \in \Theta(1)$.
+The best-case query cost for generalized BSM for a decomposable
+search problem with a cost of $\mathscr{Q}_S(n)$ is $\mathscr{Q}(n)
+\in \Theta(\mathscr{Q}_S(n))$.
 \end{theorem}
 \begin{proof}
+The best case scenario for queries in BSM occurs when a new level is
+added, which results in every record in the structure being compacted
+into a single structure. In this case, there is only a single data
+structure in the dynamization, and so the query cost over the dynamized
+structure is identical to the query cost of a single static instance of
+the structure. Thus, the best case query cost in BSM is,
+\begin{equation*}
+\mathscr{Q}_B(n) \in \Theta \left( 1 \cdot \mathscr{Q}_S(n) \right) \in \Theta\left(\mathscr{Q}_S(n)\right)
+\end{equation*}
 
 \end{proof}
 
 
 \subsection{Leveling}
 
+Our leveling layout policy is described in
+Algorithm~\ref{alg:design-level}. Each level contains a single structure
+with a capacity of $N_B\cdot s^i$ records. When a reconstruction occurs,
+the first level $i$ that has enough space to have the records in the
+level $i-1$ stored inside of it is selected as the target, and then a new
+structure is built at level $i$ containing the records in it and level
+$i-1$. Then, all levels $j < (i - 1)$ are shifted by one level to level
+$j+1$. This process clears space in level $0$ to contain the buffer flush.
+
+\begin{theorem}
+The amortized insertion cost of leveling with a scale factor of $s$ is
+\begin{equation*}
+I_A(n) \in \Theta\left(\frac{B(n)}{n} \cdot \frac{1}{2}(s+1)\log_s n\right)
+\end{equation*}
+\end{theorem}
+\begin{proof}
+Similarly to generalized BSM, the records in each level will be rewritten
+up to $s$ times before they move down to the next level. Thus, the
+amortized insertion cost for leveling can be found by determining how
+many times a record is expected to be rewritten on a single level, and
+how many levels there are in the structure.
+
+On any given level, the total number of writes require to fill the level
+is given by the expression,
+\begin{equation*}
+B(s + (s - 1) + (s - 2) + \ldots + 1)
+\end{equation*}
+where $B$ is the number of records added to the level during each
+reconstruction (i.e., $N_B$ for level $0$ and $N_B\cdots^{i-1}$ for any
+other level).
+
+This is because the first batch of records entering the level will be
+rewritten each of the $s$ times that the level is rebuilt before the
+records are merged into the level below. The next batch will be rewritten
+one fewer times, and so on. Thus, the total number of writes is,
+\begin{equation*}
+B\sum_{i=0}^{s-1} (s - i) = B\left(s^2 + \sum_{i=0}^{i-1} i\right) = B\left(s^2 + \frac{(s-1)s}{2}\right)
+\end{equation*}
+which can be simplied to get,
+\begin{equation*}
+\frac{1}{2}s(s+1)\cdot B
+\end{equation*}
+writes occuring on each level.\footnote{
+	This write count is not cummulative over the entire structure. It only
+	accounts for the number of writes occuring on this specific level.
+}
+
+To obtain the total number of times records are rewritten, we need to
+calculate the average number of times a record is rewritten per level,
+and sum this over all of the levels.
+\begin{equation*}
+\sum_{i=0}^{\log_s n} \frac{\frac{1}{2}B_i s (s+1)}{s B_i} = \frac{1}{2} \sum_{i=0}^{\log_s n} (s + 1) = \frac{1}{2} (s+1) \log_s n 
+\end{equation*}
+To calculate the amortized insertion cost, we multiply this write amplification
+number of the cost of rebuilding the structures, and divide by the total number
+of records,
+\begin{equation*}
+I_A(n) \in \Theta\left(\frac{B(n)}{n}\cdot \frac{1}{2} (s+1) \log_s n\right)
+\end{equation*}
+\end{proof}
+
+\begin{theorem}
+The worst-case insertion cost for leveling with a scale factor of $s$ is
+\begin{equation*}
+\Theta\left(\frac{s-1}{s} \cdot B(n)\right)
+\end{equation*}
+\end{theorem}
+\begin{proof}
+Unlike in BSM, where the worst case reconstruction involves all of the
+records within the structure, in leveling it only includes the records
+in the last two levels. In particular, the worst case behavior occurs
+when the last level is one reconstruction away from its capacity, and the
+level above it is full. In this case, the reconstruction will involve,
+\begin{equation*}
+\left(s^{\log_s n} - s^{\log_s n - 1}\right) + s^{\log_s n - 1}
+\end{equation*}
+records, where the first parenthesized term represents the records in
+the last level, and the second the records in the level above it.
+\end{proof}
+
+\begin{theorem}
+The worst-case query cost for leveling for a decomposable search
+problem with cost $\mathscr{Q}_S(n)$ is
+\begin{equation*}
+O\left(\mathscr{Q}_S(n) \cdot \log_s n \right)
+\end{equation*}
+\end{theorem}
+\begin{proof}
+The worst-case scenario for leveling is right before the structure gains
+a new level, at which point there will be $\log_s n$ data structures
+each with $O(n)$ records. Thus the worst-case cost will be the cost
+of querying each of these structures,
+\begin{equation*}
+O\left(\mathscr{Q}_S(n) \cdot \log_s n \right)
+\end{equation*}
+\end{proof}
+
+\begin{theorem}
+The best-case query cost for leveling for a decomposable search
+problem with cost $\mathscr{Q}_S(n)$ is
+\begin{equation*}
+\mathscr{Q}_B(n) \in O(\mathscr{Q}_S(n) \cdot \log_s n)
+\end{equation*}
+
+\end{theorem}
+\begin{proof}
+Unlike BSM, leveling will never have empty levels. The policy ensures
+that there is always a data structure on every level. As a result, the
+best-case query still must query $\log_s n$ structures, and so has a
+best-case cost of,
+\begin{equation*}
+\mathscr{Q}_B(n) \in O\left(\mathscr{Q}_S(n) \cdot \log_s n\right)
+\end{equation*}
+\end{proof}
+
 \subsection{Tiering}
 
+\begin{theorem}
+The amortized insertion cost of tiering with a scale factor of $s$ is,
+\begin{equation*}
+I_A(n) \in \Theta\left(\frac{B(n)}{n} \cdot \log_s n \right)
+\end{equation*}
+\end{theorem}
+\begin{proof}
+For tiering, each record is written \emph{exactly} one time per
+level. As a result, each record will be involved in exactly $\log_s n$
+reconstructions over the lifetime of the structure. Each reconstruction
+will have cost $B(n)$, and thus the amortized insertion cost must be,
+\begin{equation*}
+I_A(n) \in \Theta\left(\frac{B(n)}{n} \cdot \log_s n\right)
+\end{equation*}
+\end{proof}
+
+\begin{theorem}
+The worst-case insertion cost of tiering with a scale factor of $s$ is,
+\begin{equation*}
+I(n) \in \Theta\left(B(n)\right)
+\end{equation*}
+\end{theorem}
+\begin{proof}
+The worst-case reconstruction in tiering involves performing a
+reconstruction on each level. Of these, the largest level will
+contain $\Theta(n)$ records, and thus dominates the cost of the
+reconstruction. More formally, the total cost of this reconstruction
+will be,
+\begin{equation*}
+I(n) = \sum_{i=0}{\log_s n} B(s^i) = B(1) + B(s) + B(s^2) + \ldots B(s^{\log_s n})
+\end{equation*}
+Of these, the final term $B(s^{\log_s n}) = B(n)$ dominates the others,
+resulting in an asymptotic worst-case cost of,
+\begin{equation*}
+I(n) \in \Theta\left(B(n)\right)
+\end{equation*}
+\end{proof}
+
+\begin{theorem}
+The worst-case query cost for tiering for a decomposable search
+problem with cost $\mathscr{Q}_S(n)$ is
+\begin{equation*}
+\mathscr{Q}(n) \in O( \mathscr{Q}_S(n) \cdot s \log_s n)
+\end{equation*}
+\end{theorem}
+\begin{proof}
+As with the previous two policies, the worst-case query occurs when the
+structure is completely full. In case of tiering, that means that there
+will be $\log_s n$ levels, each containing $s$ shards with a size bounded
+by $O(n)$. Thus, there will be $s \log_s n$ structures to query, and the
+query cost must be,
+\begin{equation*}
+\mathscr{Q}(n) \in O \left(\mathscr{Q}_S(n) \cdot s \log_s n \right)
+\end{equation*}
+\end{proof}
+
+\begin{theorem}
+The best-case query cost for tiering for a decomposable search problem
+with cost $\mathscr{Q}_S(n)$ is $O(\log_s n)$.
+\end{theorem}
+\begin{proof}
+The tiering policy ensures that there are no internal empty levels, and
+as a result the best case scenario for tiering occurs when each level is
+populated by exactly $1$ shard. In this case, there will only be $\log_s n$
+shards to query, resulting in,
+\begin{equation*}
+\mathscr{Q}_B(n) \in O\left(\mathscr{Q}_S(n) \cdot \log_S n \right)
+\end{equation*}
+best-case query cost.
+\end{proof}
 \section{General Observations}
 
+The asymptotic results from the previous section are summarized in
+Table~\ref{tab:policy-comp}. When the scale factor is accounted for
+in the analysis, we can see that possible trade-offs begin to manifest
+within the space. We've seen some of these in action directly in
+the experimental sections of previous chapters.
 
-\begin{table*}[!t]
+\begin{table*}
 \centering
-\begin{tabular}{|l l l l l|}
+\small
+\renewcommand{\arraystretch}{1.6}
+\begin{tabular}{|l l l l|}
 \hline
-\textbf{Policy} & \textbf{Worst-case Query Cost} & \textbf{Worst-case Insert Cost} & \textbf{Best-cast Insert Cost} & \textbf{Amortized Insert Cost} \\ \hline
-Gen. Bentley-Saxe &$\Theta\left(\log_s(n) \cdot Q(n)\right)$  &$\Theta\left(B(n)\right)$           &$\Theta\left(1\right)$ &$\Theta\left(\frac{B(n)}{n} \cdot  \frac{1}{2}(s-1) \cdot ( (s-1)\log_s n + s)\right)$ \\
-Leveling          &$\Theta\left(\log_s(n) \cdot Q(n)\right)$  &$\Theta\left(B(\frac{n}{s})\right)$ &$\Theta\left(1\right)$ &$\Theta\left(\frac{B(n)}{n} \cdot \frac{1}{2} \log_s(n)(s + 1)\right)$ \\
-Tiering           &$\Theta\left(s\log_s(n) \cdot Q(n)\right)$ &$\Theta\left(B(n)\right)$           &$\Theta\left(1\right)$ &$\Theta\left(\frac{B(n)}{n} \cdot \log_s(n)\right)$ \\\hline
+& \textbf{Gen. BSM} & \textbf{Leveling} & \textbf{Tiering} \\ \hline
+$\mathscr{Q}(n)$ &$O\left(\log_s n \cdot \mathscr{Q}_S(n)\right)$ & $O\left(\log_s n \cdot \mathscr{Q}_S(n)\right)$ & $O\left(s \log_s n  \cdot \mathscr{Q}_S(n)\right)$\\ \hline
+$\mathscr{Q}_B(n)$ & $\Theta(\mathscr{Q}_S(n))$ & $O(\log_s n \cdot \mathscr{Q}_S(n))$ & $O(\log_s n \cdot \mathscr{Q}_S(n))$ \\ \hline
+$I(n)$ & $\Theta(B(n))$ & $\Theta(\frac{s - 1}{s}\cdot B(n))$ & $\Theta(B(n))$\\ \hline
+$I_A(n)$ & $\Theta\left(\frac{B(n)}{n} \frac{1}{2}(s-1)\cdot((s-1)\log_s n +s)\right)$ & $\Theta\left(\frac{B(n)}{n} \frac{1}{2}(s-1)\log_s n\right)$& $\Theta\left(\frac{B(n)}{n} \log_s n\right)$ \\ \hline
 \end{tabular}
-\caption{Comparison of cost functions for various reconstruction policies for DSPs}
+
+\caption{Comparison of cost functions for various layout policies for DSPs}
 \label{tab:policy-comp}
 \end{table*}
 
+% \begin{table*}[!t]
+% \centering
+% \begin{tabular}{|l l l l l|}
+% \hline
+% \textbf{Policy} & \textbf{Worst-case Query Cost} & \textbf{Worst-case Insert Cost} & \textbf{Best-cast Insert Cost} & \textbf{Amortized Insert Cost} \\ \hline
+% Gen. Bentley-Saxe &$\Theta\left(\log_s(n) \cdot Q(n)\right)$  &$\Theta\left(B(n)\right)$           &$\Theta\left(1\right)$ &$\Theta\left(\frac{B(n)}{n} \cdot  \frac{1}{2}(s-1) \cdot ( (s-1)\log_s n + s)\right)$ \\
+% Leveling          &$\Theta\left(\log_s(n) \cdot Q(n)\right)$  &$\Theta\left(B(\frac{n}{s})\right)$ &$\Theta\left(1\right)$ &$\Theta\left(\frac{B(n)}{n} \cdot \frac{1}{2} \log_s(n)(s + 1)\right)$ \\
+% Tiering           &$\Theta\left(s\log_s(n) \cdot Q(n)\right)$ &$\Theta\left(B(n)\right)$           &$\Theta\left(1\right)$ &$\Theta\left(\frac{B(n)}{n} \cdot \log_s(n)\right)$ \\\hline
+% \end{tabular}
+% \caption{Comparison of cost functions for various reconstruction policies for DSPs}
+% \label{tab:policy-comp-old}
+% \end{table*}
+
+% \begin{table*}[!t]
+% \centering
+% \begin{tabular}{|l l l l|}
+% %stuff &\textbf{Gen. BSM} & \textbf{Leveling} & \textbf{Tiering} \\ 
+% % \textbf{Worst-case Query} &$O\left(\log_s(n)\cdot \mathscr{Q}_S(n)\right)$ & $O\left(\log_s(n) \mathscr{Q}_S(n)\right)$ & $O\left(s \log_s(n) \mathscr{Q}_S(n)\right)$\\ \hline
+% % \textbf{Best-case Query} & & & \\ \hline
+% % \textbf{Worst-case Insert} & & & \\ \hline
+% % \textbf{Amortized Insert} & & & \\ \hline
+
+% \caption{Comparison of cost functions for various reconstruction policies for DSPs}
+% \label{tab:policy-comp}
+% \end{table*}
+
+\section{Experimental Evaluation}
+
+In the previous sections, we mathematically proved various claims about
+the performance characteristics of our three layout policies to assess
+the trade-offs that exist within the design space. While this analysis is
+useful, the effects we are examining are at the level of constant factors,
+and so it would be useful to perform experimental testing to validate
+that these claimed performance characteristics manifest in practice. In
+this section, we will do just that, running various benchmarks to explore
+the real-world performance implications of the configuration parameter
+space of our framework.
+
+
+\subsection{Asymptotic Insertion Performance}
+
+We'll begin by validating our results for the insertion performance
+characteristics of the three layout policies. For this test, we
+consider two data structures: the ISAM tree and the VP tree. The ISAM
+tree structure is merge-decomposable using a sorted-array merge, with
+a build cost of $B_M(n) \in \Theta(n \log k)$, where $k$ is the number
+of structures being merged. The VPTree, by constrast, is \emph{not}
+merge decomposable, and is built in $B(n) \in \Theta(n \log n)$ time. We
+use the $200,000,000$ record SOSD \texttt{OSM} dataset~\cite{sosd} for
+ISAM testing, and the $1,000,000$ record, $300$-dimensional Spanish
+Billion Words (\texttt{SBW}) dataset~\cite{sbw} for VPTree testing.
+
+For our first experiment, we will examine the latency distribution for
+inserts into our structures. We tested the three layout policies, using a
+common scale factor of $s=2$. This scale factor was selected to minimize
+its influence on the results (we've seen before in Sections~\ref{}
+and \ref{} that scale factor affects leveling and tiering in opposite
+ways) and isolate the influence of the layout policy alone to as great
+a degree as possible. We used a buffer size of $N_b=12000$ for the ISAM
+tree structure, and $N_B=1000$ for the VPTree.
+
+We generated this distribution by inserting $30\%$ of the records from
+the set to ``warm up'' the dynamized structure, and then measuring the
+insertion latency for each individual insert for the remaining $70\%$
+of the data.  Note that, due to timer resolution issues at nanosecond
+scales, the specific latency values associated with the faster end of
+the insertion distribution are not precise. However, it is our intention
+to examine the latency distribution, not the values themselves, and so
+this is not a significant limitation for our analysis.
+
+\begin{figure}
+\subfloat[ISAM Tree Insertion Latencies]{\includegraphics[width=.5\textwidth]{img/design-space/isam-insert-dist.pdf} \label{fig:design-isam-ins-dist}} 
+\subfloat[VPTree Insertion Latencies]{\includegraphics[width=.5\textwidth]{img/design-space/vptree-insert-dist.pdf} \label{fig:design-vptree-ins-dist}} \\
+\caption{Insertion Latency Distributions for Layout Policies}
+\label{fig:design-policy-ins-latency}
+\end{figure}
+
+The resulting distributions are shown in
+Figure~\ref{design-policy-ins-latency}. These distributions are
+representing using a "reversed" CDF with log scaling on both axes. This
+representation has proven very useful for interpretting the latency
+distributions that we see in evaluating dynamization, but are slightly
+unusual, and so we've included a guide to interpretting these charts
+in Appendix\ref{append:rcdf}.
+
+The first notable point is that, for both the ISAM tree
+in Figure~\ref{fig:design-isam-ins-dist} and VPTree in
+Figure~\ref{fig:design-vptree-ins-dist}, the Leveling
+policy results in a measurable lower worst-case insertion
+latency. This result is in line with our theoretical analysis in
+Section~\ref{ssec:design-leveling-proofs}. However, there is a major
+deviation from theoretical in the worst-case performance of Tiering
+and BSM. Both of these should have similar worst-case latencies, as
+the worst-case reconstruction in both cases involves every record in
+the structure. Yet, we see tiering consistently performing better,
+particularly for the ISAM tree.
+
+The reason for this has to do with the way that the records are
+partitioned in these worst-case reconstructions. In Tiering, with a scale
+factor of $s$, the worst-case reconstruction consists of $\Theta(\log_2
+n)$ distinct reconstructions, each involving exactly $2$ structures. BSM,
+on the other hand, will use exactly $1$ reconstruction involving
+$\Theta(\log_2 n)$ structures. This explains why ISAM performs much better
+in Tiering than BSM, as the actual reconstruction cost function there is
+$\Theta(n \log_2 k)$. For tiering, this results in $\Theta(n)$ cost in
+the worst case. BSM, on the other hand, has $\Theta(n \log_2 \log_2 n)$,
+as many more distinct structures must be merged in the reconstruction,
+and is thus asymptotically worse-off. VPTree, on the other hand, sees
+less of a difference because it is \emph{not} merge decomposable, and so
+the number of structures playing a role in the reconstructions plays less
+of a role. Having the records more partitioned still hurts performance,
+due to cache effects most likely, but less so than in the MDSP case.
+
+\begin{figure}
+
+\caption{Insertion Throughput for Layout Policies}
+\label{fig:design-ins-tput}
+\end{figure}
+
+Next, in Figure~\ref{fig:design-ins-tput}, we show the overall
+insertion throughput for the three policies. This result should
+correlate with the amorized insertion costs for each policy derived in
+Section~\ref{sec:design-asym}. As expected, tiering has the highest
+throughput.
+
+
+\subsection{General Insert vs. Query Trends}
+
+For our next experiment, we will consider the trade-offs between insertion
+and query performance that exist within this design space. We benchmarked
+each layout policy for a range of scale factors, measuring both their
+respective insertion throughputs and query latencies for both ISAM Tree
+and VPTree.
+
+\begin{figure}
+
+\caption{Insertion Throughput vs. Query Latency}
+\label{fig:design-tradeoff}
+\end{figure}
+
+\subsection{Query Size Effects}
+
+One potentially interesting aspect of decomposition-based dynamization
+techniques is that, asymptotically, the additional cost added by
+decomposing the data structure vanished for sufficiently expensive
+queries. Bentley and Saxe proved that for query costs of the form
+$\mathscr{Q}_B(n) \in \Omega(n^\epsilon)$ for $\epsilon > 0$, the
+overal query cost is unaffected (asymptotically) by the decomposition.
+This would seem to suggest that, as the cost of the query over a single
+shard increases, the effectiveness of our design space for tuning query
+performance should reduce. This is because our tuning space consists
+of adjusting the number of shards within the structure, and so as the
+effects of decomposition on the query cost reduce, we should see all
+configurations approaching a similar query performance.
+
+In order to evaluate this effect, we tested the query latency of range
+queries of varying selectivity against various configurations of our
+framework to see at what points the query latencies begin to converge. We
+also tested $k$-NN queries with varying values of $k$.
+
+\begin{figure}
+\caption{Query "Size" Effect Analysis}
+\label{fig:design-query-sze}
+\end{figure}
+
+\section{Asymptotically Relevant Trade-offs}
+
+Thus far, we have considered a configuration system that trades in
+constant factors only. In general asymptotic analysis, all possible
+configurations of our framework in this scheme collapse to the same basic
+cost functions when the constants are removed. While we have demonstrated
+that, in practice, the effects of this configuration are measurable, there
+do exist techniques in the classical literature that provide asympotically
+relevant trade-offs, such as the equal block method~\cite{maurer80} and
+the mixed method~\cite[pp. 117-118]{overmars83}.  These techniques have
+cost functions that are derived from arbitrary, positive, monotonically
+increasing functions of $n$ that govern various ways in which the data
+structure is partitioned, and changing the selection of function allows
+for "tuning" the performance. However, to the best of our knowledge,
+these techniques have never been implemented, and no useful guidance in
+the literature exists for selecting these functions. 
+
+However, it is useful to consider the general approach of these
+techniques.  They accomplish asymptotically relevant trade-offs by tying
+the decomposition of the data structure directly to a function of $n$,
+the number of records, in a user-configurable way. We can import a similar
+concept into our already existing configuration framework for dynamization
+to enable similar trade-offs, by replacing the constant scale factor,
+$s$, with some function $s(n)$. However, we must take extreme care when
+doing this to select a function that doesn't catastrophically impair
+query performance.
+
+Recall that, generally speaking, our dynamization technique requires
+multiplying the cost function for the data structure being dynamized by
+the number of shards that the data structure has been decomposed into. For
+search problems that are solvable in sub-polynomial time, this results in
+a worst-case query cost of,
+\begin{equation}
+\mathscr{Q}(n) \in O(S(n) \cdot \mathscr{Q}_S(n))
+\end{equation}
+where $S(n)$ is the number of shards and, for our framework, is $S(n) \in
+O(s \log_s n)$. The user can adjust $s$, but this tuning does not have
+asymptotically relevant consequences. Unfortunately, there is not much
+room, practically, for adjustment. If, for example, we were to allow the
+user to specify $S(n) \in \Theta(n)$, rather than $\Theta(\log n)$, then
+query performance would be greatly impaired. We need a function that is
+sub-linear to ensure useful performance.
+
+To accomplish this, we proposed adding a second scaling factor, $k$, such
+that the number of records on level $i$ is given by,
+\begin{equation}
+\label{eqn:design-k-expr}
+N_B \cdot \left(s \log_2^k(n)\right)^{i}
+\end{equation}
+with $k=0$ being equivalent to the configuration space we have discussed
+thus far. The addition of $k$ allows for the dependency of the number of
+shards on $n$ to be slightly biased upwards or downwards, in a way that
+\emph{does} show up in the asymptotic analysis for inserts and queries,
+but also ensures sub-polynomial additional query cost.
+
+In particular, we prove the following asymptotic properties of this
+configuration.
+\begin{theorem}
+The worst-case query latency of a dynamization scheme where the
+capacity of each level is provided by Equation~\ref{eqn:design-k-expr} is
+\begin{equation}
+\mathscr{Q}(n) \in O\left(\left(\frac{\log n}{\log (k \log n))}\right) \cdot \mathscr{Q}_S(n)\right)
+\end{equation}
+\end{theorem}
+\begin{proof}
+The number of levels within the structure is given by $\log_s (n)$,
+where $s$ is the scale factor. The addition of $k$ to the parameterization
+replaces this scale factor with $s \log^k n$, and so we have
+\begin{equation*}
+\log_{s \log^k n}n = \frac{\log n}{\log\left(s \log^k n\right)} = \frac{\log n}{\log s + \log\left(k \log n\right)} \in O\left(\frac{\log n}{\log (k \log n)}\right)
+\end{equation*}
+by the application of various logarithm rules and change-of-base formula.
+
+The cost of a query against a decomposed structure is $O(S(n) \cdot \mathscr{Q}_S(n))$, and
+there are $\Theta(1)$ shards per level. Thus, the worst case query cost is
+\begin{equation*}
+\mathscr{Q}(n) \in O\left(\left(\frac{\log n}{\log (k \log n))}\right) \cdot \mathscr{Q}_S(n)\right)
+\end{equation*}
+\end{proof}
+
+\begin{theorem}
+The amortized insertion cost of a dynamization scheme where the capacity of
+each level is provided by Equation~\ref{eqn:design-k-expr} is,
+\begin{equation*}
+I_A(n) \in \Theta\left(\frac{B(n)}{n} \cdot \frac{\log n}{\log ( k \log n)}\right)
+\end{equation*}
+\end{theorem}
+\begin{proof}
+\end{proof}
+
+\subsection{Evaluation}
+
+In this section, we'll access the effect that modifying $k$ in our
+new parameter space has on the insertion and query performance of our
+dynamization framework.
+
+
+\section{Conclusion}
+
+
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