From 0fff4753fac809a6ba17f428df3a041cebe692e0 Mon Sep 17 00:00:00 2001
From: "Douglas B. Rumbaugh" <doug@douglasrumbaugh.com>
Date: Sun, 6 Jul 2025 17:20:34 -0400
Subject: updates

---
 chapters/dynamization.tex | 20 +++++++++++---------
 1 file changed, 11 insertions(+), 9 deletions(-)

(limited to 'chapters/dynamization.tex')

diff --git a/chapters/dynamization.tex b/chapters/dynamization.tex
index 738a436..1012597 100644
--- a/chapters/dynamization.tex
+++ b/chapters/dynamization.tex
@@ -581,10 +581,10 @@ entire structure is compacted into a single block.
 \label{fig:dyn-kbin}
 \end{figure}
 
-One of the significant limitations of the logarithmic method is that it
-is incredibly rigid. In our earlier discussion of decomposition we noted
-that there exists a clear trade-off between insert and query performance
-for half-dynamic structures mediate by the number of blocks into which the
+One of the significant limitations of the logarithmic method is that it is
+incredibly rigid. In our earlier discussion of decomposition we noted that
+there exists a clear trade-off between insert and query performance for
+half-dynamic structures, mediated by the number of blocks into which the
 structure is decomposed. However, the logarithmic method does not allow
 any navigation of this trade-off. In their original paper on the topic,
 Bentley and Saxe proposed a different decomposition scheme that does
@@ -758,9 +758,11 @@ results in a data structure with the following performance characteristics,
 \text{Worst-case Query Cost:}& \quad \mathscr{Q}(n) \in O\left(f(n) \cdot \mathscr{Q}_S\left(\frac{n}{f(n)}\right)\right) \\
 \end{align*}
 The equal block method is generally \emph{worse} in terms of insertion
-performance than the logarithmic and $k$-binomial decompositions, because
-the sizes of reconstructions are typically much larger for an equivalent
-block count, due to all the blocks having approximately the same size.
+performance than the logarithmic and $k$-binomial decompositions.  This
+happens because, for a given number of blocks, the reconstructions will
+typically be larger in the equal block method due to each block having
+approximately the same size. This results in larger reconstructions on
+average than the logarithmic method.
 
 \subsection{Optimizations}
 
@@ -1232,8 +1234,8 @@ two decomposition approaches that expose some form of performance tuning
 to the user, these techniques are targeted as asymptotic results, which
 results in poor results in practice. Finally, most decomposition schemes
 have poor worst-case insertion performance, resulting in extremely poor
-tail latency relative to native dynamic structures. While there do exist
-decomposition schemes that have exhibit better worst-case performance,
+tail latency relative to native dynamic structures. While there do
+exist decomposition schemes that have better worst-case performance,
 they are impractical.  This section will discuss these limitations in
 more detail, and the rest of the document will be dedicated to proposing
 solutions to them.
-- 
cgit v1.2.3

