From 0dc1a8ea20820168149cedaa14e223d4d31dc4b6 Mon Sep 17 00:00:00 2001
From: "Douglas B. Rumbaugh" <doug@douglasrumbaugh.com>
Date: Sun, 6 Jul 2025 18:21:32 -0400
Subject: updates

---
 chapters/dynamization.tex | 15 ++++++++++++++-
 1 file changed, 14 insertions(+), 1 deletion(-)

(limited to 'chapters/dynamization.tex')

diff --git a/chapters/dynamization.tex b/chapters/dynamization.tex
index 1012597..5e4cdec 100644
--- a/chapters/dynamization.tex
+++ b/chapters/dynamization.tex
@@ -33,7 +33,7 @@ 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 (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
@@ -302,6 +302,19 @@ blocks, we represent it with the notation $\mathscr{I} = \{\mathscr{I}_1,
 \ldots, \mathscr{I}_m\}$, where $\mathscr{I}_i$ is the $i$th block.
 \end{example}
 
+In this example, the decomposition resulted in a reduction of the 
+worst-case insert cost. However, many decomposition schemes that we will
+examine do not affect the worst-case cost, despite having notably better
+performance in practice. As a result, it is more common to consider the
+\emph{amortized} insertion cost, $I_A(n)$ when examining
+dynamization. This cost function has the form,
+\begin{equation*}
+    I_A(n) = \frac{B(n)}{n} \cdot \text{A}
+\end{equation*}
+where $\text{A}$ is the number of times that a record within the
+structure participates in a reconstruction, often called the write
+amplification. 
+
 Much of the existing work on dynamization has considered different
 decomposition methods for static data structures, and the effects that
 these methods have on insertion and query performance. However, before
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