From fcdbcbcd45dc567792429bb314df53b42ed9f22e Mon Sep 17 00:00:00 2001
From: Douglas Rumbaugh <dbr4@psu.edu>
Date: Fri, 27 Jun 2025 15:21:38 -0400
Subject: updates

---
 chapters/sigmod23/examples.tex | 10 +++++-----
 1 file changed, 5 insertions(+), 5 deletions(-)

(limited to 'chapters/sigmod23/examples.tex')

diff --git a/chapters/sigmod23/examples.tex b/chapters/sigmod23/examples.tex
index 4e7f9ac..32807e1 100644
--- a/chapters/sigmod23/examples.tex
+++ b/chapters/sigmod23/examples.tex
@@ -74,7 +74,7 @@ makes progress towards removing it.
 \subsection{Independent Range Sampling (ISAM Tree)} 
 \label{ssec:irs-struct}
 We will next considered independent range sampling. For this decomposable
-sampling problem, we use the ISAM Tree for the SSI. Because our shards are
+sampling problem, we use the ISAM tree for the SSI. Because our shards are
 static, we can build highly compact and efficient ISAM trees by storing
 the records directly in a sorted array. So long as the leaf node size is
 a multiple of the record size, this array can be treated as a sequence of
@@ -106,7 +106,7 @@ operations are,
     \text{Worst-case Tagged Delete:} \quad &O\left(\log_s n \log_f n\right)
 \end{align*}
 where $R(n) \in \Theta(1)$ for tagging and $R(n) \in \Theta(\log_s n \log_f n)$
-for tombstones and $f$ is the fanout of the ISAM Tree.
+for tombstones and $f$ is the fanout of the ISAM tree.
 
 
 \subsection{Weighted Independent Range Sampling (Alias-augmented B+Tree)} 
@@ -114,13 +114,13 @@ for tombstones and $f$ is the fanout of the ISAM Tree.
 \label{ssec:wirs-struct}
 As a final example of applying this framework, we consider WIRS. This
 is a decomposable sampling problem that can be answered using the
-alias-augmented B+Tree structure~\cite{tao22, afshani17,hu14}. This
+alias-augmented B+tree structure~\cite{tao22, afshani17,hu14}. This
 data structure is built over sorted data, but can be bulk-loaded from
 this data in linear time, resulting in costs of $B(n) \in \Theta(n \log n)$
 and $B_M(n) \in \Theta(n)$, though the constant factors associated with
 these functions are quite high, as each bulk-loading requires multiple
-linear-time operations for building both the B+Tree and the alias
-structures, among other things. As it is built on a B+Tree, the structure
+linear-time operations for building both the B+tree and the alias
+structures, among other things. As it is built on a B+tree, the structure
 supports $L(n) \in \Theta(\log n)$ point lookups. Answering sampling
 queries requires $P(n) \in \Theta(\log n)$ pre-processing time to
 establish the query interval, during which the weight of the interval
-- 
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