A bit more work.

bibliography.bib

@@ -0,0 +1,18 @@
+@article{10.1145/321812.321815,
+author = {Brent, Richard P.},
+title = {The Parallel Evaluation of General Arithmetic Expressions},
+year = {1974},
+issue_date = {April 1974},
+publisher = {Association for Computing Machinery},
+address = {New York, NY, USA},
+volume = {21},
+number = {2},
+issn = {0004-5411},
+url = {https://doi.org/10.1145/321812.321815},
+doi = {10.1145/321812.321815},
+abstract = {It is shown that arithmetic expressions with n ≥ 1 variables and constants; operations of addition, multiplication, and division; and any depth of parenthesis nesting can be evaluated in time 4 log2n + 10(n - 1)/p using p ≥ 1 processors which can independently perform arithmetic operations in unit time. This bound is within a constant factor of the best possible. A sharper result is given for expressions without the division operation, and the question of numerical stability is discussed.},
+journal = {J. ACM},
+month = apr,
+pages = {201–206},
+numpages = {6}
+}

cost-models.tex

@@ -22,14 +22,48 @@ \chapter{Cost Models}
 \end{definition}
 
 \begin{definition}[Work]
-  The total amount of steps in an execution.
+  A cost measure denoting the total amount of steps in an execution.
 \end{definition}
 
 \begin{definition}[Span/depth]
-  The length of the longest sequential dependency chain in an
-  execution.
+  A cost measure denoting the length of the longest sequential
+  dependency chain in an execution.
 \end{definition}
 
+On an infinitely parallel computer, the span of a program is the
+number of sequential steps needed to execute the program. Since
+infinitely parallel computers are somewhat unlikely, we must wonder
+whether the span makes sense on our tragically finite computer. The
+answer is \emph{yes}: we can simulate an infinitely parallel computer
+on a finitely parallel computer with overhead proportional to the
+amount of ``missing'' parallelism. The theoretical justification for
+this is \emph{Brent's Theorem}~\cite{10.1145/321812.321815}.
+
+  \begin{theorem}[Brent's Theorem]
+    Writing $T_{i}$ for the time taken to execute a program on $i$
+    processors,
+  \[
+    \frac{T_{1}}{p} \leq T_{p} \leq T_{\infty} + \frac{T_{1}-T_{\infty}}{p}
+  \]
+  where $p$ is the number of available processors.
+
+  % \begin{proof}
+  %   At level $j$ of the computation DAG there are $M_{j}$ independent
+  %   operations, which can clearly be executed by $p$ processors in
+  %   time
+  % \[
+  %   \Bigl\lceil{\frac{M_{j}}{p}}\Bigr\rceil
+  % \]
+  % Now we have
+  % \begin{align*}
+  %   \frac{T_{1}}{p} &= \frac{\sum _{j} {M_{j}}}{p} & \text{(By definition)} \\
+  %   & \leq \sum _{j} \Bigl\lceil{\frac{M_{j}}{p}}\Bigr\rceil \\
+  %   & \leq \sum _{j} \Bigl\lceil{M_{j}}\Bigr\rceil
+  % \end{align*}
+  % \end{proof}
+
+  \end{theorem}
+
 
 %%% Local Variables:
 %%% mode: LaTeX

preample.tex

@@ -3,12 +3,8 @@
 
 \usepackage[utf8]{inputenc}
 
-\usepackage{ntheorem}
-\usepackage{amsmath}[ntheorem]
-\usepackage{mathtools}
-\usepackage{amsfonts}
+\usepackage{amsthm,amsmath,amssymb,amsfonts}
 \usepackage{thmtools}
-
 \input{notation.tex}
 
 \usepackage[table]{xcolor}
@@ -54,5 +50,5 @@
 
 %%% Local Variables:
 %%% mode: latex
-%%% TeX-master: "notes"
+%%% TeX-master: "dpp-notes"
 %%% End: