Minor changes requested by publishers

alga.tex

@@ -1,5 +1,4 @@
-\documentclass[sigplan,authorversion]{acmart}
-
+\documentclass[sigplan,screen,9pt]{acmart}
 \usepackage{booktabs}
 \usepackage{subcaption}
 
@@ -15,29 +14,31 @@
 
 \makeatletter
 
+%%% The following is specific to Haskell'17 and the paper
+%%% 'Algebraic Graphs with Class (Functional Pearl)'
+%%% by Andrey Mokhov.
+%%%
 \setcopyright{acmlicensed}
+\acmPrice{15.00}
 \acmDOI{10.1145/3122955.3122956}
-\acmISBN{978-1-4503-5182-9/17/09}
-\acmConference[Haskell'17]{10th ACM SIGPLAN International Haskell Symposium}{September 7-8, 2017}{Oxford, UK}
 \acmYear{2017}
 \copyrightyear{2017}
-\acmPrice{15.00}
+\acmISBN{978-1-4503-5182-9/17/09}
+\acmConference[Haskell'17]{10th ACM SIGPLAN International Haskell Symposium}{September 7-8, 2017}{Oxford, UK}
 
 \bibliographystyle{ACM-Reference-Format}
 \citestyle{acmauthoryear}
 
 \begin{document}
-\settopmatter{printacmref=false}
-\title{Algebraic Graphs with Class\vspace{-1.5mm}}
-\subtitle{Functional Pearl}
+\title{Algebraic Graphs with Class (Functional Pearl)}
 
 \author{Andrey Mokhov}
 \affiliation{
-  \institution{Newcastle University, United Kingdom\vspace{-1mm}}
+  \institution{Newcastle University, United Kingdom}
 }
 
 \begin{abstract}
-% \vspace{-1mm}
+\vspace{-0.5mm}
 The paper presents a minimalistic and elegant approach to working
 with graphs in Haskell. It is built on a rigorous
 mathematical foundation --- an algebra of graphs --- that allows us to apply
@@ -51,23 +52,23 @@
 and transitive graphs, as well as hypergraphs, by appropriately choosing
 the set of underlying axioms. The flexibility of the approach is
 demonstrated by developing a library for constructing
-and transforming polymorphic graphs.
+and transforming polymorphic graphs.\vspace{-1mm}
 \end{abstract}
 
 %% 2012 ACM Computing Classification System (CSS) concepts
 %% Generate at 'http://dl.acm.org/ccs/ccs.cfm'.
 \begin{CCSXML}
 <ccs2012>
 <concept>
-<concept_id>10002950.10003624.10003633</concept_id>
-<concept_desc>Mathematics of computing~Graph theory</concept_desc>
+<concept_id>10002950</concept_id>
+<concept_desc>Mathematics of computing</concept_desc>
 <concept_significance>500</concept_significance>
 </concept>
+</ccs2012>
 \end{CCSXML}
 
-\ccsdesc[500]{Mathematics of computing~Graph theory}
-%\keywords{\vspace{-2mm}Haskell, algebra, graph theory}
-\keywords{Haskell, algebra, graph theory}
+\ccsdesc[500]{Mathematics of computing}
+\keywords{\vspace{-2mm}Haskell, algebra, graph theory}
 
 \maketitle
 

algebra.tex

@@ -10,10 +10,11 @@
 \caption{Decomposition: $1 \rightarrow 2 \rightarrow 3 = 1 \rightarrow 2 +
 1 \rightarrow 3 + 2 \rightarrow 3$}
 \end{subfigure}
+\vspace{-1mm}
 \caption{Two axioms of the algebra of graphs.\label{fig-axioms}}
 \end{figure*}
 
-\section{Algebraic structure}\label{sec-algebra}
+\section{Algebraic Structure}\label{sec-algebra}
 
 The functions \hs{edge}, \hs{vertices} and \hs{clique} defined in the previous
 section \S\ref{sec-core} satisfy a few properties that we can intuitively write down
@@ -38,7 +39,7 @@ \section{Algebraic structure}\label{sec-algebra}
 generally useful for formal verification, as well as automated testing of graph library
 APIs.
 
-\subsection{Axiomatic characterisation}\label{sub-laws}
+\subsection{Axiomatic Characterisation}\label{sub-laws}
 
 The definitions of \hs{vertices} and \hs{clique} in \S\ref{sub-class}
 use $\varepsilon$ as the identity for both overlay $+$ and connect $\rightarrow$
@@ -161,7 +162,7 @@ \subsection{Axiomatic characterisation}\label{sub-laws}
 \caption{Equational reasoning with algebraic graphs.\label{fig-proof}}
 \end{figure*}
 
-\subsection{Partial order on graphs}\label{sub-partial-order}
+\subsection{Partial Order on Graphs}\label{sub-partial-order}
 
 It is fairly standard to define $x \preceq y$ as $x + y = y$ for an
 idempotent operation~$+$, since it gives a partial order on the elements
@@ -206,7 +207,7 @@ \subsection{Partial order on graphs}\label{sub-partial-order}
 
 \end{itemize}
 
-\subsection{Equational reasoning}\label{sub-reasoning}
+\subsection{Equational Reasoning}\label{sub-reasoning}
 
 In this subsection we show how to use equational reasoning and the laws
 of the algebra to prove properties of functions on graphs.

core.tex

@@ -1,36 +1,6 @@
 %!TEX root = alga.tex
-\begin{figure*}
-\begin{subfigure}[b]{0.2\linewidth}
-\centerline{\includegraphics[scale=0.27]{fig/ex-a.pdf}}
-\vspace{-2.4mm}
-\caption{$1 + 2$}
-\centerline{\includegraphics[scale=0.27]{fig/ex-b.pdf}}
-\vspace{-2.4mm}
-\caption{$1 \rightarrow 2$}
-\end{subfigure}
-\hspace{11mm}
-\begin{subfigure}[b]{0.17\linewidth}
-\centerline{\includegraphics[scale=0.27]{fig/ex-c.pdf}}
-\vspace{-1mm}
-\caption{$1 \rightarrow (2 + 3)$}
-\end{subfigure}
-\hspace{12mm}
-\begin{subfigure}[b]{0.15\linewidth}
-\centerline{\includegraphics[scale=0.27]{fig/ex-d.pdf}}
-\vspace{2.4mm}
-\caption{$1 \rightarrow 1$}
-\end{subfigure}
-\hspace{12mm}
-\begin{subfigure}[b]{0.2\linewidth}
-\centerline{\includegraphics[scale=0.27]{fig/ex-e-new.pdf}}
-\vspace{-1mm}
-\caption{$1 \rightarrow 2 + 2 \rightarrow 3$}
-\end{subfigure}
-\caption{Examples of graph construction. The overlay and connect operations are denoted
-by $+$ and $\rightarrow$, respectively.\label{fig-construction}}
-\end{figure*}
-
-\section{The core}\label{sec-core}
+\vspace{-0.5mm}
+\section{The Core}\label{sec-core}
 In this section we define the \emph{core} of algebraic graphs comprising
 four graph construction primitives. We describe the semantics of the primitives
 using the common representation of graphs by sets of vertices and edges, and
@@ -48,7 +18,7 @@ \section{The core}\label{sec-core}
 solution is to define an abstract interface that encapsulates the data structure and
 provides a set of safe construction primitives. This is exactly the approach we take.
 
-\subsection{Constructing graphs}\label{sub-constructing}
+\subsection{Constructing Graphs}\label{sub-constructing}
 
 The simplest possible graph is the \emph{empty} graph. We denote it by
 $\varepsilon$, therefore $\varepsilon = (\varnothing, \varnothing)$ and
@@ -84,11 +54,10 @@ \subsection{Constructing graphs}\label{sub-constructing}
 \begin{itemize}
   \item $1 + 2$ is the graph with two isolated vertices 1 and 2.
   \item $1 \rightarrow 2$ is the graph with an edge between vertices 1 and 2.
-  \item $1 \rightarrow (2 + 3)$ is the graph with three vertices $\{1, 2, 3\}$
-  and two edges $(1, 2)$ and $(1, 3)$.
+  \item $1 \rightarrow (2 + 3)$ comprises vertices $\{1, 2, 3\}$
+  and edges $\{(1, 2), (1, 3)\}$.
   \item $1 \rightarrow 1$ is the graph with vertex 1 and the \emph{self-loop}.
-  \item $1 \rightarrow 2 + 2 \rightarrow 3$ is the graph with three vertices $\{1, 2, 3\}$
-  and two edges $(1, 2)$ and $(2, 3)$.
+  \item $1 \rightarrow 2 + 2 \rightarrow 3$ is the \emph{path graph} on vertices $\{1, 2, 3\}$.
 \end{itemize}
 
 As shown in~\S\ref{sec-intro}, the core can be represented by a simple
@@ -97,7 +66,7 @@ \subsection{Constructing graphs}\label{sub-constructing}
 has the usual inhabitants, such as the pair $(V,E)$, data types from
 \textsf{containers} and \textsf{fgl}, as well as other, stranger forms of life.
 
-\subsection{Type class}\label{sub-class}
+\subsection{Type Class}\label{sub-class}
 
 We abstract the graph construction primitives defined in \S\ref{sub-constructing}
 as the type class \hs{Graph}\footnote{The name collision (\hs{data Graph}

discussion.tex

@@ -1,6 +1,5 @@
 %!TEX root = alga.tex
-\vspace{-1mm}
-\section{Discussion and future research opportunities}\label{sec-discussion}
+\section{\hspace{-1mm}Discussion~and~Future~Research~Opportunities}\label{sec-discussion}
 
 The paper presented a new algebraic foundation for working with graphs.
 It is particularly well-suited for functional programming languages

graphs.tex

@@ -1,5 +1,5 @@
 %!TEX root = alga.tex
-\section{Graphs \`{a} la carte}\label{sec-a-la-carte}
+\section{Graphs \`{a} la Carte}\label{sec-a-la-carte}
 
 In this section we define several useful \hs{Graph} instances, and
 show that the algebra presented in the previous section~\S\ref{sec-algebra} is
@@ -28,14 +28,14 @@ \section{Graphs \`{a} la carte}\label{sec-a-la-carte}
                 connect x y = R (domain x `union` domain y) (relation x `union` relation y `union`
                     fromAscList [ (a, b) | a <- elems (domain x), b <- elems (domain y) ])
 \end{minted}
-\vspace{-2mm}
+\vspace{-2.5mm}
 \caption{Implementing the \hs{Graph} type class by a binary relation
 and the core graph construction primitives
 defined in~\S\ref{sub-constructing}.\label{fig-relation}}
-\vspace{-2mm}
+\vspace{-2.5mm}
 \end{figure*}
 
-\subsection{Binary relation}\label{sub-relation}
+\subsection{Binary Relation}\label{sub-relation}
 
 We start by a direct encoding of the graph construction primitives defined
 in~\S\ref{sub-constructing} into the abstract data type \hs{Relation} isomorphic
@@ -103,7 +103,7 @@ \subsection{Binary relation}\label{sub-relation}
 The last example highlights the fact that the \hs{Relation@\,\,\blk{a}@} instance allows vertices
 of any type \hs{@\blk{a}@} that satisfies the~\hs{Ord@\,\,\blk{a}@} constraint.
 
-\subsection{Deep embedding}\label{sub-embedding}
+\subsection{Deep Embedding}\label{sub-embedding}
 
 We can embed the core graph construction primitives into a simple data type
 (excuse and ignore the name clash with the type class):
@@ -273,7 +273,7 @@ \subsection{Deep embedding}\label{sub-embedding}
 % which motivates us to study polymorphic graph transformations that can be reused by
 % all law-abiding citizens of the \hs{Graph} world -- see \S\ref{sec-transformations}.
 
-\subsection{Undirected graphs}\label{sub-undirected}
+\subsection{Undirected Graphs}\label{sub-undirected}
 
 As hinted at in \S\ref{sub-laws}, to switch from directed to undirected graphs it
 is sufficient to add the axiom of commutativity for the connect operation. For
@@ -354,7 +354,7 @@ \subsection{Undirected graphs}\label{sub-undirected}
     \label{fig-3-decomposition}}
 \end{figure*}
 
-\subsection{Reflexive graphs}\label{sub-reflexive}
+\subsection{Reflexive Graphs}\label{sub-reflexive}
 
 A graph is \emph{reflexive} if every vertex of the graph is connected to itself,
 i.e. has a self-loop. An example of a reflexive graph is the graph corresponding
@@ -382,7 +382,7 @@ \subsection{Reflexive graphs}\label{sub-reflexive}
 \hs{class Graph@\,\,\blk{g}@ => ReflexiveGraph@\,\,\blk{g}@}
 to increase the type safety of functions that rely on the self-loop axiom.
 
-\subsection{Transitive graphs}\label{sub-transitive}
+\subsection{Transitive Graphs}\label{sub-transitive}
 
 In many applications graphs satisfy the \emph{transitivity} property: if a vertex $x$ is
 connected to $y$, and $y$ is connected to $z$, then the edge between $x$ and $z$ can be
@@ -418,7 +418,7 @@ \subsection{Transitive graphs}\label{sub-transitive}
 A subclass \hs{class Graph@\,\,\blk{g}@ => TransitiveGraph@\,\,\blk{g}@} can be
 defined to distinguish algebraic graphs with the closure axiom from others.
 
-\subsection{Preorders and equivalence relations}\label{sub-preorder}
+\subsection{Preorders and Equivalence Relations}\label{sub-preorder}
 
 By combining reflexive and transitive graphs, one can obtain \emph{preorders}.
 For example, $(1 + 2 + 3) \rightarrow (2 + 3 + 4)$

intro.tex

@@ -1,7 +1,7 @@
 %!TEX root = alga.tex
-%\vspace{-2mm}
+\vspace{-2mm}
 \section{Introduction}\label{sec-intro}
-%\vspace{-0.5mm}
+\vspace{-0.5mm}
 
 Graphs are ubiquitous in computing, yet working with graphs often requires
 painfully low-level fiddling with sets of vertices and edges. Building high-level
@@ -47,12 +47,14 @@ \section{Introduction}\label{sec-intro}
 Algebraic graphs have a safe and minimalistic core of four graph construction primitives,
 as captured by the following data type:
 
+\vspace{-0.59mm}
 \begin{minted}{haskell}
   data Graph a = Empty
                | Vertex a
                | Overlay (Graph a) (Graph a)
                | Connect (Graph a) (Graph a)
 \end{minted}
+\vspace{-0.59mm}
 
 \noindent
 Here \hs{Empty} and \hs{Vertex} construct the \emph{empty} and \emph{single-vertex} graphs,
@@ -84,22 +86,56 @@ \section{Introduction}\label{sec-intro}
   i.e. malformed graphs cannot be constructed.
 
   \item Under the basic set of axioms, algebraic graphs correspond to directed
-  graphs with no edge labels. As we show in~\S\ref{sec-a-la-carte}, by extending
-  the algebra   with additional axioms, it is possible to also represent undirected,
-  reflexive and transitive graphs, their combinations, as well as hypergraphs.
+  graphs. As we show in~\S\ref{sec-a-la-carte}, by extending
+  the algebra with additional axioms, we can represent undirected,
+  reflexive, transitive graphs, their combinations, and hypergraphs.
   Importantly, the core remains unchanged, which allows us to define highly
   reusable polymorphic functions on graphs.
 
   \item We develop a library\footnote{The library is on Hackage:
   \url{http://hackage.haskell.org/package/algebraic-graphs}.}
-  for constructing and transforming algebraic graphs and demonstrate its
-  flexibility in \S\ref{sec-transformations}.
+  for constructing and transforming algebraic graphs
+  and demonstrate its flexibility in \S\ref{sec-transformations}.
+
   % Although the development of efficient algorithms for algebraic
   % graphs is outside the scope of this paper, we show that the library can cope
   % with graphs comprising billions of edges in the matter of
   % seconds, which is sufficiently fast for many applications.
 \end{itemize}
-\vspace{-0.25mm}
+\vspace{-2mm}
+
+\begin{figure*}
+\begin{subfigure}[b]{0.2\linewidth}
+\centerline{\includegraphics[scale=0.27]{fig/ex-a.pdf}}
+\vspace{-2.4mm}
+\caption{$1 + 2$}
+\centerline{\includegraphics[scale=0.27]{fig/ex-b.pdf}}
+\vspace{-2.4mm}
+\caption{$1 \rightarrow 2$}
+\end{subfigure}
+\hspace{11mm}
+\begin{subfigure}[b]{0.17\linewidth}
+\centerline{\includegraphics[scale=0.27]{fig/ex-c.pdf}}
+\vspace{-1mm}
+\caption{$1 \rightarrow (2 + 3)$}
+\end{subfigure}
+\hspace{12mm}
+\begin{subfigure}[b]{0.15\linewidth}
+\centerline{\includegraphics[scale=0.27]{fig/ex-d.pdf}}
+\vspace{2.4mm}
+\caption{$1 \rightarrow 1$}
+\end{subfigure}
+\hspace{12mm}
+\begin{subfigure}[b]{0.2\linewidth}
+\centerline{\includegraphics[scale=0.27]{fig/ex-e-new.pdf}}
+\vspace{-1mm}
+\caption{$1 \rightarrow 2 + 2 \rightarrow 3$}
+\end{subfigure}
+\vspace{-1.5mm}
+\caption{Examples of graph construction. The overlay and connect operations are denoted
+by $+$ and $\rightarrow$, respectively.\label{fig-construction}}
+\vspace{-3mm}
+\end{figure*}
 
 Graphs and functional programming have a long history. We review related
 work in \S\ref{sec-related}. Limitations of the presented approach and future