Applied edits to network flow handout

Advanced/Graph Algorithms/parts/01 flow.tex

@@ -1,23 +1,23 @@
 \section{Network Flow}
 
 \generic{Networks}
-Say have a network: a sequence of pipes, a set of cities and highways, an electrical circuit, server infrastructure, etc.
+Say have a network: a sequence of pipes, a set of cities and highways, an electrical circuit, etc.
 
 \vspace{1ex}
 
-We'll represent our network with a connected directed weighted graph. If we take a city, edges will be highways and cities will be nodes. There are a few conditions for a valid network graph:
+We can draw this network as a directed weighted graph. If we take a transporation network, for example, edges will represent highways and nodes will be cities. There are a few conditions for a valid network graph:
 
 \begin{itemize}
-	\item The weight of each edge represents its capacity, the number of lanes in the highway.
+	\item The weight of each edge represents its capacity, e.g, the number of lanes in the highway.
 	\item Edge capacities are always positive integers.\hspace{-0.5ex}\footnotemark{}
 	\item Node $S$ is a \textit{source}: it produces flow.
 	\item Node $T$ is a \textit{sink}: it consumes flow.
-	\item All other nodes \textit{conserve} flow. In other words, the sum of flow coming in must equal the sum of flow going out.
+	\item All other nodes \textit{conserve} flow. The sum of flow coming in must equal the sum of flow going out.
 \end{itemize}
 
 \footnotetext{An edge with capacity zero is equivalent to an edge that does not exist; An edge with negative capacity is equivalent to an edge in the opposite direction.}
 
-Here is an example of a such a graph:
+Here is an example of such a graph:
 \begin{center}
 	\begin{tikzpicture}
 
@@ -43,7 +43,7 @@ Here is an example of a such a graph:
 \hrule{}
 
 \generic{Flow}
-In our city example, cars represent \textit{flow}. Let's send one unit of cars along the topmost highway:
+In our city example, traffic represents \textit{flow}. Let's send one unit of traffic along the topmost highway:
 
 \vspace{2ex}
 
@@ -89,11 +89,9 @@ In our city example, cars represent \textit{flow}. Let's send one unit of cars a
 
 \vspace{1ex}
 
-The \textit{magnitude} of a flow\footnotemark{} is the number of \say{flow-units} that go from $S$ to $T$. \\
+The \textit{magnitude} of a flow is the number of \say{flow-units} that go from $S$ to $T$. \\
 
-We are interested in the \textit{maximum flow} through this network: what is the greatest amount of flow we can push from $S$ to $T$?
-
-\footnotetext{you could also think of \say{flow} as a directed weighted graph on top of our network.}
+We are interested in the \textit{maximum flow} through this network: what is the greatest amount of flow we can get from $S$ to $T$?
 
 \problem{}
 What is the magnitude of the flow above?
@@ -156,7 +154,8 @@ Find a maximal flow on the graph below. \\
 \pagebreak
 
 \section{Combining Flows}
-It is fairly easy to combine two flows on a graph. All we need to do is add the flows along each edge. For example, consider the following flows:
+It is fairly easy to combine two flows on a graph. All we need to do is add the flows along each edge. \\
+Consider the following flows:
 
 \vspace{2ex}
 
@@ -266,11 +265,7 @@ It is fairly easy to combine two flows on a graph. All we need to do is add the
 
 	\vspace{1ex}
 
-	For example, we could not add these graphs if the magnitude of flow in the right graph above was 2.
-
-	\vspace{1ex}
-
-	This is because the capacity of the top-right edge is 2, and $2 + 1 > 2$.
+	For example, we could not add these graphs if the magnitude of flow in the right graph above was 2, since the capacity of the top-right edge is 2, and $2 + 1 > 2$.
 \end{minipage}
 
 \vspace{2ex}
@@ -278,7 +273,7 @@ It is fairly easy to combine two flows on a graph. All we need to do is add the
 \vspace{2ex}
 
 \problem{}
-Combine the following flows and ensure that the flow along all edges remains within capacity.
+Combine the flows below, ensuring that the flow along each edge remains within capacity.
 
 \vspace{2ex}