Applied edits to network flow handout

Advanced/Graph Algorithms/parts/00 review.tex

@@ -2,7 +2,7 @@
 
 \definition{}
 A \textit{graph} consists of a set of \textit{nodes} $\{A, B, ...\}$ and a set of edges $\{ (A,B), (A,C), ...\}$ connecting them.
-A \textit{directed graph} is a graph where edges have direction. In such a graph, $(A, B)$ and $(B, A)$ are distinct edges.
+A \textit{directed graph} is a graph where edges have direction. In such a graph, edges $(A, B)$ and $(B, A)$ are distinct.
 A \textit{weighted graph} is a graph that features weights on its edges. \\
 A weighted directed graph is shown below.
 
@@ -28,8 +28,7 @@ A weighted directed graph is shown below.
 \vfill
 
 \definition{}
-We say a graph is \textit{bipartite} if its nodes can be split into two groups $L$ and $R$ so that no two nodes in the same group are connected by an edge. \\
+We say a graph is \textit{bipartite} if its nodes can be split into two groups $L$ and $R$, where no two nodes in the same group are connected by an edge: \\
-The following graph is bipartite:
 
 \begin{center}
 	\begin{tikzpicture}
@@ -39,9 +38,9 @@ The following graph is bipartite:
 			\node[main] (B) at (0mm, -10mm) {$B$};
 			\node[main] (C) at (0mm, -20mm) {$C$};
 
-			\node[main] (D) at (20mm, 0mm) {$D$};
+			\node[main, hatch] (D) at (20mm, 0mm) {$D$};
-			\node[main] (E) at (20mm, -10mm) {$E$};
+			\node[main, hatch] (E) at (20mm, -10mm) {$E$};
-			\node[main] (F) at (20mm, -20mm) {$F$};
+			\node[main, hatch] (F) at (20mm, -20mm) {$F$};
 		\end{scope}
 
 		% Edges
@@ -55,30 +54,5 @@ The following graph is bipartite:
 	\end{tikzpicture}
 \end{center}
 
-\vfill
-
-\definition{}
-We say two nodes $A$ ane $B$ are \textit{connected} if we can reach $A$ from $B$ and $B$ from $A$ by walking along (possibly directed) edges. We say a graph is connected if all its nodes are connected to each other.\\
-
-The bipartite graph above and the directed graph below are not connected.
-
-\begin{center}
-	\begin{tikzpicture}[node distance = 20mm]
-		% Nodes
-		\begin{scope}
-			\node[main] (A) {$A$};
-			\node[main] (B) [below right of = A] {$B$};
-			\node[main] (C) [below left of = A] {$C$};
-		\end{scope}
-
-		% Edges
-		\draw[->]
-			(A) edge[bend right] (B)
-			(B) edge[bend right] (A)
-			(B) edge (C)
-		;
-	\end{tikzpicture}
-\end{center}
-
 \vfill
 \pagebreak

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 $T$?
+We are interested in the \textit{maximum flow} through this network: what is the greatest amount of flow we can get from $S$ to $T$?
-
-\footnotetext{you could also think of \say{flow} as a directed weighted graph on top of our network.}
 
 \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.
+	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 $2 + 1 > 2$.
-
-	\vspace{1ex}
-
-	This is because 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}
 

Advanced/Graph Algorithms/parts/02 residual.tex

@@ -1,9 +1,10 @@
 \section{Residual Graphs}
-As our network gets bigger, finding a maximum flow by hand becomes much more difficult. It will be convenient to have an algorithm that finds a maximal flow in any network.
+It is hard to find a maximum flow for a large network by hand. \\
+We need to create an algorithm to accomplish this task.
 
 \vspace{1ex}
 
-The first thing we'll need to construct such an algorithm is a \textit{residual graph}.
+The first thing we'll is the notion of a \textit{residual graph}.
 
 \vspace{2ex}
 \hrule
@@ -202,11 +203,9 @@ If it isn't, find a maximal flow. \\
 \problem{}
 Show that...
 \begin{enumerate}
-	\item A maximal flow exists in every network with integral\footnotemark{} edge weights.
+	\item A maximal flow exists in every network with integral edge weights.
 	\item Every edge in this flow carries an integral amount of flow
 \end{enumerate}
 
-\footnotetext{Integral = \say{integer} as an adjective.}
-
 \vfill
 \pagebreak

Advanced/Graph Algorithms/parts/03 fulkerson.tex

@@ -108,7 +108,7 @@ There is extra space on the next page.
 \pagebreak
 
 \problem{}
-You are given a large network. How would you quickly find an upper bound for the number of iterations the Ford-Fulkerson algorithm will need to find a maximum flow?
+You are given a large network. How can you quickly find an upper bound for the number of iterations the Ford-Fulkerson algorithm will need to find a maximum flow?
 
 \begin{solution}
 	Each iteration adds at least one unit of flow. So, we will find a maximum flow in at most $\min(\text{flow out of } S,~\text{flow into } T)$ iterations.

Advanced/Graph Algorithms/parts/04 applications.tex

@@ -86,8 +86,9 @@ A matching is \textit{maximal} if it has more edges than any other matching.
 
 \vspace{5mm}
 
-Devise an algorithm to find a maximal matching in any bipartite graph. \\
+Create an algorithm that finds a maximal matching in any bipartite graph. \\
-Find an upper bound for its runtime.
+Find an upper bound for its runtime. \\
+\hint{Can you modify an algorithm we already know?}
 
 \begin{solution}
 	Turn this into a maximum flow problem and use FF. \\
@@ -101,7 +102,7 @@ Find an upper bound for its runtime.
 \vfill
 \pagebreak
 
-\problem{Circulations with Demand}
+\problem{Supply and Demand}
 
 Say we have a network of cities and power stations. Stations produce power; cities consume it.
 
@@ -146,7 +147,7 @@ We'll represent station capacity with a negative number, since they \textit{cons
 	\end{tikzpicture}
 \end{center}
 
-We'd like to know if there exists a \textit{feasible circulation} in this network---that is, can we supply our cities with the energy they need without exceeding the capacity of power plants or transmission lines?
+We'd like to know if there exists a \textit{feasible circulation} in this network---that is, can we supply our cities with the energy they need without exceeding the capacity of our power plants and transmission lines?
 
 \vspace{2ex}
 

Advanced/Graph Algorithms/parts/05 reductions.tex

@@ -8,11 +8,7 @@ An \textit{independent set} is a set of vertices\footnotemark{} in which no two
 
 \begin{center}
 	\begin{tikzpicture}[
-		node distance = 12mm,
+		node distance = 12mm
-		hatch/.style = {
-			pattern=north west lines,
-			pattern color=gray
-		}
 	]
 		% Nodes
 		\begin{scope}[layer = nodes]
@@ -48,11 +44,7 @@ A \textit{vertex cover} is a set of vertices that includes at least one endpoint
 
 \begin{center}
 	\begin{tikzpicture}[
-		node distance = 12mm,
+		node distance = 12mm
-		hatch/.style = {
-			pattern=north west lines,
-			pattern color=gray
-		}
 	]
 		% Nodes
 		\begin{scope}[layer = nodes]

Advanced/Graph Algorithms/parts/06 bonus.tex

@@ -1,6 +1,6 @@
-\section{Crosses (Bonus Problem)}
+\section{Crosses}
 
-You are given an $n \times n$ grid. Some of its squares are white, and some are gray. Your goal is to place $n$ crosses on white cells so that each row and each column contains exactly one cross.
+You are given an $n \times n$ grid. Some of its squares are white, some are gray. Your goal is to place $n$ crosses on white cells so that each row and each column contains exactly one cross.
 
 \vspace{2ex}
 

Advanced/Graph Algorithms/tikxset.tex

@@ -49,5 +49,9 @@
 		% completely under nodes.
 		line cap = rect,
 		opacity = 0.3
+	},
+	hatch/.style = {
+		pattern=north west lines,
+		pattern color=gray
 	}
 }