Add writings from 3rd of January

thesis/sections/2_background.tex

@@ -325,7 +325,7 @@ \subsection{Distributed computing} \label{sec:distributed_computing}
 
 
 \subsection{Port number model} \label{sec:port_number_model}
-This section is based on the course material from \cite{HirvonenSuomelaDistAlg2020} unless otherwise mentioned.
+This section is based on the excellent textbook \cite{HirvonenSuomelaDistAlg2020} unless otherwise mentioned.
 
 Port number model (PN model) is a rather weak model of computation that inherits from the message passing model.
 In the model, nodes do not have identification.
@@ -359,7 +359,7 @@ \subsection{Port number model} \label{sec:port_number_model}
 \subsection{Formalized port number model}
 In the section \ref{sec:port_number_model} we briefly and informally introduced the PN model.
 Now in this section and in the following subsections we give a more formal definition to the PN model.
-As in the previous section, this section is based on the course material from \cite{HirvonenSuomelaDistAlg2020} unless otherwise mentioned.
+As in the previous section, this section is based on the textbook \cite{HirvonenSuomelaDistAlg2020} unless otherwise mentioned.
 
 PN network is a 3-element tuple $N = (V, P, p)$, where $V$ and $P$ are the sets of vertices and ports respectively, and $p: P \rightarrow P$ is a function that maps a port to another port, forming a communication channel.
 A port, an element of $P$, is a pair $(v, i)$ where $v \in V$ and $i \in \{1, 2, ...\}$.
@@ -660,7 +660,7 @@ \subsection{Covering map}
 \footnote{Note that this is not related to the vertex cover problem even though they share the word \emph{cover}.}.
 Later in this work, we use the concept to show
 
-First we want to show a definition of \emph{covering} for graphs, from an excellent paper
+First we want to show a definition of \emph{covering} for graphs, from  paper
 \cite{DBLP:conf/stoc/Angluin80}:
 \begin{displayquote}
 A graph $H$ is a \emph{covering} of a graph $G$ if there is a way to label the nodes of $H$ with the names of nodes in $G$ in such a way that if a node $x$ of $H$ is labelled "v" then the labels of the neighbors of $x$ are precisely the neighbors of $v$ in G.
@@ -678,14 +678,17 @@ \subsection{Covering map}
 % \end{enumerate}
 In order for us to use covering maps with PN networks, we need to additionally consider the port numbers of nodes so that they are also preserver by the covering map.
 
-We define now the covering map for PN networks similiary it is defined in the excellent course book \cite{HirvonenSuomelaDistAlg2020}.
-Let $N=(V, P, p)$ and $N'=(V', P', p')$ be PN networks and let $\phi: V \rightarrow V'$.
-The function $\phi$ is a covering map from $N$ to $N'$ if all of the following hold:
-\begin{enumerate}
-  \item $\phi$ is a surjection i.e. for every $v' \in V'$ there exists at least one $v \in V$ such that $\phi(v) = v'$.
-  \item $\phi$ preserves the degrees of a node i.e. $\deg_N(v) = \deg_{N'}(\phi(v))$, for all $v \in V$.
-  \item $\phi$ preserves port numbers and connections i.e. if $p((u, i)) = (v, j)$ then $p((\phi(u), i)) = (\phi(v), j)$.
-\end{enumerate}
+We define now the covering map for PN networks similiary it is defined in the excellent textbook \cite{HirvonenSuomelaDistAlg2020}.
+
+\begin{definition} \label{def:covering_map}
+  Let $N=(V, P, p)$ and $N'=(V', P', p')$ be PN networks and let $\phi: V \rightarrow V'$.
+  The function $\phi$ is a covering map from $N$ to $N'$ if all of the following hold:
+  \begin{enumerate}
+    \item $\phi$ is a surjection i.e. for every $v' \in V'$ there exists at least one $v \in V$ such that $\phi(v) = v'$.
+    \item $\phi$ preserves the degrees of a node i.e. $\deg_N(v) = \deg_{N'}(\phi(v))$, for all $v \in V$.
+    \item $\phi$ preserves port numbers and connections i.e. if $p((u, i)) = (v, j)$ then $p'((\phi(u), i)) = (\phi(v), j)$.
+  \end{enumerate}
+\end{definition}
 
 \begin{figure}[H]
   \subcaptionbox{

thesis/sections/4_prior_work.tex

@@ -5,7 +5,7 @@ \section{Prior work} \label{sec:prior_work}
 
 %TODO I'll draft some sections here that I probably should consider writing about.
 % Some might be really similiar or even identical.
-\subsection{\todo{these sections are WIP}}
+\subsection{\todo{these sections are WIP and they do not necessarily stay this way and each probably does not stay under their own section.}}
 \subsection{Lower and upper bounds of LCL's}
 \subsection{Complexity classifications of LCL's}
 \subsection{Computation of lower and upper bounds}

thesis/sections/5_algorithm.tex

@@ -83,6 +83,7 @@ \section{Algorithm} \label{sec:algorithm}
 %TODO Do we need to talk about trees and how these biregular multigraphs relate to them? 'Informally, the idea is to prove negative results for the case in which "we are in the middle of a very large tree, far away from the leaves".'
 
 \subsection{\todo{Couple implications}}
+\footnote{Where should I put these theorems? Do they fit under Section \ref{sec:algorithm}?}
 In this section we show that if an LCL problem is not solvable in PN networks with multiple connections, then the problem is also not solvable in simple PN networks.
 We try to split this into couple of theorems and at the end utilize the shown theorems to show this to be true.
 
@@ -100,35 +101,49 @@ \subsection{\todo{Couple implications}}
 
 % Network with multiple connections ----k-lift--->  simple network
 \begin{theorem} \label{thm:lcl_nonsolvability:3}
-    If there is a PN network $N$ with multiple connections with $k$ being the highest count of multiple connections between any two nodes, then there exists a $k$-lift $N'$ of $N$ such that $N'$ is a simple PN network.
+    If there is a PN network $N_2$ with multiple connections with $k$ being the highest count of multiple connections between any two nodes, then there exists a $k$-lift $N_1$ of $N_2$ such that $N_1$ is a simple PN network.
 \end{theorem}
 \begin{proof}
-    Let $N=(V, P, p)$ be a PN network with multiple connections.
-    Let $\operatorname{mul}(u, v)$ be the number of connections between any nodes $u, v \in V$.
-    Let $k=\max {m(u, v) | u, v \in V}$ i.e. the maximum number of multiple connections.
+    Let $N_2=(V_2, P_2, p_2)$ be a PN network with multiple connections.
+    Let $\operatorname{mul}(u, v)$ be the number of connections between any nodes $u, v \in V_2$.
+    Let $k=\max {m(u, v) | u, v \in V_2}$ i.e. the maximum number of multiple connections.
     \footnote{\todo{Is the "maximum number of multiple connections" ambiguous? Does it appear as the highest count of parallel connections or as the total number of multiple connections in the network?}}
 
-    Let there be another network $N'=(V', P', p')$ such that:
+    Let there be another network $N_1=(V_1, P_1, p_1)$ such that:
     \begin{itemize}
-        \item For each $v \in V$, there are $k$ clones, namely the nodes $v_1, v_2, ..., v_k \in V'$.
-            Thus the size $|V'|$ is $k|V|$.
-        \item For each port $(v, i) \in P$, we have port $(v_x, i) \in P'$ where $x=1, 2, ..., k$.
-        \item For each non multiple connection $p((v, i)) = (u, j)$, we have a connections $p'((v_x, i)) = (u_x, j)$, where $x=1, 2, ..., k$.
-        \item For each multiple connections $p((v, i_a)) = (u, j_a)$ and each $a = 1, 2, ..., \operatorname{mul}(u, v)$, we have $p'((v_{x}, i_a)) = (u_{x+a-1}, j_a)$.
+        \item For each $v \in V_2$, there are $k$ clones in $V_1$, namely the nodes $v_2, v_1, ..., v_k \in V_1$.
+            Thus the size $|V_1|$ is $k|V_2|$.
+        \item For each port $(v, i) \in P_2$, we have each port $(v_x, i) \in P_1$ where $x=1, 2, ..., k$.
+        \item For each non multiple connection $p_2((v, i)) = (u, j)$, we have connections $p_1((v_x, i)) = (u_x, j)$, where $x=1, 2, ..., k$.
+        \item For each multiple connections $p_2((v, i_a)) = (u, j_a)$ where $a = 1, 2, ..., \operatorname{mul}(u, v)$, we have $p_1((v_{x}, i_a)) = (u_{x+a-1 \mod k}, j_a)$.
     \end{itemize}
 
-    Now we show that $N'$ is a simple PN network.
+    Now we show that there is a covering map $\phi: V_1 \rightarrow V_2$.
+    Let $\phi(v_i) = v \in V_2$ for each $v_i \in V_1$ where $i=1, 2, ..., k$.
+    We will show that $\phi$ is a covering map using the Definition \ref{def:covering_map}:
+    \begin{itemize}
+        \item By the definition of $\phi$, it is surjective.
+        \item For each connection in $N_2$, we have $k$ similiar connections in $N_1$, therefore degrees of each node are preserved.
+        \item For each non multiple connection $p_1((v_x, i)) = (u_x, j)$, where $x=1, 2, ..., k$, we have $p_2((v, i)) = (u, j)$.
+        From our definition of $\phi$ we can see that the mapping preserves port numbers and connections in non multiple connections.
+        \item For each multiple connection $p_1((v_{x}, i_a)) = (u_{x+a-1 \mod k}, j_a)$ we have
+        \begin{align*}
+            p_2((\phi(v_{x}), i_a)) &= (\phi(u_{x+a-1 \mod k}), j_a)\\
+            \Leftrightarrow p_2((v, i_a)) &= (u, j_a)
+        \end{align*}
+        Both $(v, i_a)$ and $(u, j_a)$ are in $P_2$ and $p_2((v, i_a)) = (u, j_a)$, therefore for each multiple connection, the port numbers and connections are preserved.
+    \end{itemize}
+    Each condition holds, hence the function $\phi$ is a covering map from $V_1$ to $V_2$
+    therefore $N_1$ is a $k$-lift of $N_2$.
 
-    Finally we need to show that there is a covering map $\phi: V' \rightarrow V$.
+    Finally we need to show that $N_1$ is a simple PN network i.e. it does not have multiple edges.
     \todo{Complete this proof}
-    Function $\phi$ is a covering map, therefore $V'$ is a k-lift of $V$.
-
 \end{proof}
 
 \begin{figure}[H]
  \centering
  \includegraphics[]{example-image-duck}
- \caption{Illustration of k-lift using Theorem \ref{thm:lcl_nonsolvability:3} }
+ \caption{\todo{Illustration of k-lift using Theorem \ref{thm:lcl_nonsolvability:3}}}
  \label{fig:duck2}
 \end{figure}
 %% This is more like an instruction on how to build the network, not a proof or is it?