From 89dcb7b6f687ecdf9a2a1f674add3e40f16f5dc3 Mon Sep 17 00:00:00 2001 From: Andrew Date: Mon, 13 Jan 2025 21:08:32 +0000 Subject: [PATCH] [CT420]: Set up notes directory --- .../CT420: Real-Time Systems/notes/CT420.tex | 100 +++ .../report/images/uniog.jpg | Bin 109772 -> 0 bytes .../report/report.pdf | Bin 204431 -> 0 bytes .../report/report.tex | 649 ------------------ 4 files changed, 100 insertions(+), 649 deletions(-) create mode 100644 year4/semester2/CT420: Real-Time Systems/notes/CT420.tex delete mode 100644 year4/semester2/CT420: Real-Time Systems/report/images/uniog.jpg delete mode 100644 year4/semester2/CT420: Real-Time Systems/report/report.pdf delete mode 100644 year4/semester2/CT420: Real-Time Systems/report/report.tex diff --git a/year4/semester2/CT420: Real-Time Systems/notes/CT420.tex b/year4/semester2/CT420: Real-Time Systems/notes/CT420.tex new file mode 100644 index 00000000..ad784684 --- /dev/null +++ b/year4/semester2/CT420: Real-Time Systems/notes/CT420.tex @@ -0,0 +1,100 @@ +% ! TeX program = lualatex +\documentclass[a4paper,11pt]{article} +% packages +\usepackage{censor} +\StopCensoring +\usepackage{fontspec} +\setmainfont{EB Garamond} +% for tironian et fallback +% % \directlua{luaotfload.add_fallback +% % ("emojifallback", +% % {"Noto Serif:mode=harf"} +% % )} +% % \setmainfont{EB Garamond}[RawFeature={fallback=emojifallback}] + +\setmonofont[Scale=MatchLowercase]{Deja Vu Sans Mono} +\usepackage[a4paper,left=2cm,right=2cm,top=\dimexpr15mm+1.5\baselineskip,bottom=2cm]{geometry} +\setlength{\parindent}{0pt} + +\usepackage{fancyhdr} % Headers and footers +\fancyhead[R]{\normalfont \leftmark} +\fancyhead[L]{} +\pagestyle{fancy} + +\usepackage{microtype} % Slightly tweak font spacing for aesthetics +\usepackage[english]{babel} % Language hyphenation and typographical rules +\usepackage{xcolor} +\definecolor{linkblue}{RGB}{0, 64, 128} +\usepackage[final, colorlinks = false, urlcolor = linkblue]{hyperref} +% \newcommand{\secref}[1]{\textbf{§~\nameref{#1}}} +\newcommand{\secref}[1]{\textbf{§\ref{#1}~\nameref{#1}}} + +\usepackage{changepage} % adjust margins on the fly + +\usepackage{minted} +\usemintedstyle{algol_nu} + +\usepackage{pgfplots} +\pgfplotsset{width=\textwidth,compat=1.9} + +\usepackage{caption} +\newenvironment{code}{\captionsetup{type=listing}}{} +\captionsetup[listing]{skip=0pt} +\setlength{\abovecaptionskip}{5pt} +\setlength{\belowcaptionskip}{5pt} + +\usepackage[yyyymmdd]{datetime} +\renewcommand{\dateseparator}{--} + +\usepackage{enumitem} + +\usepackage{titlesec} + +\author{Andrew Hayes} + +\begin{document} +\begin{titlepage} + \begin{center} + \hrule + \vspace*{0.6cm} + \Huge \textsc{ct420} + \vspace*{0.6cm} + \hrule + \LARGE + \vspace{0.5cm} + Real-Time Systems + \vspace{0.5cm} + \hrule + + \vfill + + \hrule + \begin{minipage}{0.495\textwidth} + \vspace{0.4em} + \raggedright + \normalsize + \begin{tabular}{@{}l l} + Name: & Andrew Hayes \\ + Student ID: & 21321503 \\ + E-mail: & \href{mailto://a.hayes18@universityofgalway.ie}{a.hayes18@universityofgalway.ie} \\ + \end{tabular} + \end{minipage} + \begin{minipage}{0.495\textwidth} + \raggedleft + \vspace*{0.8cm} + \Large + \today + \vspace*{0.6cm} + \end{minipage} + \medskip\hrule + \end{center} +\end{titlepage} + +\pagenumbering{roman} +\newpage +\tableofcontents +\newpage +\setcounter{page}{1} +\pagenumbering{arabic} + +\end{document} diff --git a/year4/semester2/CT420: Real-Time Systems/report/images/uniog.jpg b/year4/semester2/CT420: Real-Time Systems/report/images/uniog.jpg deleted file mode 100644 index e25e5fe8b3006b61f969e15c924a686adc9a546e..0000000000000000000000000000000000000000 GIT binary patch literal 0 HcmV?d00001 literal 109772 zcmbT7WmFtN*XIZK;I0X-!QF!s+y@Ju83^t!0fM^?E`z&!AOvR^oZt?@-3i$|@9uf` zoc*-Bx2itX?e6-WKHay@{onfg>+dE2Penml0f2yj0QmH;0sg)M#6D=t%jjxoi_+Oy zbAp|1?C3msxVY(9JlsHzc67=rigYTP@``kx!rZJJbS@qi4j^|QIte-vZp^_9 zAfWt1{SWQ`G6ckbYeYpu$H2t;=g@)&Ktw=7LPSRT&#M1<2mG4{AmgJDyycNWCDgJ& 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zKK)!aK1y4F#|hiq6wY^zBkOF|C!MqyJ;FiWs0Im*6mVY-XNsr5_ne*8+oj>%Pr8z? zS=oHN$kgJya|b$a{o1c!x=c~fRmfEc4U~Etr)B24v=L&~XM{9D(g(_1;E4B(hRDWG z9#uJ2xt|AK96-9(bOEyjEr?I3gppNFvw0p*~BhumxSypwcnv+Y2$}`u!;_?|v z$@TDL5Wj3L>kq_(>4p~#+wGi}QkWdGva@)MJ}cB=cp$pgpmOR43n(?EwU-2jX&RG0 z)pedJUMcsSd{?f#SgefpU1^$ER?484GQ&g#pedXyNy2N2UO&LM4c`>*c graphs; -} - -public class Graph { - public LocalDateTime timeGenerated; // need to import java.time.LocalDateTime for this to work - - // set of Edge objects encapsulated within the Graph object - public Set edges; -} - -public class Edge { - public float distance; - - // set of Player objects encapsulated within the Edge object - there should be no more than 2 - public Set nodes; - -} - -public class Player { - // potential data fields that would be contained within the Player class - public String name; - public int playerId; // assuming each player in the league has a unique ID -} -\end{minted} -\caption{Sample Java Code to Represent the Proposed Data Structure} -\end{code} - -\subsection{Alternative Data Structures} -Another good (perhaps more conventional) choice of data structure to represent a graph would be to use an Adjacency Matrix. -This has the benefit of being very simple: it is simply a 2D array with a row \& column for each node. -Nodes that share an edge should have the length of this edge recorded in the cells at the intersections between their rows \& columns. -One drawback of the adjacency matrix is the data duplication: we store each value twice. -It also doesn't make it very easy to store information on the nodes in an easily retrievable manner. -It also is not very easy to change the size of dynamically. -However, it is simple, and highly programming-paradigm agnostic, so it too would be a good choice. - -\section{Algorithm to Measure the Similarity of Two Graphs} -I think that the most appropriate algorithm for measuring the similarity of two graphs representing this kind of data is -Graph Edit Distance. -Graph Edit Distance measures the similarity of two graphs by counting the number of primitive graph operations that would -be required to transform one of the graphs into the other. -I feel that this approach is particularly appropriate in the context of sports teams, as we can compare the number of -graph operations required to turn one graph into another to the number of player movements required to turn one formation -into another. -\\\\ -Getting the GED of two graphs is greatly simplified if the nodes can be set up in a one-to-one correspondence. -If such a relationship between nodes could be established for our data, then the task of computing the number of operations -required to transform one graph into the other would be much easier. -Getting the identity of player nodes is more or less difficult depending on the sport in question: -\begin{itemize} - \item For most team sports, players have set positions on a team. - This makes it easy to establish equivalenced between nodes: if we were comparing two rugby team graphs, we - could compare the scrum-half node from one graph to the scrum-half node of the other. - I am making the simplifying assumption that when we compare two graphs, the teams represented in those two - graphs will be teams playing the same sport. - - \item For some team sports however, such as soccer, there are no set positions. - There are common positions and formations, but these are not set in stone. - This makes it much harder to establish a correspondence between equivalent nodes in different graphs. - My proposed solution for establishing node identity for this type of sport is to identify players by their - distance from some mostly stationary player. - In the case of soccer, players would be identified by their distance from the goalkeeper of their team; - in the majority of cases, the closest players will be the backs, the furthest will be the forwards, and the - middle players would be the midfielders. - This is a less precise correspondence than the one for sports with set positions, as the left back and the - right back for example might be of essentially the same distance from the goalkeeper causing the identities to - be confused, but without having set positions, it is difficult to compare the graphs of two potentially entirely - unrelated teams. -\end{itemize} - -Before calculating the Graph Edit Distance, we must also define what we consider to be primitive graph operations in this -context. -Since we are dealing with a very specific type of data with weighted edges, I will be using some specific graph operations. -I will not be using ``substitution'' operations, as this is not something that I think is important to this analysis, i.e. -if we are comparing two team graphs, and the formation of the two is identical, but one has John Smith playing as goalkeeper -and the other has Joe Murphy, we don't care about the specific identity of the nodes. -Since the formation is the same, we should consider the graphs to be identical without having to substitute the John Smith -node for the Joe Murphy node. -The primitive graph operations that I will be considering are the following: -\begin{itemize} - \item Node insertion \& deletion: if a player is missing from one graph, to transform one graph into the other we - will need to insert or delete a node. - - \item Distance lengthening \& shortening: if the weight of an edge between two nodes is greater in one graph, to - transform one graph into the other we will need to lengthen or shorten the distance between the nodes. - I am assuming here that we are dealing only with complete graphs, i.e. each node is adjacent to every other node. - For a simple measurement of how different the two graphs are, I am going to treat the distances between nodes - as if one can be changed without changing all the others. - Of course, on a real pitch, if the distance between two players was to get shorter, it would mean that one or both - of them had moved, in turn changing the distances between them and the other players. - I am going to count changing the length of an edge by 1 unit to be a simple operation, so if an edge were - to be transformed to be 5 units shorter, that would be 5 operations, and if it were to be made 10 units longer, - that would be 10 units. -\end{itemize} - -Essentially, my approach can be boiled down to establishing a one-to-one correspondence between the nodes, and then comparing -the different edge lengths that those equivalent nodes have, and summing up the absolute value of the differences to get a -dissimilarity score, with the higher the score, the more dissimilar the graphs. -The algorithm would be as follows: -\begin{enumerate} - \item Establish a one-to-one correspondence between the nodes in the graph, by position if applicable for the sport, - or by distance from some root player (such as the goalkeeper) otherwise. - If one graph has more nodes than the other, note this. - - \item Consider how many nodes would need to be added/deleted to/from Graph 1 to transform it into Graph 2. - Take the insertion/deletion of a node to be one operation, ignoring the length of those nodes' edges. - Add the count of these operations to the overall dissimilarity score of the graphs. - - \item For each pair of equivalent nodes in the two graphs, identify the equivalent edges by the ones that link to - equivalent nodes. - For each of these equivalent edges, get the absolute value of the difference between the two edge lengths. - Add this absolute value to the overall dissimilarity score of the graph. -\end{enumerate} - -This gives a simple yet meaningful measure of the differences between the two graphs. -One way that this could be improved is to use some algorithm to calculate how to re-arrange the graph such that each -edge was the correct length to reflect the physical reality of potential distances between players. - -\subsection{Alternative Algorithms} -There are, of course, other algorithms that could be employed to calculate the similarity of two graphs. -One such algorithm that could be applicable in this context is the Global Clustering Coefficient of the graph, which would tell us how likely nodes within that -graph are to form clusters, i.e. how likely it is that a node's neighbours are also connected to each other. -This would give us an idea of how spread out the team is on the pitch, which would likely be useful for certain types of analysis. -\\\\ -My initial idea for comparing the similarity of two graphs was to represent them as vectors and use Cosine Similarity. -This was particularly appealing to me because it's a consistent and meaningful measure of the similarity of two vectors, -giving a score between $-1$ and $1$, and doesn't require re-inventing the wheel or any complex computations. -Furthermore, Cosine Similarity doesn't consider the magnitude of a vector, so we could consider the shape of the graph and -ignore the exact distance between nodes if our vectorisation approach was appropriate. -Even more appealing was the fact that Cosine Similarity can compare vectors with different numbers of elements, meaning -that we could compare a team with 11 players to one that was missing a player and only had 10. -Everything about the Cosine Similarity approach was extremely appealing to me, and I spent a lot of time trying to get it -to work, but I failed to define an algorithm that could translate a graph into a vector in a meaningful manner, such that -the direction of the vector in $N$ dimensions (with $N$ being the number of nodes in the graph) said something meaningful -about the overall shape of the graph. -I still feel that there is a lot of potential in the Cosine Similarity approach, if only I had a meaningful way of -representing a graph as a vector. -My attempts usually resulted in vectors that pointed in roughly the same direction regardless of the shape of the graph, as -I calculated the vectors by having one element per node, and each element to be the ``value'' of the node, defined by -$\mathrm{value}(v) = \alpha \cdot \mathrm{Degree}(v) - \sum^n_{i = 1}{d_i}$ where $v$ is a vertex in the graph, -$\alpha$ is some weight to emphasise the importance of the degree of the vertex, from which we subtract the sum of the -length of each \textbf{d}istance value from 1 to $n$ for an $n$-node graph. -The issue with this approach is that it generally generated quite similar vectors regardless of the actually similarity -of the graphs, much to my disappointment. - - -\section{Degree \& In-Betweenness} -\subsection{Calculate the Degree of Each Node} -Given a snapshot graph wherein edges shorter than some length $k$ are discarded, and the data structure outlined -in \secref{sec:data_structure}, we can calculate the degree -of each node by iterating over the set of edges and incrementing a variable for each player when it is contained -by an \mintinline{java}{Edge} object. -This can be achieved with the following Java code: - -\begin{code} -\begin{minted}[texcl, mathescape, linenos, breaklines, frame=single]{Java} -// Depending on how often we intend to do this calculation and how we intend to do it, it would likely be better to put this method in the `Graph` class. -// However, to keep in line with the simplicity of the classes as defined previously, I have opted to pass the Graph to the method as an argument rather than change the Graph class. - -// Input: A snapshot Graph object. -// Output: A HashMap data structure in which the key is a Player (node) object and the value accessible by that key is the degree of that Player (node) object. -public HashMap calculateDegrees(Graph graph) { - HashMap returnValues = new HashMap<>(); - - // looping over each Edge in the Graph - for (Edge edge: graph.edges) { - // loop over each Player in the Edge's `nodes` Set - // there should really be no more than two, practically speaking, and no less than 2 assuming that we are not dealing with multigraphs but the number of nodes an edge joins doesn't matter from an algorithmic perspective - for (Player player : edge.nodes) { - // get the current degree count for the player if it is already defined - Integer degree = returnValues.get(player); - - // if the degree is not yet defined, set it to 1 - if (degree == null) { - degree = 1; - } - // otherwise increment it by 1 - else { - degree++; - } - - // set the player's degree in returnValues to the updated value - returnValues.put(player, degree); - } - } -} -\end{minted} -\caption{Java Code to Calculate the Degree of Each Node in a Graph Snapshot} -\end{code} - -\subsection{Determine Which Node(s) is/are On the Most Paths} -Before we can determine which node is on the most paths, we must first define what we mean by that. -I will take a path in this context to be a finite sequence of nodes, with no repetition of nodes. -Here, I am making the assumption that each pair of nodes can only have one edge joining them, which is true for this kind of data. -I am also making the simplifying assumption that each path is a ``simple path'' and that a node cannot appear in a path twice, i.e. I am excluding cyclic paths. -I am also not considering single nodes to be a path, i.e. a path must contain more than one node to be considered a path. -We must note that since we are dealing with undirected graphs, the paths too are undirected, so a path $A \rightarrow B \rightarrow C$ is the same as the path -$C \rightarrow B \rightarrow A$. -\\\\ -Firstly, we must find each extant simple path in the graph. -Since we are dealing only with simple paths, we know that no node can be repeated in a path, i.e. each path contains at most $N$ nodes, where $N$ is the -number of nodes in the graph. -We will represent a path as a Java \mintinline{java}{ArrayList} of \mintinline{java}{Player} objects. -Each path of length $i$ will be stored with the other paths of that length in an \mintinline{Java}{Set} of \mintinline{Java}{ArrayList}s, which is a data structure that allows no -duplicates and ignores any attempt to insert a duplicate object. -We are going to generate each path of length $i$ by appending nodes (where there is a joining edge) to paths of length $i-1$ that we have already generated. -Then we will have a \mintinline{java}{Set} of each \emph{directed} path in the graph, as we have not considered when generating the paths that a path $A \rightarrow B \rightarrow C \cong -C \rightarrow B \rightarrow A$. -Each \mintinline{java}{Set} containing each path of length $i$ actually contains double the amount of paths that it should, as it has two of each path: one forwards, one reversed. -There are two options for dealing with this: -\begin{enumerate} - \item Remove the duplicate paths somehow, most likely by looping over each one backwards. - While probably more technically correct \& robust, this seems like a lot of work. - - \item Ignore the problem. - Since we know that there are two of each path in our sets, we know that when we count how many times a given \mintinline{java}{Player} object occurs in a path, we will get a - number twice the size that it should be, and we can just half it to get the correct value. - If we wanted to be even more lazy, we could get away with not even halving the count to get the correct value, as we just want to find the largest count relative to all the others, - and all the counts being off by a factor of $2$ doesn't affect our ability to do this. -\end{enumerate} - -I have opted to ignore the problem, as justified by \secref{sec:data_structure_code}. -My proposed algorithm to determine which node is on the greatest number of paths using the data structure proposed in \secref{sec:data_structure} is as -follows: -\begin{enumerate} - \item Declare a \mintinline{java}{HashMap} where the key is the \mintinline{java}{Player} and the value stored is the number of times that \mintinline{java}{Player} occurs on - a path in the graph. - - \item Then, we must generate the ``paths'' of length $1$. - We won't consider these paths for the calculation of which node is on the most paths, but we will use these to build up our paths of length $2$. - We will loop over each \mintinline{Java}{Edge} in the graph, and insert each of its \mintinline{java}{Player} nodes into their own \mintinline{java}{ArrayList} of length $1$, - which will then each by inserted into a \mintinline{java}{Set} containing all the ``paths'' of length $1$. - The \mintinline{java}{Set} will ensure that the single-node paths have no duplicates. - - \item Loop from $i=2$ to $i=N$ building up paths of length $i$. - We will do this by looping over each path of length $i-1$ in a sub-loop, and looping over each \mintinline{java}{Edge} in the graph in a sub-sub-loop. - While having nested loops is usually a bad sign, I feel justified in making use of them as finding each path requires it to be traversed, meaning that this will be a costly - process no matter what. - For each path of length $i-1$, we will check if its final node is also one of the nodes in each \mintinline{java}{Edge} object. - If it is, we know that the \mintinline{java}{Edge} object's other node could be appended to this path to make a path of length $i$. - Before doing this however, we will check if this other node is already in the path (as we are not allowing duplicate nodes) by using the \mintinline{java}{contains()} method - (which is computationally equivalent to looping over the whole list and comparing each object it contains to the node we want to append). - If the node is already in the path, we ignore it; otherwise, we make a new \mintinline{java}{ArrayList} by appending the node (\mintinline{java}{Player}) to the path and inserting - it into the \mintinline{java}{Set} of all paths of length $i$. - Then we increment the count for that \mintinline{Java}{Player} object in the \mintinline{Java}{HashMap}. - - \item Return the \mintinline{java}{Player} object(s) from the \mintinline{Java}{HashMap} that has/have the highest count. -\end{enumerate} - -\subsubsection{Java Code to Determine Which Node(s) is/are On the Most Paths} -\begin{code} -\begin{minted}[texcl, mathescape, linenos, breaklines, frame=single]{Java} -// Input: A snapshot Graph object -// Output: The a Set of Player objects that are on the most paths -public Set getMostInfluentialNode(Graph graph) { - HashMap counts = new HashMap<>(); - - // an ArrayList to hold the Sets of paths generated - ArrayList>> setsOfPaths = new ArrayList<>(); - - // get a set of all the nodes in the graph - Set nodes = new HashSet<>(); - for (Edge edge : graph.edges) { - for (Player player : edge.nodes) { - nodes.add(player); - } - } - - // generate the "paths" of length 1 - Set> oneNodePaths = new HashSet<>(); - for (Player player : nodes) { - ArrayList path = new ArrayList<>(); - path.add(player); - oneNodePaths.add(path); - } - - // loop from i = 2 to i = N building up paths of length i - for (int i = 2; i <= oneNodePaths.size(); i+=) { - // get the Set of paths of length i-1 - Set> iMinusOneLengthPaths = setsOfPaths.get(i-1); - - // create the Set of paths of length i - Set> iLengthPaths = new HashSet<>(); - - // loop over each path of length i-1 - for (ArrayList path : iMinusOneLengthPaths) { - // loop over each edge in the graph - for (Edge edge : graph.edges) { - // check if the last node of the path is in the Edge - Player lastNode = path.get(path.size()-1); - - // convert Set to Array so we can refer to the nodes by indices - // assuming here that each edge contains only two nodes -- more robust code would check for this - Player[] players = edge.nodes.toArray(new Player[2]); - - // the code repetition here is not ideal, but it allows everything to be kept in one single method for readability - - // if the 0th node of players is the same as lastNode, the 1st node of players can be appended - if (lastNode.equals(players[0])) { - // if players[1] is not already in the path - if (!path.contains(players[1])) { - // create new ArrayList to represent the path - ArrayList newPath = new ArrayList<>(path); - newPath.append(players[1]); - - // increment the count for players[1] - if (counts.containsKey(players[1])) { - int newCount = counts.get(players[1])++; - } - else { - int newCount = 1; - } - counts.put(players[1], newCount); - } - } - // else if the 1st node of players is the same as lastNode, the 0th node of players can be appended - else if (lastNode.equals(players[1])) { - // if players[0] is not already in the path - if (!path.contains(players[0])) { - // create new ArrayList to represent the path - ArrayList newPath = new ArrayList<>(path); - newPath.append(players[0]); - - // increment the count for players[0] - if (counts.containsKey(players[0])) { - int newCount = counts.get(players[0])++; - } - else { - int newCount = 1; - } - counts.put(players[0], newCount); - } - } - } - } - } - - // find the highest value in the hashmap - // will be double what it technically should be for an undirected graph but it doesn't matter - int highestCount = 0; - for (int count : counts.values) { - highestCount = (count > highestCount) ? count : highestCount; - } - - // Set of Players with the highest counts to return - Set mostInfluential = new HashSet<>(); - for (Player player : counts.keySet()) { - if (counts.get(player) == highestCount) { - mostInfluential.add(player); - } - } - - return mostInfluential; -} -\end{minted} -\caption{Java Code to Determine Which Node(s) is/are On the Most Paths} -\end{code} -\end{document}