Add second year
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||||
|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
[14] (./CT2109-Notes.aux)
|
||||
|
||||
Package rerunfilecheck Warning: File `CT2109-Notes.out' has changed.
|
||||
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|
||||
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|
||||
|
||||
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|
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|
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|
||||
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|
||||
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|
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|
||||
texlive/2023/texmf-dist/fonts/type1/urw/times/utmri8a.pfb>
|
||||
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|
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|
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|
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|
||||
|
||||
29
second/semester2/CT2109/Notes/CT2109-Notes.out
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|
||||
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|
||||
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second/semester2/CT2109/Notes/CT2109-Notes.pdf
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second/semester2/CT2109/Notes/CT2109-Notes.pdf
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second/semester2/CT2109/Notes/CT2109-Notes.tex
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|
||||
\documentclass[11pt]{article}
|
||||
|
||||
\usepackage{report}
|
||||
\usepackage{algpseudocode}
|
||||
\usepackage{algorithm}
|
||||
\usepackage[utf8]{inputenc} % allow utf-8 input
|
||||
\usepackage[T1]{fontenc} % use 8-bit T1 fonts
|
||||
\usepackage[colorlinks=true, linkcolor=black, citecolor=blue, urlcolor=blue]{hyperref} % hyperlinks
|
||||
\usepackage{url} % simple URL typesetting
|
||||
\usepackage{booktabs} % professional-quality tables
|
||||
\usepackage{amsfonts} % blackboard math symbols
|
||||
\usepackage{nicefrac} % compact symbols for 1/2, etc.
|
||||
\usepackage{microtype} % microtypography
|
||||
\usepackage{lipsum} % Can be removed after putting your text content
|
||||
\usepackage{graphicx}
|
||||
\graphicspath{ {./images/} }
|
||||
\usepackage{natbib}
|
||||
\usepackage{doi}
|
||||
\setcitestyle{aysep={,}}
|
||||
\usepackage{array}
|
||||
|
||||
\usepackage{listings}
|
||||
|
||||
\usepackage{xcolor}
|
||||
\definecolor{codegreen}{rgb}{0,0.6,0}
|
||||
\definecolor{codegray}{rgb}{0.5,0.5,0.5}
|
||||
\definecolor{codeorange}{rgb}{1,0.49,0}
|
||||
\definecolor{backcolour}{rgb}{0.95,0.95,0.96}
|
||||
|
||||
\lstdefinestyle{mystyle}{
|
||||
backgroundcolor=\color{backcolour},
|
||||
commentstyle=\color{codegray},
|
||||
keywordstyle=\color{codeorange},
|
||||
numberstyle=\tiny\color{codegray},
|
||||
stringstyle=\color{codegreen},
|
||||
basicstyle=\ttfamily\footnotesize,
|
||||
breakatwhitespace=false,
|
||||
breaklines=true,
|
||||
captionpos=b,
|
||||
keepspaces=true,
|
||||
numbers=left,
|
||||
numbersep=5pt,
|
||||
showspaces=false,
|
||||
showstringspaces=false,
|
||||
showtabs=false,
|
||||
tabsize=2,
|
||||
xleftmargin=10pt,
|
||||
}
|
||||
|
||||
\lstset{style=mystyle}
|
||||
|
||||
|
||||
\title{CT2109 - Object Oriented Programming: Data Structures \& Algorithms}
|
||||
|
||||
\author{Andrew Hayes\\
|
||||
\AND
|
||||
\AND
|
||||
\AND
|
||||
\AND
|
||||
\AND
|
||||
2BCT\\
|
||||
\AND
|
||||
University of Galway\\
|
||||
}
|
||||
|
||||
% Uncomment to remove the date
|
||||
% \date{February 2022}
|
||||
|
||||
% Uncomment to override the `A preprint' in the header
|
||||
\renewcommand{\headeright}{Object Oriented Programming: Data Structures \& Algorithms}
|
||||
\renewcommand{\undertitle}{Object Oriented Programming: Data Structures \& Algorithms}
|
||||
\renewcommand{\shorttitle}{}
|
||||
|
||||
%%% Add PDF metadata to help others organize their library
|
||||
%%% Once the PDF is generated, you can check the metadata with
|
||||
%%% $ pdfinfo template.pdf
|
||||
% \hypersetup{
|
||||
% pdftitle={A template for the arxiv style},
|
||||
% pdfsubject={q-bio.NC, q-bio.QM},
|
||||
% pdfauthor={David S.~Hippocampus, Elias D.~Striatum},
|
||||
% pdfkeywords={First keyword, Second keyword, More},
|
||||
% }
|
||||
|
||||
\begin{document}
|
||||
\maketitle
|
||||
|
||||
\newpage
|
||||
\tableofcontents
|
||||
\thispagestyle{empty}
|
||||
\setcounter{page}{1}
|
||||
|
||||
\newpage
|
||||
\section{Abstract Data Types}
|
||||
An \textbf{Abstract Data Type (ADT)} is an abstract model of a data structure that specifies the data stored \& oeprations that may be performed on the data.
|
||||
An ADT specifies \textit{what} each operation does, but not \textit{how}.
|
||||
In object-oriented languages such as Java, this naturally corresponds to an \textbf{interface definition}.
|
||||
An ADT is \textit{realised} as a concrete data structure.
|
||||
In Java, this is a class that \textbf{implements} the interface.
|
||||
|
||||
\begin{center}
|
||||
\includegraphics[width=0.3\textwidth]{adt.png}
|
||||
\end{center}
|
||||
|
||||
\textbf{Composite ADTs} are used manage \textit{collections} of data, e.g., Arrays, Lists, Stacks, Queues, Hash Tables, etc.
|
||||
|
||||
\subsection{Stacks \& Queues}
|
||||
\textbf{Stacks} \& \textbf{Queues} are linearly ordered ADTs for list-structured data.
|
||||
|
||||
\subsubsection{Stacks}
|
||||
A \textbf{Stack} is a last in, first out (LIFO) data structure.
|
||||
No sort order is assumed.
|
||||
Items can only enter or leave via the \textit{top} of the stack.
|
||||
Items can be \textbf{pushed} \& \textbf{popped} to add \& remove.
|
||||
Example applications of a stack include processing nested structures or the ``undo'' operation in an editor.
|
||||
Objects stored in a stack are a \textit{finite sequence} of elements of the \textbf{same type}.
|
||||
|
||||
Stacks have few operations.
|
||||
For a stack \verb|s|, node \verb|n|, \& boolean value \verb|b|:
|
||||
\begin{itemize}
|
||||
\item \verb|s.push(n)| - Place item \verb|n| on top of the stack.
|
||||
\item \verb|s.pop()| $\rightarrow$ \verb|n| - Remove top item from the stack \& return it.
|
||||
\item \verb|s.top| $\rightarrow$ \verb|n| - Examine the top item on the stack without removing it.
|
||||
\item \verb|s.isEmpty()| $\rightarrow$ \verb|b| - Returns \verb|b = true| if the stack is empty.
|
||||
\item \verb|s.isFull()| $\rightarrow$ \verb|b = true| if the stack is full (relevant if storage is limited).
|
||||
\end{itemize}
|
||||
|
||||
Java has a built-in stack interface from \verb|java.util.Stack|.
|
||||
However, we will look at making our own for the sake of learning.
|
||||
Our stack implementation may look something like this:
|
||||
\begin{lstlisting}[language=java]
|
||||
public interface Stack {
|
||||
public void push(Object n);
|
||||
public Oject pop();
|
||||
public Object top();
|
||||
public boolean isEmpty();
|
||||
public boolean isFull();
|
||||
\end{lstlisting}
|
||||
|
||||
Other stack operations include \verb|size()| \& \verb|makeEmpty()|.
|
||||
We could implement this stack using an array, linked list, or other storage type.
|
||||
|
||||
\subsubsection{Queues}
|
||||
A \textbf{Queue} is a first in, first out (FIFO) data structure.
|
||||
No sort order is assumed.
|
||||
Items enter at the rear of the queue, and leave at the front of the queue.
|
||||
Items can be \textbf{enqueued} \& \textbf{dequeued} to add \& remove them from the queue.
|
||||
Example applications of a queue include ensuring ``fair treatment'' to each of a list of pending tasks (first come, first served)
|
||||
or simulation: modelling \& analysis of real-world problems.
|
||||
Objects stored in a queue are a finite sequence of elements of the same type.
|
||||
The item at the front of the queue has been in the queue the longest, while the item at the rear has entered the queue most recently.
|
||||
|
||||
Queues have few operations.
|
||||
For a queue \verb|q|, element \verb|e|, \& boolean value \verb|b|:
|
||||
\begin{itemize}
|
||||
\item \verb|q.enqueue(e)| - Place \verb|e| at the rear of \verb|q|, assuming there is space.
|
||||
\item \verb|q.dequeue()| $\rightarrow$ \verb|e| - Remove fron item \verb|e| from \verb|q| and return it.
|
||||
\item \verb|q.front()| $\rightarrow$ \verb|e| - Returns front element \verb|e| without removing it.
|
||||
\item \verb|q.isEmpty()| $\rightarrow$ \verb|b| - Returns \verb|b = true| if the queue is empty.
|
||||
\item \verb|q.isFull()| $\rightarrow$ \verb|b| - Returns \verb|b = true| if the queue is full.
|
||||
\end{itemize}
|
||||
|
||||
With an array implementation of a queue, items must be ``shuffled'' towards the front after a \textit{dequeue}.
|
||||
Note that with an array implementation, once \verb|rear| becomes equal to $N-1$, no further items can be enqueued (array space limitation).
|
||||
|
||||
\subsection{Linked Lists}
|
||||
A \textbf{Linked List} is an abstract data type which stores an arbitrary-length list of data objects as a sequence of \textit{nodes}.
|
||||
Each node consists of data and has a \textbf{link} to the next node.
|
||||
Each node, excepting the last, is links to a \textbf{successor node}.
|
||||
|
||||
\begin{center}
|
||||
\includegraphics[width=0.6\textwidth]{ll.png}
|
||||
\end{center}
|
||||
|
||||
Characteristics of Linked Lists:
|
||||
\begin{itemize}
|
||||
\item \textbf{Self-referential} structure type - Every node has a pointer to a node of the same type.
|
||||
\item Very useful for \textbf{dynamically} growing/shrinking lists of data.
|
||||
\item Compared to arryas, drastically reduces the effort required to add/remove items from the middle of the list.
|
||||
\item Solves the potential problem of \textbf{overflow} that arrays have.
|
||||
\item \textbf{Sequential access} - It is \textbf{inefficient} to retrieve an element at an arbitrary position, relative to an array.
|
||||
\end{itemize}
|
||||
|
||||
\subsubsection{Implementation of Linked Lists}
|
||||
We define a \textbf{Node} class, with members \textbf{data} (whichever variables are required) \& \textbf{next} (reference to another Node object).
|
||||
|
||||
Each node occcurrence is linked to a succeeding occurence by way of the member \textbf{next}.
|
||||
If \verb|next| is \verb|null|, then there is no item after this node in the list (termed the \textbf{tail} node).
|
||||
The starting point for the list is the \textbf{head} node.
|
||||
We can trace from the head node to any other node.
|
||||
|
||||
\begin{center}
|
||||
\includegraphics[width=0.3\textwidth]{node.png}
|
||||
\end{center}
|
||||
|
||||
\begin{lstlisting}[language=Java]
|
||||
public class Node {
|
||||
// instance variables
|
||||
private Object element;
|
||||
private Node next;
|
||||
|
||||
// creates node with null refs to its element \& next node
|
||||
public Node() {
|
||||
this(null, null);
|
||||
}
|
||||
|
||||
// creates node with the given element & next node
|
||||
public Node(Object e, Node n) {
|
||||
element = e;
|
||||
next = n;
|
||||
}
|
||||
|
||||
// accessor methods
|
||||
public Object getElement() {
|
||||
return element;
|
||||
}
|
||||
public Node getNext() {
|
||||
return next;
|
||||
}
|
||||
|
||||
// mutator methods
|
||||
public void setElement(Object newElem) {
|
||||
element = newElem;
|
||||
}
|
||||
public void setNext(Node newNext) {
|
||||
next = newNext;
|
||||
}
|
||||
}
|
||||
\end{lstlisting}
|
||||
|
||||
Generally, we don't create nodes manually, rather we just supply element data to a method which keeps track of the current position in the list.
|
||||
|
||||
Typical methods in a Linked List ADT include:
|
||||
\begin{itemize}
|
||||
\item \verb|long size()| - Returns the size of the list.
|
||||
\item \verb|boolean isEmpty()| - Returns \verb|true| if the list is empty, \verb|false| otherwise.
|
||||
\item \verb|Object getCurr()| - Returns the element at the current position.
|
||||
\item \verb|boolean gotoHead()| - Sets the current position to \verb|head|, returning \verb|true| if successful.
|
||||
\item \verb|boolean gotoNext()| - Moves to the next position, returning \verb|true| if successful.
|
||||
\item \verb|void insertNext(Object el| - Creates a new node after the current node.
|
||||
\item \verb|void deleteNext()| - Removes the node after the current node.
|
||||
\item \verb|void insertHead(Object el)| - Creates a new node at the head.
|
||||
\item \verb|void deleteHead()| - Removes the head node.
|
||||
\end{itemize}
|
||||
|
||||
\subsubsection{Singly Linked List Class}
|
||||
A \textbf{singly linked list} is one in which each node links to a \textit{single} other node.
|
||||
|
||||
The Singly Linked List Class should store the head of the list \& the current position.
|
||||
For efficiency, it also keeps track of the current size of the list (alternatively, we could just count its nodes when needed).
|
||||
|
||||
\begin{lstlisting}[language=Java]
|
||||
public class SLinkedList {
|
||||
protected Node head; // head node of the list
|
||||
protected Node curr; // current position in list
|
||||
protected long size; // number of nodes in the list
|
||||
|
||||
// default constructor which creates an empty list
|
||||
public SLinkedList() {
|
||||
curr = head = null;
|
||||
size = 0;
|
||||
}
|
||||
|
||||
// insert, remove, & search methods go here
|
||||
}
|
||||
\end{lstlisting}
|
||||
|
||||
\section{Algorithm Analysis}
|
||||
All algorithms take CPU \textbf{time} \& \textbf{memory} space.
|
||||
Often, we can make tradeoffs, choosing algorithm variants that either user more memory, or more CPU.
|
||||
If the memory space requirements of an algorithm are large, the program may use disk sawp space rather than RAM, which is much slower.
|
||||
If the memory requirements are too large, then the program cannot run.
|
||||
Often, we identify the algorithms that don't require ``too much'' spaace, and then choose the one with the lowest
|
||||
CPU requirements.
|
||||
The purpose of algorithm analysis is comparing the time \& space requirements of various algorithms.
|
||||
|
||||
``Why not just run the algorithm and measure the time \& space used?'' -
|
||||
While this is sometimes done when theoretical analysis is difficult, it is better to be able to evaluate
|
||||
algorithms ``on paper'' without first having to implement, debug, and test them all.
|
||||
It's important to have a measure that's \textit{independent} of particular computer configurations and to be able to
|
||||
compare algorithms reliably, without being influenced by variations in implementation.
|
||||
We want to understand how an algorithm will perform on large problems and identify ``hot spots'' to give our attention
|
||||
to when developing \& optimising programs.
|
||||
|
||||
\subsection{Algorithm Analysis Basics}
|
||||
\textbf{Theoretical Analysis} uses a high-level \textit{pseudocode} description of the algorithm instead of a real
|
||||
implementation, and characterises run-time as a function of input size $n$.
|
||||
This function specifies the \textbf{order of growth} of rate of runtime as $n$ increases.
|
||||
Theoretical analysis takes into account all possible inputs and evaluates speed independent of hardware or software.
|
||||
|
||||
\subsubsection{Counting Primitive Operations}
|
||||
The basic approach is deriving the function for the \textbf{count of the primitive operations}.
|
||||
The primitive operations are the individual steps performed by a program.
|
||||
We assume that each step takes the same amount of time and examine any terms that control repetition.
|
||||
|
||||
Example: Algorithm to find the largest element of an array.
|
||||
We count the maximum number of operations as function of array size $n$.
|
||||
\begin{lstlisting}
|
||||
Algorithm arrayMax(A, n) // Number of Operations
|
||||
currentMax = A[0] // 2
|
||||
for i = 1 to n -1 do // 2n
|
||||
if A[i] > currentMax then // 2(n-1)
|
||||
currentMax = A[i] // 2(n-1)
|
||||
{increment coutner i} // 2(n-1)
|
||||
return currentMax // 1
|
||||
// Total: 8n-3
|
||||
\end{lstlisting}
|
||||
|
||||
We could consider the \textbf{average}, \textbf{best}, or \textbf{worst} case.
|
||||
Usually, we analyse the worst case, as we want our algorithms to work well even in bad cases.
|
||||
The average case is quite important too, if different from the worst case.
|
||||
These counts are the basis of (big) O notation.
|
||||
|
||||
\subsection{O Notation}
|
||||
The basic approach to \textbf{O Notation} involves deriving an expression for the count of basic operations (as
|
||||
discussed).
|
||||
We focus on the \textit{dominant term}, and ignore constants.
|
||||
E.g., $O(5n^2 + 1000n -3) \rightarrow O(n^2)$.
|
||||
Since contants \& low-order terms are eventually dropped, we can disregard them when counting primitive operations.
|
||||
|
||||
O Notation is used for \textbf{asymptotic analysis of complexity} - the trend in the algorithms runtime as $n$
|
||||
gets large.
|
||||
We look at the \textbf{order of magnitude} of the number of actions, independent of computer/compiler/etc.
|
||||
|
||||
Note: We specifically care about the \textbf{tightest} upper bound.
|
||||
Technically speaking, an algorithm that is $O(n^2)$ is also $O(n^3)$, but the former is more informative.
|
||||
The function specified in O notation is the \textbf{upper bound} on the behaviour of the algorithm being analysed.
|
||||
This can be the best/average/worst case behaviour.
|
||||
|
||||
Example: Let $f(n) = 6n^4 -2n^3 + 5$.
|
||||
Apply the following rules:
|
||||
\begin{itemize}
|
||||
\item If $f(n)$ is a sum of several terms, then only the one with largest rate of growth is kept.
|
||||
\item If $f(n)$ is a product of several factors, any constants that do not depend on $n$ are ommitted.
|
||||
\end{itemize}
|
||||
Thus, we say that $f(n)$ has a ``big-oh'' of $(n^4)$.
|
||||
We can write $f(n)$ is $O(n^4)$.
|
||||
|
||||
\subsubsection{Important Functions Used in O Notation}
|
||||
Functions commonly used include:
|
||||
\begin{itemize}
|
||||
\item \textbf{Constant:} $O(1)$.
|
||||
\item \textbf{Logarithmic:} $O($log$n)$.
|
||||
\item \textbf{Linear:} $O(n)$.
|
||||
\item \textbf{n-Log-n:} $O(n$log$n)$.
|
||||
\item \textbf{Quadratic:} $O(n^2)$.
|
||||
\item \textbf{Cubic:} $O(n^3)$.
|
||||
\item \textbf{Exponential:} $(1^n)$.
|
||||
\end{itemize}
|
||||
|
||||
Notes:
|
||||
\begin{itemize}
|
||||
\item By convention, all logs are base 2 unless otherwise stated.
|
||||
\item Two algorithms having the same complexity doesn't been that they are exactly the same, it means that their
|
||||
running times will be \textit{proportional}.
|
||||
\end{itemize}
|
||||
|
||||
|
||||
\begin{center}
|
||||
\includegraphics[width=0.8\textwidth]{comparisonoffunctions1.png}
|
||||
\end{center}
|
||||
|
||||
\begin{center}
|
||||
\includegraphics[width=0.8\textwidth]{comparisonoffunctions2.png}
|
||||
\end{center}
|
||||
|
||||
\subsubsection{Efficiency \& O Notation}
|
||||
\begin{itemize}
|
||||
\item \textbf{Constant:} Most efficient possible, but only applicable to simple jobs.
|
||||
\item \textbf{Logarithmic, Linear, \& n-Log-n:} If an algorithm is described as ``efficient'', this usually means
|
||||
$O($log$n)$ or better.
|
||||
\item \textbf{Quadratic \& Cubic:} Not very efficient, but polynomial algorithms are usually considered ``tractable''
|
||||
(acceptable for problems of reasonable size).
|
||||
\item \textbf{Exponential:} Very inefficient. Problems that (provably) require an algorithm of O greater than
|
||||
polynomial complexity are called ``\textbf{hard}''.
|
||||
\end{itemize}
|
||||
|
||||
\subsection{Recursion Review}
|
||||
Methods can call other methods, but they can also call themselves, either directly, or indirectly, via
|
||||
another method.
|
||||
This creates a type of loop called \textbf{recursion}.
|
||||
|
||||
Iteration can be used anywhere that you can use recursion.
|
||||
Sometimes, recursion can be a more elegant solution, if it reflects the way that the problem is usually thought
|
||||
about, as we aim to use the most intuitive representation of the problem.
|
||||
Recursion can make complexity analysis easier in some cases.
|
||||
|
||||
The drawbacks of recursion include:
|
||||
\begin{itemize}
|
||||
\item Inefficient use of the function: Large amount of concurrent, deeply nested method calls.
|
||||
\item If done naively, the number of calls can explode.
|
||||
\item Depending on the algorithm, we need to take care not to recompute values unnecessarily.
|
||||
\end{itemize}
|
||||
|
||||
\section{Dynamic Programming}
|
||||
The basic idea of \textbf{Dynamic Programming} is to solve complex problems by breaking them into simpler
|
||||
sub-problems.
|
||||
When a solution to a sub-problem is found, store it (``memo-ize'') so that it can be re-used without
|
||||
recomputing it.
|
||||
Combine the solutions to the sub-problems to get the overall solution.
|
||||
|
||||
This is particularly useful when the number of repeating sub-problems grows exponentially with the problem
|
||||
size.
|
||||
|
||||
In general, dynamic programming takes problems that appear exponential and produces polynomial-time algorithms
|
||||
for them.
|
||||
The trade-off in dynamic programming is between \textit{storage} \& \textit{speed}.
|
||||
Dynamic programming is widely used in heuristic optimisation problems.
|
||||
|
||||
For Dynamic Programming, the problem structure requires three components:
|
||||
\begin{enumerate}
|
||||
\item \textbf{Simple sub-problems:} Must be able to break the overall problem into indexed sub-problems
|
||||
\& sub-sub-problems.
|
||||
\item \textbf{Sub-problem decomposition:} Optimal/correct solution to the overall problem must be
|
||||
composed from sub-problems.
|
||||
\item \textbf{Sub-problem overlap:} So that elements can be re-used.
|
||||
\end{enumerate}
|
||||
|
||||
The basic steps in the approach to DynProg:
|
||||
\begin{enumerate}
|
||||
\item Set up the overall problem as one that is decomposable into overlapping sub-problems that can
|
||||
be indexed.
|
||||
\item Solve the sub-problems as they arise and \textbf{store solutions} in a table.
|
||||
\item Derive the overall solution from the solutions in the table.
|
||||
\end{enumerate}
|
||||
|
||||
\subsection{More Big Greek Letters}
|
||||
$O(n$log$n)$ (``Big Oh''):
|
||||
\begin{itemize}
|
||||
\item Upper bound on asymptotic complexity.
|
||||
\item In this case, there is a constant $c_2$ such that $c_2 n$log$n$ is an upper bound on
|
||||
asymptotic complexity.
|
||||
\end{itemize}
|
||||
|
||||
$\Omega(n$log$n)$ (``Big Omega''):
|
||||
\begin{itemize}
|
||||
\item Specifies a lower bound on asymptotic complexity.
|
||||
\item In this case, the algorithm has a lower bound of $c_1 n$log$n$.
|
||||
\end{itemize}
|
||||
|
||||
$\Theta(n$log$n)$ (``Big Theta''):
|
||||
\begin{itemize}
|
||||
\item Specifies the upper \& lower bounds.
|
||||
\item In this case, there exist two constants, $c_1$ \& $c_2$, such that $c_1n$log$n < f(n) < c_2n$log$n$.
|
||||
\end{itemize}
|
||||
|
||||
Of these, $\Theta()$ makes the strongest claims: It specifies that the rate of growth is no better and no
|
||||
worse than some level.
|
||||
Requires additional analysis relative to $O$.
|
||||
|
||||
There are also some others that are common in Mathematics but not in Computer Science:
|
||||
\begin{itemize}
|
||||
\item \textbf{Little o:} $o(g(n))$ specifies a function $g(n)$ that grows much faster than the one
|
||||
that we are analysing.
|
||||
\item \textbf{Little omega:} $\omega(g(n))$ specifies a function $g(n)$ that grows much slower than
|
||||
the one that we are analysing.
|
||||
\end{itemize}
|
||||
|
||||
Don't confuse upper/lower bounds with best/worst case: all cases have bounds.
|
||||
|
||||
\newpage
|
||||
\subsection{P, NP, \& NP-Complete Problems}
|
||||
\textbf{P Problems} are those for which there is a \textbf{deterministic} algorithm that solves it in
|
||||
\textbf{Polynomial Time}.
|
||||
In other words, the algorithm's complexity is $O(p(n))$ where $p(n)$ is a polynomial function.
|
||||
|
||||
A (trivial) example of a P problem is searching an array of integers for a certain value.
|
||||
|
||||
Problems that can be solved in polynomial time are termed \textbf{tractable}, while worse problems are
|
||||
termed \textbf{intractable}.
|
||||
|
||||
\textbf{NP Problems (Non-deterministic Polynomial)} are those algorithms which have two repeating steps:
|
||||
\begin{itemize}
|
||||
\item Generate a \textit{potential solution}, either randomly or systematically.
|
||||
\item Verify whether the potential solution is right, and if not, repeat.
|
||||
\end{itemize}
|
||||
|
||||
If the verification step is \textbf{polynomial}, the algorithm \& associated problem are \textbf{NP}.
|
||||
|
||||
An example of an NP problem is factoring large integers as used in RSA encryption.
|
||||
Another example of an NP problem is the \textbf{subset problem}:
|
||||
Given a set of integers, does some non-empty subset of them sum to 0?
|
||||
There is no polynomial algorithm to solve this problem.
|
||||
However, verification of a potential solution is polynomial ($O(n)$ (just add up the numbers in the
|
||||
potential solution)).
|
||||
|
||||
Note that P is a subset of NP.
|
||||
|
||||
\textbf{NP-Complete} problems are those that are ``\textbf{as hard as} all others'' in NP, i.e. algorithms that
|
||||
are comparable to (``\textbf{polynomially reducible} to'') others in NP but not reducible to P.
|
||||
If an algorithm is \textbf{polynomially reducible}, there is some polynomial-time transformation that
|
||||
converts the inputs for Problem $X$ to inputs for Problem $Y$.
|
||||
|
||||
NP-Complete is a complexity class which represents the set of all problems $X$ in NP for which it is
|
||||
possible to reduce any other NP problem $Y$ to $X$ in polynomial time.
|
||||
Intuitively, this means that we can solve $Y$ quickly if we know how to solve $X$ quickly.
|
||||
What makes NP-complete problems inportant is that if a deterministic polynomial time algorithm can be
|
||||
found to solve one of them, every NP problem is solvable in polynomial time.
|
||||
|
||||
\textbf{NP-Hard} problems are those that are ``\textbf{as hard or harder}'' than all others in NP.
|
||||
A problem $X$ is Np-Hard if NP-Complete problems are polynomially reducible to it.
|
||||
|
||||
Intuitively, NP-hard problems are problems that are at least as hard as the NP-complete problems.
|
||||
Note that NP-hard problems do not have to be in NP, and they do not have to be decision problems.
|
||||
The precise definition here is that ``a problem $X$ is NP-hard, if there is an NP-complete problem $Y$,
|
||||
such that $Y$ is reducible to $X$ in polynomial time''.
|
||||
An example of an NP-hard problem is \textit{the halting problem}: Given a program $P$ and input $I$,
|
||||
will it halt?
|
||||
|
||||
The ``P versus NP'' problem is a major unsolved problem in computer science:
|
||||
``If the solution to a problem is easy to verify, is the problem also easy to solve?'' or ``whether
|
||||
every problem whose solution can be \textbf{quickly verified} by a computer can also be \textbf{quickly solved}
|
||||
by a computer''.
|
||||
``Quickly'' here means that there exists an algorithm to solve the task that runs in polynomial time.
|
||||
An answer to the $P = NP$ question would determine whether all problems that can be verified in polynomial time can also be solved in
|
||||
polynomial time.
|
||||
If it turned out that $P \neq NP$, then it would mean that there are problems in NP (such as NP-complete problems) that are harder to compute
|
||||
than to verify.
|
||||
We already know that $P \subseteq NP$.
|
||||
|
||||
In theoretical computer science, the problems considered for P \& NP are \textbf{decision problems}, i.e. problems that don't produce
|
||||
numeric results but yes or no answers.
|
||||
|
||||
\begin{center}
|
||||
\includegraphics[width=0.6\textwidth]{pvsnp.png}
|
||||
\end{center}
|
||||
|
||||
\begin{itemize}
|
||||
\item \textbf{P} (Polynomial): Solvable in polynomial time.
|
||||
\item \textbf{NP} (Non-Deterministic Polynomial): Only \textit{verifiable} in polynomial time.
|
||||
\end{itemize}
|
||||
|
||||
\section{Searching \& Sorting}
|
||||
\subsection{Keys \& Values}
|
||||
Each object to be sorted can be considered to have a \textbf{key} \& a \textbf{value}, e.g. A Student has properties Name, ID, \& grade.
|
||||
|
||||
\subsection{Java Interface: Comparator}
|
||||
The \verb|Comparator| interface compares two objects to say which should come first.
|
||||
In Java, any class that implements \verb|java.util.Comparator| interface is only required to implement one method:
|
||||
\begin{lstlisting}[language=java]
|
||||
int compare(Object ob1, Object ob2);
|
||||
\end{lstlisting}
|
||||
|
||||
This returns a negative number if \verb|ob1| is less than \verb|ob2|, a positive number if \verb|ob1| is greater than \verb|ob2|, and \verb|0| if \verb|ob1| is equal to \verb|ob2|.
|
||||
|
||||
\subsection{Java Interface: Comparable}
|
||||
The \verb|Comparable| interface compares a given object to another to see which object should come first.
|
||||
The two objects that are being compared must be of a class that implements \verb|java.lang.Comparable|, which has just one method to implement:
|
||||
\begin{lstlisting}[language=java]
|
||||
int compareTo(Object other);
|
||||
\end{lstlisting}
|
||||
|
||||
Standard classes such as \verb|String| implement this.
|
||||
|
||||
\subsection{Insertion Sort}
|
||||
Consider sorting a bookshelf using \textbf{Insertion Sort}:
|
||||
\begin{enumerate}
|
||||
\item Remove the next unsorted book.
|
||||
\item Slide the sorted books to the right one by one until you find the right sport for for the removed book.
|
||||
\item Insert the book into its new position once it is found.
|
||||
\end{enumerate}
|
||||
|
||||
\begin{algorithm}
|
||||
\caption{Insertion Sort Pseudocode}
|
||||
\begin{algorithmic}
|
||||
\Require{$A[0 \dots N-1]$} \Comment{Unsorted Array}
|
||||
\Ensure{$A[0 \dots N-1]$} \Comment{Sorted Array}
|
||||
|
||||
\Procedure{Insertion Sort}{$A[0 \dots N-1]$}
|
||||
\For{$ToSort \gets 1$ to $N-1$ Step 1}
|
||||
\State $Index = ToSort -1$
|
||||
\State $ToSortEl = A[ToSort]$
|
||||
\\
|
||||
\While{$Index \geq First$ AND $A[Index] > ToSortEl$}
|
||||
\State $A[Index+1] \gets A[Index]$ \Comment{Shuffle elements to the right}
|
||||
\State $Index \gets Index - 1$
|
||||
\EndWhile
|
||||
\\
|
||||
\State $A[Index+1] = ToSortEl$ \Comment{Insert the element to sort in its appropriate place}
|
||||
\EndFor
|
||||
|
||||
\State \textbf{return} $A[]$ \Comment{Return the sorted array}
|
||||
\EndProcedure
|
||||
\end{algorithmic}
|
||||
\end{algorithm}
|
||||
|
||||
The worst case efficiency of Insertion Sort is $O(n^2)$.
|
||||
The best case efficiency of Insertion Sort is $O(n)$.
|
||||
If the array is closer to sorted order, the algorithm does less work, and the operation is more efficient.
|
||||
This leads to a related algorithm: Shell Sort.
|
||||
|
||||
\newpage
|
||||
\subsection{Shell Sort}
|
||||
\textbf{Shell Sort} is more efficient than Selection Sort or Insertion Sort.
|
||||
It works by comparing distant items first, and works its way down to nearby items.
|
||||
The interval is called the \textbf{gap}.
|
||||
The gap begins at one half of the length of the list and is successively halved until each item has been compared with its neighbour.
|
||||
|
||||
Insertion Sort only tests elements in \textit{adjacent} locations - it might take several steps to get to the final location.
|
||||
Insertion Sort is more efficient if an array is partially sorted.
|
||||
By making larger jumps, Shell Sort makes the array become ``more sorted'' more quickly.
|
||||
|
||||
\begin{algorithm}
|
||||
\caption{Shell Sort Pseudocode}
|
||||
\begin{algorithmic}
|
||||
\Require{$A[0 \dots N-1]$} \Comment{Unsorted Array}
|
||||
\Ensure{$A[0 \dots N-1]$} \Comment{Sorted Array}
|
||||
|
||||
\Procedure{Shell Sort}{$A[0 \dots N-1]$}
|
||||
\State $Gap \gets floor(\frac{Gap}{2})$ \Comment{Round to the nearest odd number, as it's best with an odd-sized gap}
|
||||
\\
|
||||
\While{$Gap \geq 1$}
|
||||
\For{$ToSort \gets 1$ to $N-1$ Step 1}
|
||||
\State $Index = ToSort -1$
|
||||
\State $ToSortEl = A[ToSort]$
|
||||
\\
|
||||
\While{$Index \geq First$ AND $A[Index] > ToSortEl$}
|
||||
\State $A[Index+1] \gets A[Index]$ \Comment{Shuffle elements to the right}
|
||||
\State $Index \gets Index - 1$
|
||||
\EndWhile
|
||||
\\
|
||||
\State $A[Index+1] = ToSortEl$ \Comment{Insert the element to sort in its appropriate place}
|
||||
\EndFor
|
||||
\\
|
||||
\State $Gap \gets floor(\frac{Gap}{2})$
|
||||
\EndWhile
|
||||
\EndProcedure
|
||||
\end{algorithmic}
|
||||
\end{algorithm}
|
||||
|
||||
The worst-case complexity of Shell Sort is $O(n^2)$.
|
||||
However, this is because the gap is sometimes even which results in sub-arrays that include all the elements of an array that was already sorted.
|
||||
To avoid this, we round the gap up to the nearest odd number which gives us a worst-case complexity of $O(n^{1.5})$.
|
||||
Other gap sequences can improve performance a little more, but this is beyond the scope of this topic.
|
||||
|
||||
\subsection{Quick Sort}
|
||||
\textbf{Quick Sort} is a divide-and-conquer algorithm.
|
||||
\begin{enumerate}
|
||||
\item Firstly, it partitions the array into two sub-arrays that are \textbf{partially sorted}.
|
||||
\item Then, it picks a \textbf{pivot value}, and re-arranges the elements such that all elements less than or equal to the pivot value are on the left of the pivot, and all elements that are greater than
|
||||
it are on the right.
|
||||
\item The array is now divided into sub-arrays and a pivot value.
|
||||
\item This procedure is then repeated \textbf{recursively} for each sub-array, to further sort each of them.
|
||||
\item When the algorithm has reached the level of a sub-array with just one element, that sub-array is sorted.
|
||||
All sub-arrays are sorted relative to each other, so the whole array is sorted when all the sub-arrays are.
|
||||
\end{enumerate}
|
||||
|
||||
\section{Trees}
|
||||
The data structures that we looked at previously placed data in linear order, but we sometimes need to organise data into groups \& sub-groups.
|
||||
This is called \textbf{hierarchical classification}, where data items appear at various levels within the organisation.
|
||||
|
||||
In Computer Science, a tree is an abstract model of a hierarchical structure which consists of nodes with a parent-child relation.
|
||||
Trees have applications in organisation charts, file systems, programming environments, \& more.
|
||||
A \textbf{tree} is a set of nodes, connected by edges.
|
||||
The edges indicate relationships between nodes.
|
||||
|
||||
Nodes are arranged in \textbf{levels}, which indicate the node's position in the hierarchy.
|
||||
Nodes with the same parent node are called \textbf{siblings}.
|
||||
The only node with no parent is the \textbf{root} node.
|
||||
All other nodes have one parent each.
|
||||
A node with no child nodes is called a \textbf{leaf} node or an \textbf{external} node.
|
||||
All other nodes are referred to as \textbf{internal} nodes.
|
||||
|
||||
A node is reached from the root by a \textbf{path}.
|
||||
The length of a path is the number of edges that compose it.
|
||||
This is also referred to as the depth of the node.
|
||||
The \textbf{height} of a tree is the number of levels in the tree.
|
||||
The number of nodes along the longest path is equal to the maximum depth plus one.
|
||||
Note that we talk about the depth of a node, but the height of a tree.
|
||||
|
||||
The ancestors of a node include its parent, grandparent, great-grandparent, etc.
|
||||
The descendants of a node include its children, grandchildren, great-grandchildren, etc.
|
||||
A \textbf{subtree} of a node is a tree that has that node as its root, including the node, all its descendants, and the arcs connecting them.
|
||||
|
||||
\subsection{Binary Trees}
|
||||
A \textbf{binary tree} is one in which each internal node has at most two children (\& exactly two if it is a ``\textbf{proper}'' binary tree).
|
||||
The children of a node are an \textbf{ordered pair}.
|
||||
We refer to the children of an internal node as the left child \& the right child.
|
||||
Non-binary trees are termed general trees.
|
||||
|
||||
An alternative (recursive) definition for binary trees is a tree that is either just single node or a tree whose root has an ordered pair of children, each of which is a binary tree.
|
||||
|
||||
A \textbf{full binary tree} is one in which is a proper binary tree, and in which all the leaves are on the same level.
|
||||
This is only achievable for certain numbers of nodes.
|
||||
A \textbf{complete binary tree} is a binary tree which is full to the penultimate level and the leaves on the last level are filled frmo left to right.
|
||||
This is achievable for any number of nodes.
|
||||
|
||||
The height of either a complete or full binary tree with $n$ nodes is log$_2(n+1)$.
|
||||
|
||||
\begin{center}
|
||||
\includegraphics[width=0.7\textwidth]{binarytrees.png}
|
||||
\end{center}
|
||||
|
||||
\subsection{Generics in Java}
|
||||
The \verb|< >| operators relate to the concept of \textbf{generics}.
|
||||
Generics are used to specify a specific type parameter for a generic collection class.
|
||||
This saves us from having to cast objects in methods such as \verb|add()|, \verb|set|, \& \verb|remove|.
|
||||
This is a big advantage, as type checking is now done at compile time.
|
||||
Without generics, compile-time type-checking is impossible, since we don't have a type specification for the list.
|
||||
Other object-oriented programming languages have similar concepts, such as templates in C++.
|
||||
|
||||
\verb|ArrayList|s are part of the Java Collections framework, a standard library of pre-built data structures.
|
||||
The underlying storage of an \verb|ArrayList| is an array.
|
||||
The \verb|ArrayList| class looks after resizing it as required.
|
||||
\verb|ArrayLists| can be used with or without generics notation.
|
||||
|
||||
\begin{lstlisting}[language=Java]
|
||||
// ArrayList code without generics:
|
||||
ArrayList words = new ArrayList(); // holds objects
|
||||
words.add("hello");
|
||||
String a = (String) words.get(0); // return type of get() is object, so must cast to String
|
||||
|
||||
// ArrayList parameterised to specifically hold Strings
|
||||
ArrayList<String> words = new ArrayList<String>();
|
||||
words.add("hello");
|
||||
String a = words.get(0); // no cast needed
|
||||
\end{lstlisting}
|
||||
|
||||
Creating a Generics collection:
|
||||
\begin{lstlisting}[language=Java]
|
||||
|
||||
public interface List<E> {
|
||||
void add(E x);
|
||||
Iterator<E> iterator();
|
||||
}
|
||||
public interface Iterator<E> {
|
||||
E next();
|
||||
boolean hasNext();
|
||||
}
|
||||
\end{lstlisting}
|
||||
|
||||
\section{Search Trees}
|
||||
A \textbf{Search Tree} organises its data so that searching it is efficient.
|
||||
|
||||
\subsection{Binary Search Trees}
|
||||
A \textbf{Binary Search Tree (BST)} is a binary tree with nodes that contain \verb|Comparable| objects.
|
||||
A node's data is greater than the data in the left subtree and less than the data in the right subtree.
|
||||
Usually, no duplicates are allowed.
|
||||
|
||||
An \textbf{in-order} traversal of a BST will visit all nodes in ascending order.
|
||||
|
||||
BSTs are not uniquely structured - The structure of a BST depends on what node is chosen as the root of the tree and the order in which all the other nodes are added.
|
||||
|
||||
\bibliographystyle{unsrtnat}
|
||||
\bibliography{references}
|
||||
\end{document}
|
||||
29
second/semester2/CT2109/Notes/CT2109-Notes.toc
Normal file
29
second/semester2/CT2109/Notes/CT2109-Notes.toc
Normal file
@ -0,0 +1,29 @@
|
||||
\contentsline {section}{\numberline {1}Abstract Data Types}{2}{section.1}%
|
||||
\contentsline {subsection}{\numberline {1.1}Stacks \& Queues}{2}{subsection.1.1}%
|
||||
\contentsline {subsubsection}{\numberline {1.1.1}Stacks}{2}{subsubsection.1.1.1}%
|
||||
\contentsline {subsubsection}{\numberline {1.1.2}Queues}{2}{subsubsection.1.1.2}%
|
||||
\contentsline {subsection}{\numberline {1.2}Linked Lists}{3}{subsection.1.2}%
|
||||
\contentsline {subsubsection}{\numberline {1.2.1}Implementation of Linked Lists}{3}{subsubsection.1.2.1}%
|
||||
\contentsline {subsubsection}{\numberline {1.2.2}Singly Linked List Class}{4}{subsubsection.1.2.2}%
|
||||
\contentsline {section}{\numberline {2}Algorithm Analysis}{5}{section.2}%
|
||||
\contentsline {subsection}{\numberline {2.1}Algorithm Analysis Basics}{5}{subsection.2.1}%
|
||||
\contentsline {subsubsection}{\numberline {2.1.1}Counting Primitive Operations}{5}{subsubsection.2.1.1}%
|
||||
\contentsline {subsection}{\numberline {2.2}O Notation}{5}{subsection.2.2}%
|
||||
\contentsline {subsubsection}{\numberline {2.2.1}Important Functions Used in O Notation}{6}{subsubsection.2.2.1}%
|
||||
\contentsline {subsubsection}{\numberline {2.2.2}Efficiency \& O Notation}{7}{subsubsection.2.2.2}%
|
||||
\contentsline {subsection}{\numberline {2.3}Recursion Review}{7}{subsection.2.3}%
|
||||
\contentsline {section}{\numberline {3}Dynamic Programming}{7}{section.3}%
|
||||
\contentsline {subsection}{\numberline {3.1}More Big Greek Letters}{8}{subsection.3.1}%
|
||||
\contentsline {subsection}{\numberline {3.2}P, NP, \& NP-Complete Problems}{9}{subsection.3.2}%
|
||||
\contentsline {section}{\numberline {4}Searching \& Sorting}{10}{section.4}%
|
||||
\contentsline {subsection}{\numberline {4.1}Keys \& Values}{10}{subsection.4.1}%
|
||||
\contentsline {subsection}{\numberline {4.2}Java Interface: Comparator}{10}{subsection.4.2}%
|
||||
\contentsline {subsection}{\numberline {4.3}Java Interface: Comparable}{10}{subsection.4.3}%
|
||||
\contentsline {subsection}{\numberline {4.4}Insertion Sort}{10}{subsection.4.4}%
|
||||
\contentsline {subsection}{\numberline {4.5}Shell Sort}{12}{subsection.4.5}%
|
||||
\contentsline {subsection}{\numberline {4.6}Quick Sort}{12}{subsection.4.6}%
|
||||
\contentsline {section}{\numberline {5}Trees}{13}{section.5}%
|
||||
\contentsline {subsection}{\numberline {5.1}Binary Trees}{13}{subsection.5.1}%
|
||||
\contentsline {subsection}{\numberline {5.2}Generics in Java}{13}{subsection.5.2}%
|
||||
\contentsline {section}{\numberline {6}Search Trees}{14}{section.6}%
|
||||
\contentsline {subsection}{\numberline {6.1}Binary Search Trees}{14}{subsection.6.1}%
|
||||
BIN
second/semester2/CT2109/Notes/images/adt.png
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|
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|
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second/semester2/CT2109/Notes/images/comparisonoffunctions1.png
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second/semester2/CT2109/Notes/images/comparisonoffunctions2.png
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|
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second/semester2/CT2109/Notes/images/ll.png
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second/semester2/CT2109/Notes/images/ll.png
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|
After Width: | Height: | Size: 5.8 KiB |
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second/semester2/CT2109/Notes/images/node.png
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|
After Width: | Height: | Size: 3.2 KiB |
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second/semester2/CT2109/Notes/images/pvsnp.png
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second/semester2/CT2109/Notes/images/pvsnp.png
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|
After Width: | Height: | Size: 121 KiB |
BIN
second/semester2/CT2109/Notes/images/sllpart1.png
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BIN
second/semester2/CT2109/Notes/images/sllpart1.png
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|
After Width: | Height: | Size: 12 KiB |
264
second/semester2/CT2109/Notes/report.sty
Normal file
264
second/semester2/CT2109/Notes/report.sty
Normal file
@ -0,0 +1,264 @@
|
||||
\NeedsTeXFormat{LaTeX2e}
|
||||
|
||||
\ProcessOptions\relax
|
||||
|
||||
% fonts
|
||||
\renewcommand{\rmdefault}{ptm}
|
||||
\renewcommand{\sfdefault}{phv}
|
||||
|
||||
% set page geometry
|
||||
\usepackage[verbose=true,letterpaper]{geometry}
|
||||
\AtBeginDocument{
|
||||
\newgeometry{
|
||||
textheight=9in,
|
||||
textwidth=6.5in,
|
||||
top=1in,
|
||||
headheight=14pt,
|
||||
headsep=25pt,
|
||||
footskip=30pt
|
||||
}
|
||||
}
|
||||
|
||||
\widowpenalty=10000
|
||||
\clubpenalty=10000
|
||||
\flushbottom
|
||||
\sloppy
|
||||
|
||||
|
||||
|
||||
\newcommand{\headeright}{Ph.D. Confirmation Report}
|
||||
\newcommand{\undertitle}{Ph.D. Confirmation Report}
|
||||
\newcommand{\shorttitle}{\@title}
|
||||
|
||||
\usepackage{fancyhdr}
|
||||
\fancyhf{}
|
||||
\pagestyle{fancy}
|
||||
\renewcommand{\headrulewidth}{0.4pt}
|
||||
\fancyheadoffset{0pt}
|
||||
\rhead{\scshape \footnotesize \headeright}
|
||||
\chead{\shorttitle}
|
||||
\cfoot{\thepage}
|
||||
|
||||
|
||||
%Handling Keywords
|
||||
\def\keywordname{{\bfseries \emph{Keywords}}}%
|
||||
\def\keywords#1{\par\addvspace\medskipamount{\rightskip=0pt plus1cm
|
||||
\def\and{\ifhmode\unskip\nobreak\fi\ $\cdot$
|
||||
}\noindent\keywordname\enspace\ignorespaces#1\par}}
|
||||
|
||||
% font sizes with reduced leading
|
||||
\renewcommand{\normalsize}{%
|
||||
\@setfontsize\normalsize\@xipt\@xiipt
|
||||
\abovedisplayskip 7\p@ \@plus 2\p@ \@minus 5\p@
|
||||
\abovedisplayshortskip \z@ \@plus 3\p@
|
||||
\belowdisplayskip \abovedisplayskip
|
||||
\belowdisplayshortskip 4\p@ \@plus 3\p@ \@minus 3\p@
|
||||
}
|
||||
\normalsize
|
||||
\renewcommand{\small}{%
|
||||
\@setfontsize\small\@xpt\@xipt
|
||||
\abovedisplayskip 6\p@ \@plus 1.5\p@ \@minus 4\p@
|
||||
\abovedisplayshortskip \z@ \@plus 2\p@
|
||||
\belowdisplayskip \abovedisplayskip
|
||||
\belowdisplayshortskip 3\p@ \@plus 2\p@ \@minus 2\p@
|
||||
}
|
||||
\renewcommand{\footnotesize}{\@setfontsize\footnotesize\@ixpt\@xpt}
|
||||
\renewcommand{\scriptsize}{\@setfontsize\scriptsize\@viipt\@viiipt}
|
||||
\renewcommand{\tiny}{\@setfontsize\tiny\@vipt\@viipt}
|
||||
\renewcommand{\large}{\@setfontsize\large\@xiipt{14}}
|
||||
\renewcommand{\Large}{\@setfontsize\Large\@xivpt{16}}
|
||||
\renewcommand{\LARGE}{\@setfontsize\LARGE\@xviipt{20}}
|
||||
\renewcommand{\huge}{\@setfontsize\huge\@xxpt{23}}
|
||||
\renewcommand{\Huge}{\@setfontsize\Huge\@xxvpt{28}}
|
||||
|
||||
% sections with less space
|
||||
\providecommand{\section}{}
|
||||
\renewcommand{\section}{%
|
||||
\@startsection{section}{1}{\z@}%
|
||||
{-2.0ex \@plus -0.5ex \@minus -0.2ex}%
|
||||
{ 1.5ex \@plus 0.3ex \@minus 0.2ex}%
|
||||
{\large\bf\raggedright}%
|
||||
}
|
||||
\providecommand{\subsection}{}
|
||||
\renewcommand{\subsection}{%
|
||||
\@startsection{subsection}{2}{\z@}%
|
||||
{-1.8ex \@plus -0.5ex \@minus -0.2ex}%
|
||||
{ 0.8ex \@plus 0.2ex}%
|
||||
{\normalsize\bf\raggedright}%
|
||||
}
|
||||
\providecommand{\subsubsection}{}
|
||||
\renewcommand{\subsubsection}{%
|
||||
\@startsection{subsubsection}{3}{\z@}%
|
||||
{-1.5ex \@plus -0.5ex \@minus -0.2ex}%
|
||||
{ 0.5ex \@plus 0.2ex}%
|
||||
{\normalsize\bf\raggedright}%
|
||||
}
|
||||
\providecommand{\paragraph}{}
|
||||
\renewcommand{\paragraph}{%
|
||||
\@startsection{paragraph}{4}{\z@}%
|
||||
{1.5ex \@plus 0.5ex \@minus 0.2ex}%
|
||||
{-1em}%
|
||||
{\normalsize\bf}%
|
||||
}
|
||||
\providecommand{\subparagraph}{}
|
||||
\renewcommand{\subparagraph}{%
|
||||
\@startsection{subparagraph}{5}{\z@}%
|
||||
{1.5ex \@plus 0.5ex \@minus 0.2ex}%
|
||||
{-1em}%
|
||||
{\normalsize\bf}%
|
||||
}
|
||||
\providecommand{\subsubsubsection}{}
|
||||
\renewcommand{\subsubsubsection}{%
|
||||
\vskip5pt{\noindent\normalsize\rm\raggedright}%
|
||||
}
|
||||
|
||||
% float placement
|
||||
\renewcommand{\topfraction }{0.85}
|
||||
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|
||||
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|
||||
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|
||||
|
||||
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|
||||
\newlength{\@belowcaptionskip}\setlength{\@belowcaptionskip}{\z@}
|
||||
|
||||
\setlength{\abovecaptionskip}{\@abovecaptionskip}
|
||||
\setlength{\belowcaptionskip}{\@belowcaptionskip}
|
||||
|
||||
% swap above/below caption skip lengths for tables
|
||||
\renewenvironment{table}
|
||||
{\setlength{\abovecaptionskip}{\@belowcaptionskip}%
|
||||
\setlength{\belowcaptionskip}{\@abovecaptionskip}%
|
||||
\@float{table}}
|
||||
{\end@float}
|
||||
|
||||
% footnote formatting
|
||||
\setlength{\footnotesep }{6.65\p@}
|
||||
\setlength{\skip\footins}{9\p@ \@plus 4\p@ \@minus 2\p@}
|
||||
\renewcommand{\footnoterule}{\kern-3\p@ \hrule width 12pc \kern 2.6\p@}
|
||||
\setcounter{footnote}{0}
|
||||
|
||||
% paragraph formatting
|
||||
\setlength{\parindent}{\z@}
|
||||
\setlength{\parskip }{5.5\p@}
|
||||
|
||||
% list formatting
|
||||
\setlength{\topsep }{4\p@ \@plus 1\p@ \@minus 2\p@}
|
||||
\setlength{\partopsep }{1\p@ \@plus 0.5\p@ \@minus 0.5\p@}
|
||||
\setlength{\itemsep }{2\p@ \@plus 1\p@ \@minus 0.5\p@}
|
||||
\setlength{\parsep }{2\p@ \@plus 1\p@ \@minus 0.5\p@}
|
||||
\setlength{\leftmargin }{3pc}
|
||||
\setlength{\leftmargini }{\leftmargin}
|
||||
\setlength{\leftmarginii }{2em}
|
||||
\setlength{\leftmarginiii}{1.5em}
|
||||
\setlength{\leftmarginiv }{1.0em}
|
||||
\setlength{\leftmarginv }{0.5em}
|
||||
\def\@listi {\leftmargin\leftmargini}
|
||||
\def\@listii {\leftmargin\leftmarginii
|
||||
\labelwidth\leftmarginii
|
||||
\advance\labelwidth-\labelsep
|
||||
\topsep 2\p@ \@plus 1\p@ \@minus 0.5\p@
|
||||
\parsep 1\p@ \@plus 0.5\p@ \@minus 0.5\p@
|
||||
\itemsep \parsep}
|
||||
\def\@listiii{\leftmargin\leftmarginiii
|
||||
\labelwidth\leftmarginiii
|
||||
\advance\labelwidth-\labelsep
|
||||
\topsep 1\p@ \@plus 0.5\p@ \@minus 0.5\p@
|
||||
\parsep \z@
|
||||
\partopsep 0.5\p@ \@plus 0\p@ \@minus 0.5\p@
|
||||
\itemsep \topsep}
|
||||
\def\@listiv {\leftmargin\leftmarginiv
|
||||
\labelwidth\leftmarginiv
|
||||
\advance\labelwidth-\labelsep}
|
||||
\def\@listv {\leftmargin\leftmarginv
|
||||
\labelwidth\leftmarginv
|
||||
\advance\labelwidth-\labelsep}
|
||||
\def\@listvi {\leftmargin\leftmarginvi
|
||||
\labelwidth\leftmarginvi
|
||||
\advance\labelwidth-\labelsep}
|
||||
|
||||
% create title
|
||||
\providecommand{\maketitle}{}
|
||||
\renewcommand{\maketitle}{%
|
||||
\par
|
||||
\begingroup
|
||||
\renewcommand{\thefootnote}{\fnsymbol{footnote}}
|
||||
% for perfect author name centering
|
||||
\renewcommand{\@makefnmark}{\hbox to \z@{$^{\@thefnmark}$\hss}}
|
||||
% The footnote-mark was overlapping the footnote-text,
|
||||
% added the following to fix this problem (MK)
|
||||
\long\def\@makefntext##1{%
|
||||
\parindent 1em\noindent
|
||||
\hbox to 1.8em{\hss $\m@th ^{\@thefnmark}$}##1
|
||||
}
|
||||
\thispagestyle{empty}
|
||||
\@maketitle
|
||||
\@thanks
|
||||
%\@notice
|
||||
\endgroup
|
||||
\let\maketitle\relax
|
||||
\let\thanks\relax
|
||||
}
|
||||
|
||||
% rules for title box at top of first page
|
||||
\newcommand{\@toptitlebar}{
|
||||
\hrule height 2\p@
|
||||
\vskip 0.25in
|
||||
\vskip -\parskip%
|
||||
}
|
||||
\newcommand{\@bottomtitlebar}{
|
||||
\vskip 0.29in
|
||||
\vskip -\parskip
|
||||
\hrule height 2\p@
|
||||
\vskip 0.09in%
|
||||
}
|
||||
|
||||
% create title (includes both anonymized and non-anonymized versions)
|
||||
\providecommand{\@maketitle}{}
|
||||
\renewcommand{\@maketitle}{%
|
||||
\vbox{%
|
||||
\hsize\textwidth
|
||||
\linewidth\hsize
|
||||
\vskip 0.8in
|
||||
\@toptitlebar
|
||||
\centering
|
||||
{\LARGE\sc \@title\par}
|
||||
\@bottomtitlebar
|
||||
\vskip 0.5in
|
||||
\textsc{\Large\undertitle}\\
|
||||
\vskip 2.0in
|
||||
\def\And{%
|
||||
\end{tabular}\hfil\linebreak[0]\hfil%
|
||||
\begin{tabular}[t]{c}\bf\rule{\z@}{24\p@}\ignorespaces%
|
||||
}
|
||||
\def\AND{%
|
||||
\end{tabular}\hfil\linebreak[4]\hfil%
|
||||
\begin{tabular}[t]{c}\bf\rule{\z@}{24\p@}\large\ignorespaces%
|
||||
}
|
||||
\begin{tabular}[t]{c}\bf\rule{\z@}{24\p@}\Large\@author\end{tabular}%
|
||||
\vskip 1.0in \@minus 0.1in \center{\large\@date} \vskip 0.2in
|
||||
}
|
||||
}
|
||||
|
||||
% add conference notice to bottom of first page
|
||||
\newcommand{\ftype@noticebox}{8}
|
||||
\newcommand{\@notice}{%
|
||||
% give a bit of extra room back to authors on first page
|
||||
\enlargethispage{2\baselineskip}%
|
||||
\@float{noticebox}[b]%
|
||||
\footnotesize\@noticestring%
|
||||
\end@float%
|
||||
}
|
||||
|
||||
% abstract styling
|
||||
\renewenvironment{abstract}
|
||||
{
|
||||
\centerline
|
||||
{\large \bfseries \scshape Abstract}
|
||||
\begin{quote}
|
||||
}
|
||||
{
|
||||
\end{quote}
|
||||
}
|
||||
|
||||
\endinput
|
||||
|
||||
Reference in New Issue
Block a user