diff --git a/bibliography.bib b/bibliography.bib index d9a0ae8..34bd5d0 100644 --- a/bibliography.bib +++ b/bibliography.bib @@ -104,3 +104,11 @@ @misc{futhark howpublished = {\url{https://futhark-lang.org/}}, note = {Accessed: 2021-12-22} } +@inproceedings{andoni2006efficient, +author = {Andoni, Alexandr and Indyk, Piotr}, +year = {2006}, +month = {01}, +pages = {1203-1212}, +title = {Efficient algorithms for substring near neighbor problem}, +doi = {10.1145/1109557.1109690} +} diff --git a/forside/eksempel.tex b/forside/eksempel.tex new file mode 100644 index 0000000..d68ad95 --- /dev/null +++ b/forside/eksempel.tex @@ -0,0 +1,45 @@ +\documentclass[11pt]{article} +\usepackage[a4paper, hmargin={2.8cm, 2.8cm}, vmargin={2.5cm, 2.5cm}]{geometry} % Geometri-pakke: Styrer bl.a. maginer % +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\usepackage[babel, lille]{ku-forside} % KU-forside +% +% Mini-manual til ku-forside pakken: +% +% Sprogmuligheder: da, en +% babel loader babelpakken, med det valgte sprog +% Fakultetsmuligheder: farma, hum, jur, ku, life, nat, samf, sund, teo +% Farvemuligheder: sh, farve +% Forsidemuligheder: lille, stor, titelside +% titelside er identisk med designet på ku.dk/designmanual +% lille er giver et lille logo sammen med titlen på den første side +% stor er giver et stort logo sammen med titlen på den første side +% +% Default er [da,nat,farve,titelside] +% +% Ex. \usepackage[babel, lille, jur, sh, en]{ku-forside} giver et lille logo i sorthvid for juridisk fakultet og loader babelpakken med engelsk som sprog. + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%%% Titel %%% +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\titel{Test} % +\undertitel{Test test} % +\opgave{Overspringshandling} % Findes kun under 'titelside' +\forfatter{Navnet}% +\dato{\today}% +\vejleder{Doktoren} % Findes kun under 'titelside' +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%%% Her begynder dokumentet %%% +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\begin{document} +\maketitle % LAVER TITLEN +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%%%%%%%%%%%%%%%%%%%% TEKST BEGYNDER HER %%%%%%%%%%%%%%%%%%%%%%% +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\section{Reserving on an aggregated level} +$$ +R=\sum_{i=2}^{n}R_{i}=\sum_{i=2}^{n}D_{i(n+1-i)}\left(\prod_{j=n+1-i}^{n-1}\hat{f}_{j}-1\right) +\label{(a0)} +$$ + + +\end{document} diff --git a/introduction/main.tex b/introduction/main.tex index 176848f..68d131c 100644 --- a/introduction/main.tex +++ b/introduction/main.tex @@ -1,6 +1,7 @@ \section{Introduction} The \textit{Approximate Similarity Search Problem} regards efficiently finding a set $A$ from a corpus $\mathcal{F}$ that is approximately similar to a query set $Q$ in regards to the \textit{Jaccard Similarity} metric $J(A,Q) = \frac{|A\cap Q|}{|A\cup Q|}$\cite{dahlgaard2017fast}\cite{fast-similarity-search}. Practical applications includes searching through large corpi of high-dimensional text documents like plagiarism-detection or website duplication checking among others\cite{vassilvitskii2018}. The main bottleneck in this problem is the \textit{curse of dimensionality}. Any trivial algorithm can solve this problem in $O(nd|Q|)$ time, but algorithms that query in linear time to the dimensionality of the corpus scale poorly when working with high-dimensional datasets. Text documents are especially bad in this regard since they often are encoded using \textit{$w$-shingles} ($w$ contigous words) which \citet{li2011hashing} shows easily can reach a dimensionality upwards of $d=2^{83}$ using just $5$-shingles.\\ The classic solution to this problem is the MinHash algorithm presented by \citet{broder1997minhash} to perform website duplication checking for the AltaVista search engine. It preprocesses the data once using hashing to perform effective querying in $O(n + |Q|)$ time, a significant improvement independent of the dimensionality of the corpus. -Many improvements have since been presented to both improve processing time, query time and space efficiency. Notable mentions includes (but are not limited to) \textit{b-bit minwise hashing}\cite{ping2011theory}, \textit{fast similarity sketching}\cite{dahlgaard2017fast} and \textit{parallel bit-counting}\cite{fast-similarity-search} (the latter of which is the main focus of this project). -These contributions have brought the query time down to sublinear time while keeping a constant error probability.\\ -The addition of parallel bit-counting for querying +Many improvements have since been presented to both improve processing time, query time and space efficiency. Notable mentions includes (but are not limited to) the use of \textit{tensoring}\cite{andoni2006efficient}, \textit{b-bit minwise hashing}\cite{ping2011theory}, \textit{fast similarity sketching}\cite{dahlgaard2017fast}. Simple applications of these techniques leads to efficient querying with a constant error probability. If one wishes to achieve an even better error probability such as $\varepsilon = o(1)$, it is standard practice within the field to use $O(\log_2(1/\varepsilon))$ independent data structures and return the best result, resulting in a query time of $O(\frac{1}{\epsilon} (n^\rho + |Q|))$. Recent advances by \citet{fast-similarity-search} show that it is possible to achieve an even better query time by sampling these data structures from one large sketch. The similarity between these sub-sketches and a query set needs to be evaluated efficiently when querying, which requires efficient computation of the cardinality of a bit-string. To do this, \citet{fast-similarity-search} presents a general parallel bit-counting algorithm that computes the cardinality of a list of bit-strings in sub-linear time amortized due to word-parallism. This brings the query time down to $O((\frac{n\log_2 w}{w})^\rho \log(1/\varepsilon) + |Q|)$.\\ +The main focus of this project is to analyse, prove, implement and evaluate this parallel bit-counting technique. The analysis will be based on the original paper \cite{fast-similarity-search}, but with some modifications to resolve some of the issues with the original algorithm. This will also include a pseudo-code implementation of the algorithm since the original paper only describes it through recurrences. This leads to a proof of correctness that slightly alters from the one presented in the paper, and a runtime analysis that does indeed show the sub-linear run time as claimed.\\ +This theoretical analysis will be backed up by a real-life implementation that can be benchmarked to help show this sub-linear run time in practice. At last, reflections on the results and methods will be made to back up eventual conclusions. + diff --git a/ku-forside.sty b/ku-forside.sty new file mode 100644 index 0000000..924ea54 --- /dev/null +++ b/ku-forside.sty @@ -0,0 +1,129 @@ +% KU-forside pakke. Forsider til opgaver skrevet på Københavns Universitet +% Skrevet af Christian Aastrup. Designet af forsiderne følger det på http://www.ku.dk/designmanual +% +\ProvidesPackage{ku-forside}[2007/07/07 v1.0 Frontpages with University of Cph. logos] +% +%Definerer Standard SPROG/AFDELING/FARVE +\def\SPROG{da}\def\FARVE{farve}\def\AFDELING{nat}\def\FORSIDE{titelside} +% +% Laver SPROG-mulighederne til 'if's +\newif\if@en \newif\if@da +% +% Laver AFDELINGS-mulighederne til 'if's +\newif\if@ku \newif\if@farma \newif\if@hum +\newif\if@jur \newif\if@life \newif\if@nat +\newif\if@samf \newif\if@sund \newif\if@teo +% +% Laver FARVE-mulighederne til 'if's +\newif\if@farve \newif\if@sh +% +% Laver FORSIDE-mulighederne til 'if's +\newif\if@titelside \newif\if@stor \newif\if@lille +% +\newif\if@babel \DeclareOption{babel}{\@babeltrue} +% +% Erklærer sprogene som 'options' i pakke-kaldet +\DeclareOption{en}{\@entrue} \DeclareOption{da}{\@datrue} +% +% Erklærer afdelingerne som 'options' i pakke-kaldet +\DeclareOption{ku}{\@kutrue} \DeclareOption{farma}{\@farmatrue} \DeclareOption{hum}{\@humtrue} +\DeclareOption{jur}{\@jurtrue} \DeclareOption{life}{\@lifetrue} \DeclareOption{nat}{\@nattrue} +\DeclareOption{samf}{\@samftrue} \DeclareOption{sund}{\@sundtrue} \DeclareOption{teo}{\@teotrue} +% +% Erklærer farverne som 'options' i pakke-kaldet +\DeclareOption{farve}{\@farvetrue} \DeclareOption{sh}{\@shtrue} +% +% Erklærer forsidemulighederne som 'options' i pakke-kaldet +\DeclareOption{lille}{\@lilletrue} \DeclareOption{stor}{\@stortrue} +\DeclareOption{titelside}{\@titelsidetrue} +% +\ProcessOptions\relax +% +% Definerer hvad der skal ske når sprogene er TRUE +\if@en \def\SPROG{en} \fi \if@da \def\SPROG{da} \fi +% +% Definerer hvad der skal ske når afdelingerne er TRUE +\if@ku \def\AFDELING{ku} \fi \if@farma \def\AFDELING{farma} \fi \if@hum \def\AFDELING{hum} \fi +\if@jur \def\AFDELING{jur} \fi \if@life \def\AFDELING{life} \fi \if@nat \def\AFDELING{nat} \fi +\if@samf \def\AFDELING{samf} \fi \if@sund \def\AFDELING{sund} \fi \if@teo \def\AFDELING{teo} \fi +% +% Definerer hvad der skal ske når farverne er TRUE +\if@sh \def\FARVE{sh} \fi \if@farve \def\FARVE{farve} \fi +% +% Definerer hvad der skal ske når de forskellige forsidemuligheder er TRUE +\if@stor \def\FORSIDE{stor} \fi \if@lille \def\FORSIDE{lille} \fi +\if@titelside \def\FORSIDE{titelside} \fi +% +\def\OPGAVE{$\backslash$opgave$\{\ldots\}$} +\def\FORFATTER{$\backslash$forfatter$\{\ldots\}$ el. $\backslash$author$\{\ldots\}$ } +\def\TITEL{$\backslash$titel$\{\ldots\}$ el. $\backslash$title$\{\ldots\}$} +\def\UNDERTITEL{$\backslash$undertitel$\{\ldots\}$} +\def\VEJLEDER{$\backslash$vejleder$\{\ldots\}$} +\def\AFLEVERINGSDATO{$\backslash$dato$\{\ldots\}$ el. $\backslash$date$\{\ldots\}$} +% +\renewcommand{\author}[1]{\def\FORFATTER{#1}} +\renewcommand{\title}[1]{\def\TITEL{#1}} +\renewcommand{\date}[1]{\def\AFLEVERINGSDATO{#1}} +% +\newcommand{\opgave}[1]{\def\OPGAVE{#1}} +\newcommand{\forfatter}[1]{\def\FORFATTER{#1}} +\newcommand{\titel}[1]{\def\TITEL{#1}} +\newcommand{\undertitel}[1]{\def\UNDERTITEL{#1}} +\newcommand{\vejleder}[1]{\def\VEJLEDER{#1}} +\newcommand{\dato}[1]{\def\AFLEVERINGSDATO{#1}} +% +% Pakker nødvendige for at sætte forsiden op % +% +%\RequirePackage[OT2,OT4]{fontenc} +\RequirePackage{eso-pic,graphicx,fix-cm,ae,aecompl,ifthen} % +\RequirePackage[usenames]{color} % +%% BABEL-option: Undersøger det erklærede sprog og sætter pakken Babel derefter %% +\if@babel +\ifthenelse{\equal{\SPROG}{en}}{\RequirePackage[danish,english]{babel}}{} % Engelsk ordeling, overskrifts- og kapitel struktur % +\ifthenelse{\equal{\SPROG}{da}}{\RequirePackage[english,danish]{babel}}{} % Dansk ordeling, overskrifts- og kapitel struktur % +% Bemærk at begge sprog indlæses. Rækkefølgen er vigtig, idet det er det sidste sprog som dokumnetet generelt sættes i. % +% Det andet sprogs orddeling mm. kan man få fat i ved at skrive \selectlanguage{sprog} i brødteksten % +\fi +% +%% FORSIDEN DEFINERES: % +% +% Mulighed: titelside +\ifthenelse{\equal{\FORSIDE}{titelside}}{ +\def\tyk{\fontfamily{phv}\fontseries{bx}\selectfont} %Bold extended % +\def\tynd{\fontfamily{phv}\fontseries{sb}\selectfont} % Semi-bold % +\def\maketitle{\thispagestyle{empty} % +\AddToShipoutPicture*{\put(0,0){\includegraphics*[viewport=0 0 700 600]{\AFDELING-\FARVE}}}% % +\AddToShipoutPicture*{\put(0,602){\includegraphics*[viewport=0 600 700 1600]{\AFDELING-\FARVE}}}% % +\AddToShipoutPicture*{\put(0,0){\includegraphics*{\AFDELING-\SPROG}}}% % +\AddToShipoutPicture*{\put(50,583.5){\fontsize{20 pt}{22 pt} \tyk \OPGAVE }} % % +\AddToShipoutPicture*{\put(50,555.3){\fontsize{14 pt}{16 pt} \tynd \FORFATTER }} % % +\AddToShipoutPicture*{\put(50,499){\fontsize{22 pt}{24 pt} \tynd \TITEL }} % % +\AddToShipoutPicture*{\put(50,480.5){\fontsize{14 pt}{16 pt} \tynd \UNDERTITEL }} % % +\AddToShipoutPicture*{\put(50,92){\fontsize{11 pt}{12 pt} \tynd \VEJLEDER }} % % +\AddToShipoutPicture*{\put(50,66.7){\fontsize{11 pt}{12 pt} \tynd \AFLEVERINGSDATO }} % % +\phantom{Usynlig, men nødvendig} % +\newpage \noindent}}{} % +% Mulighed: lille +\ifthenelse{\equal{\FORSIDE}{lille}}{ +\def\maketitle{\thispagestyle{plain} +\AddToShipoutPicture*{\put(035,613){\includegraphics*[viewport=0 600 700 1600, scale=0.88]{\AFDELING-\FARVE}}}% Billedet bruges +\AddToShipoutPicture*{\put(-010,613){\includegraphics*[viewport=0 600 420 1600, scale=0.88]{\AFDELING-\FARVE}}}% tre gange for at +\AddToShipoutPicture*{\put(400,613){\includegraphics*[viewport=0 600 420 1600, scale=0.88]{\AFDELING-\FARVE}}}% få stregen lang. +\AddToShipoutPicture*{\put(79,755){\large{\textbf{\TITEL}}}}% +\AddToShipoutPicture*{\put(79,733){\UNDERTITEL}}% +\AddToShipoutPicture*{\put(79,715){\tiny{\emph{\FORFATTER}}}}% +\AddToShipoutPicture*{\put(79,702){\tiny{\AFLEVERINGSDATO}}}% +\phantom{Usynlig, men nødvendig} +\vspace*{3.2cm} % +\noindent}}{} % +% Mulighed: stor +\ifthenelse{\equal{\FORSIDE}{stor}}{ +\def\maketitle{\thispagestyle{plain} +\AddToShipoutPicture*{\put(0,602){\includegraphics*[viewport=156 649 700 1600, scale=1.4]{\AFDELING-\FARVE}}} % % +\AddToShipoutPicture*{\put(79,755){\LARGE{\textbf{\TITEL}}}}% +\AddToShipoutPicture*{\put(79,723){\Large{\UNDERTITEL}}}% +\AddToShipoutPicture*{\put(79,695){\normalsize{\emph{\FORFATTER}}}}% +\AddToShipoutPicture*{\put(79,670){\footnotesize{\AFLEVERINGSDATO}}}% +\phantom{Usynlig, men nødvendig} +\vspace*{5cm} % +\noindent}}{} diff --git a/ku-forside.zip b/ku-forside.zip new file mode 100644 index 0000000..a57d49e Binary files /dev/null and b/ku-forside.zip differ diff --git a/ku-forside/Startark.pdf b/ku-forside/Startark.pdf new file mode 100644 index 0000000..bd00ce4 Binary files /dev/null and b/ku-forside/Startark.pdf differ diff --git a/ku-forside/Startark.tex b/ku-forside/Startark.tex new file mode 100644 index 0000000..fcfbf11 --- /dev/null +++ b/ku-forside/Startark.tex @@ -0,0 +1,88 @@ +\documentclass[11pt]{article} % DOKUMENTKLASSE. Mulighederne er bl.a. article, report, book, memoir % +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% PAKKER DER INDLÆSES %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\usepackage[a4paper, hmargin={2.8cm, 2.8cm}, vmargin={2.8cm, 2.8cm}]{geometry} % Geometri-pakke: Styrer bl.a. maginer % +\usepackage{amssymb} % Matematiske tegn og skrifttyper, bl.a. \mathbb{} % +\usepackage{dsfont} % Anderledes fed skrifttype i matematik-milijøer, som kaldes med $\mathds{}$ % +\usepackage{amsthm} % Opsætning der gør sætninger og beviser nemmere, se amsthdoc.pdf % +\usepackage[ansinew]{inputenc} % Lidt kodning så der ikke kommer problemer ved visse konverteringer % +\usepackage{amsmath} % Matematiske tegn % +%\usepackage[colorlinks=true]{hyperref} % Referencer som hyperlinks % +\usepackage{eurosym} % Euro (€) by typing \euro % +\def\texteuro{\euro} % Definerer € % +%\usepackage{txfonts} % Times font % +\usepackage[hang,small]{caption} % Caption-pakke % +\setlength{\captionmargin}{20pt} % Sætter caption-margin % +\usepackage{listings} % Listings. Indsætter kildekode pænt. % +\lstset{ % % +basicstyle=\ttfamily\footnotesize, % % Sætter basisstilen i listings % +showstringspaces=false, % % no special string spaces % +extendedchars=true % % Bogstaver som æ, ø og å % +} % % +\usepackage{courier} % Courierskrifttype. Slankere skrivemaskineskrift i verbatim og listings % +\usepackage{multicol} % Kolonner % +\usepackage[babel, lille]{ku-forside} % KU-forside med logoer % +\def\HyperLinks{ % Hyperlinks-pakke, der laver referencer til links og tillader links til www % +\usepackage[pdftitle={\TITEL},pdfauthor={\FORFATTER}, % % Der er foretaget et lille trick så pakken indlæses efter % +pdfsubject={\UNDERTITEL}, linkbordercolor={0.8 0.8 0.8}]{hyperref}} % titlen defineres. % +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +% MINI-MANUAL TIL ku-forside PAKKEN: % +% % +% Sprogmuligheder: da, en % +% babel loader babelpakken, med det valgte sprog % +% Fakultetsmuligheder: farma, hum, jur, ku, life, nat, samf, sund, teo % +% Farvemuligheder: sh, farve % +% Forsidemuligheder: lille, stor, titelside % +% titelside er identisk med designet på ku.dk/designmanual % +% lille er giver et lille logo sammen med titlen på den første side % +% stor er giver et stort logo sammen med titlen på den første side % +% % +% Default er [da,nat,farve,titelside] % +% % +% Ex. \usepackage[babel, lille, jur, sh, en]{ku-forside} giver et lille logo i % +% sorthvid for juridisk fakultet og loader babelpakken med engelsk som sprog. % +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% KALIGRAFISK MATEMATIKAlFABET: \mcal %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\DeclareSymbolFont{rsfscript}{OMS}{rsfs}{m}{n} % Henter alfabet % +\DeclareSymbolFontAlphabet{\mcal}{rsfscript} % Kalder dette alfabet for \mcal % +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\setlength\arraycolsep{2 pt} % Sætter kolonneafstanden i tabeller og eqnarrays til 2pt % +\setcounter{tocdepth}{2} % Dybde af indholdsfortegnelsen, 1 resulterer i at kun sections kommer med, 2 er sectons og subsec. etc.% +\setcounter{secnumdepth}{0}% Dybde af sektionsnummereringen. % +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% TITEL %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\titel{Min titel} % +\undertitel{Min undertitel, fx. faget opgaven skrives i} % +\opgave{Overspringshandling} % Findes kun under 'titelside' muligheden i ku-forside +\forfatter{Navn Navnesen} % +\dato{\today}% +\vejleder{Doktoren} % Findes kun under 'titelside' muligheden i ku-forside +\HyperLinks % Henter hyperlinks-pakke og sætter pdf-titel mm. til at svare til de just definerede +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% DOKUMENT BEGYNDER %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\begin{document} +\maketitle % Laver titlen +%%%%%%%%%%%%% Indholdsfortegnelse %%%%%%%%%%%%% +\tableofcontents % Indholdsfortegnelse +%\listoffigures % Liste over figurer \begin{figure} ... \end{figure} +%\listoftables % Liste over tabeller \begin{table} ... \end{table} +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\section{A} +\url{http://sast.larslau.dk} +\subsection{a} +\subsubsection{a} +\paragraph{a} +\subparagraph{a} + +\subsection{b} +\subsection{c} +\section{B} + +\end{document} +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% DOKUMENT SLUTTER %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% diff --git a/ku-forside/eksempel.pdf b/ku-forside/eksempel.pdf new file mode 100644 index 0000000..8b01ab4 Binary files /dev/null and b/ku-forside/eksempel.pdf differ diff --git a/ku-forside/eksempel.tex b/ku-forside/eksempel.tex new file mode 100644 index 0000000..d68ad95 --- /dev/null +++ b/ku-forside/eksempel.tex @@ -0,0 +1,45 @@ +\documentclass[11pt]{article} +\usepackage[a4paper, hmargin={2.8cm, 2.8cm}, vmargin={2.5cm, 2.5cm}]{geometry} % Geometri-pakke: Styrer bl.a. maginer % +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\usepackage[babel, lille]{ku-forside} % KU-forside +% +% Mini-manual til ku-forside pakken: +% +% Sprogmuligheder: da, en +% babel loader babelpakken, med det valgte sprog +% Fakultetsmuligheder: farma, hum, jur, ku, life, nat, samf, sund, teo +% Farvemuligheder: sh, farve +% Forsidemuligheder: lille, stor, titelside +% titelside er identisk med designet på ku.dk/designmanual +% lille er giver et lille logo sammen med titlen på den første side +% stor er giver et stort logo sammen med titlen på den første side +% +% Default er [da,nat,farve,titelside] +% +% Ex. \usepackage[babel, lille, jur, sh, en]{ku-forside} giver et lille logo i sorthvid for juridisk fakultet og loader babelpakken med engelsk som sprog. + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%%% Titel %%% +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\titel{Test} % +\undertitel{Test test} % +\opgave{Overspringshandling} % Findes kun under 'titelside' +\forfatter{Navnet}% +\dato{\today}% +\vejleder{Doktoren} % Findes kun under 'titelside' +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%%% Her begynder dokumentet %%% +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\begin{document} +\maketitle % LAVER TITLEN +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%%%%%%%%%%%%%%%%%%%% TEKST BEGYNDER HER %%%%%%%%%%%%%%%%%%%%%%% +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\section{Reserving on an aggregated level} +$$ +R=\sum_{i=2}^{n}R_{i}=\sum_{i=2}^{n}D_{i(n+1-i)}\left(\prod_{j=n+1-i}^{n-1}\hat{f}_{j}-1\right) +\label{(a0)} +$$ + + +\end{document} diff --git a/ku-forside/farma-da.pdf b/ku-forside/farma-da.pdf new file mode 100644 index 0000000..f5116e0 Binary files /dev/null and b/ku-forside/farma-da.pdf differ diff --git a/ku-forside/farma-da.ps b/ku-forside/farma-da.ps new file mode 100644 index 0000000..529e29a Binary files /dev/null and b/ku-forside/farma-da.ps differ diff --git a/ku-forside/farma-en.pdf b/ku-forside/farma-en.pdf new file 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+\usepackage[a4paper, hmargin={2.8cm, 2.8cm}, vmargin={2.5cm, 2.5cm}]{geometry} % Geometri-pakke: Styrer bl.a. maginer % +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\usepackage[en, nat, babel, titelside]{ku-forside} % KU-forside +\usepackage[utf8]{inputenc} \usepackage[ backend=biber, style=ieee, @@ -32,10 +37,36 @@ \graphicspath{ {./assets} } +% +% Mini-manual til ku-forside pakken: +% +% Sprogmuligheder: da, en +% babel loader babelpakken, med det valgte sprog +% Fakultetsmuligheder: farma, hum, jur, ku, life, nat, samf, sund, teo +% Farvemuligheder: sh, farve +% Forsidemuligheder: lille, stor, titelside +% titelside er identisk med designet pÃ¥ ku.dk/designmanual +% lille er giver et lille logo sammen med titlen pÃ¥ den første side +% stor er giver et stort logo sammen med titlen pÃ¥ den første side +% +% Default er [da,nat,farve,titelside] +% +% Ex. \usepackage[babel, lille, jur, sh, en]{ku-forside} giver et lille logo i sorthvid for juridisk fakultet og loader babelpakken med engelsk som sprog. + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%%% Titel %%% +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\titel{Parallel Bit-Counting} % +\undertitel{for Approximate Similarity Searching} % +\opgave{Bachelor's project} % Findes kun under 'titelside' +\forfatter{Rasmus Hag Løvstad, \texttt{pgq596@alumni.ku.dk}}% +\dato{\today}% +\vejleder{Supervisor: Mikkel Thorup} % Findes kun under 'titelside' \begin{document} -\title{Parallel Bit-Counting for Approximate Similarity Searching} -\author{Rasmus Hag Løvstad} -\date{Block 1+2, 2021} + +% \title{Parallel Bit-Counting for Approximate Similarity Searching} +% \author{Rasmus Hag Løvstad} +% \date{Block 1+2, 2021} \maketitle \input{abstract/main} \tableofcontents diff --git a/nat-da.pdf b/nat-da.pdf new file mode 100644 index 0000000..a3b1a36 Binary files /dev/null and b/nat-da.pdf differ diff --git a/nat-da.ps b/nat-da.ps new file mode 100644 index 0000000..cb140cf Binary files 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