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PI.tex
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\documentclass[11pt]{article}
\usepackage[breakable]{tcolorbox}
\usepackage{parskip} % Stop auto-indenting (to mimic markdown behaviour)
\usepackage{iftex}
\ifPDFTeX
\usepackage[T1]{fontenc}
\usepackage{mathpazo}
\else
\usepackage{fontspec}
\fi
% Basic figure setup, for now with no caption control since it's done
% automatically by Pandoc (which extracts  syntax from Markdown).
\usepackage{graphicx}
% Maintain compatibility with old templates. Remove in nbconvert 6.0
\let\Oldincludegraphics\includegraphics
% Ensure that by default, figures have no caption (until we provide a
% proper Figure object with a Caption API and a way to capture that
% in the conversion process - todo).
\usepackage{caption}
\DeclareCaptionFormat{nocaption}{}
\captionsetup{format=nocaption,aboveskip=0pt,belowskip=0pt}
\usepackage{float}
\floatplacement{figure}{H} % forces figures to be placed at the correct location
\usepackage{xcolor} % Allow colors to be defined
\usepackage{enumerate} % Needed for markdown enumerations to work
\usepackage{geometry} % Used to adjust the document margins
\usepackage{amsmath} % Equations
\usepackage{amssymb} % Equations
\usepackage{textcomp} % defines textquotesingle
% Hack from http://tex.stackexchange.com/a/47451/13684:
\AtBeginDocument{%
\def\PYZsq{\textquotesingle}% Upright quotes in Pygmentized code
}
\usepackage{upquote} % Upright quotes for verbatim code
\usepackage{eurosym} % defines \euro
\usepackage[mathletters]{ucs} % Extended unicode (utf-8) support
\usepackage{fancyvrb} % verbatim replacement that allows latex
\usepackage{grffile} % extends the file name processing of package graphics
% to support a larger range
\makeatletter % fix for old versions of grffile with XeLaTeX
\@ifpackagelater{grffile}{2019/11/01}
{
% Do nothing on new versions
}
{
\def\Gread@@xetex#1{%
\IfFileExists{"\Gin@base".bb}%
{\Gread@eps{\Gin@base.bb}}%
{\Gread@@xetex@aux#1}%
}
}
\makeatother
\usepackage[Export]{adjustbox} % Used to constrain images to a maximum size
\adjustboxset{max size={0.9\linewidth}{0.9\paperheight}}
% The hyperref package gives us a pdf with properly built
% internal navigation ('pdf bookmarks' for the table of contents,
% internal cross-reference links, web links for URLs, etc.)
\usepackage{hyperref}
% The default LaTeX title has an obnoxious amount of whitespace. By default,
% titling removes some of it. It also provides customization options.
\usepackage{titling}
\usepackage{longtable} % longtable support required by pandoc >1.10
\usepackage{booktabs} % table support for pandoc > 1.12.2
\usepackage[inline]{enumitem} % IRkernel/repr support (it uses the enumerate* environment)
\usepackage[normalem]{ulem} % ulem is needed to support strikethroughs (\sout)
% normalem makes italics be italics, not underlines
\usepackage{mathrsfs}
% Colors for the hyperref package
\definecolor{urlcolor}{rgb}{0,.145,.698}
\definecolor{linkcolor}{rgb}{.71,0.21,0.01}
\definecolor{citecolor}{rgb}{.12,.54,.11}
% ANSI colors
\definecolor{ansi-black}{HTML}{3E424D}
\definecolor{ansi-black-intense}{HTML}{282C36}
\definecolor{ansi-red}{HTML}{E75C58}
\definecolor{ansi-red-intense}{HTML}{B22B31}
\definecolor{ansi-green}{HTML}{00A250}
\definecolor{ansi-green-intense}{HTML}{007427}
\definecolor{ansi-yellow}{HTML}{DDB62B}
\definecolor{ansi-yellow-intense}{HTML}{B27D12}
\definecolor{ansi-blue}{HTML}{208FFB}
\definecolor{ansi-blue-intense}{HTML}{0065CA}
\definecolor{ansi-magenta}{HTML}{D160C4}
\definecolor{ansi-magenta-intense}{HTML}{A03196}
\definecolor{ansi-cyan}{HTML}{60C6C8}
\definecolor{ansi-cyan-intense}{HTML}{258F8F}
\definecolor{ansi-white}{HTML}{C5C1B4}
\definecolor{ansi-white-intense}{HTML}{A1A6B2}
\definecolor{ansi-default-inverse-fg}{HTML}{FFFFFF}
\definecolor{ansi-default-inverse-bg}{HTML}{000000}
% common color for the border for error outputs.
\definecolor{outerrorbackground}{HTML}{FFDFDF}
% commands and environments needed by pandoc snippets
% extracted from the output of `pandoc -s`
\providecommand{\tightlist}{%
\setlength{\itemsep}{0pt}\setlength{\parskip}{0pt}}
\DefineVerbatimEnvironment{Highlighting}{Verbatim}{commandchars=\\\{\}}
% Add ',fontsize=\small' for more characters per line
\newenvironment{Shaded}{}{}
\newcommand{\KeywordTok}[1]{\textcolor[rgb]{0.00,0.44,0.13}{\textbf{{#1}}}}
\newcommand{\DataTypeTok}[1]{\textcolor[rgb]{0.56,0.13,0.00}{{#1}}}
\newcommand{\DecValTok}[1]{\textcolor[rgb]{0.25,0.63,0.44}{{#1}}}
\newcommand{\BaseNTok}[1]{\textcolor[rgb]{0.25,0.63,0.44}{{#1}}}
\newcommand{\FloatTok}[1]{\textcolor[rgb]{0.25,0.63,0.44}{{#1}}}
\newcommand{\CharTok}[1]{\textcolor[rgb]{0.25,0.44,0.63}{{#1}}}
\newcommand{\StringTok}[1]{\textcolor[rgb]{0.25,0.44,0.63}{{#1}}}
\newcommand{\CommentTok}[1]{\textcolor[rgb]{0.38,0.63,0.69}{\textit{{#1}}}}
\newcommand{\OtherTok}[1]{\textcolor[rgb]{0.00,0.44,0.13}{{#1}}}
\newcommand{\AlertTok}[1]{\textcolor[rgb]{1.00,0.00,0.00}{\textbf{{#1}}}}
\newcommand{\FunctionTok}[1]{\textcolor[rgb]{0.02,0.16,0.49}{{#1}}}
\newcommand{\RegionMarkerTok}[1]{{#1}}
\newcommand{\ErrorTok}[1]{\textcolor[rgb]{1.00,0.00,0.00}{\textbf{{#1}}}}
\newcommand{\NormalTok}[1]{{#1}}
% Additional commands for more recent versions of Pandoc
\newcommand{\ConstantTok}[1]{\textcolor[rgb]{0.53,0.00,0.00}{{#1}}}
\newcommand{\SpecialCharTok}[1]{\textcolor[rgb]{0.25,0.44,0.63}{{#1}}}
\newcommand{\VerbatimStringTok}[1]{\textcolor[rgb]{0.25,0.44,0.63}{{#1}}}
\newcommand{\SpecialStringTok}[1]{\textcolor[rgb]{0.73,0.40,0.53}{{#1}}}
\newcommand{\ImportTok}[1]{{#1}}
\newcommand{\DocumentationTok}[1]{\textcolor[rgb]{0.73,0.13,0.13}{\textit{{#1}}}}
\newcommand{\AnnotationTok}[1]{\textcolor[rgb]{0.38,0.63,0.69}{\textbf{\textit{{#1}}}}}
\newcommand{\CommentVarTok}[1]{\textcolor[rgb]{0.38,0.63,0.69}{\textbf{\textit{{#1}}}}}
\newcommand{\VariableTok}[1]{\textcolor[rgb]{0.10,0.09,0.49}{{#1}}}
\newcommand{\ControlFlowTok}[1]{\textcolor[rgb]{0.00,0.44,0.13}{\textbf{{#1}}}}
\newcommand{\OperatorTok}[1]{\textcolor[rgb]{0.40,0.40,0.40}{{#1}}}
\newcommand{\BuiltInTok}[1]{{#1}}
\newcommand{\ExtensionTok}[1]{{#1}}
\newcommand{\PreprocessorTok}[1]{\textcolor[rgb]{0.74,0.48,0.00}{{#1}}}
\newcommand{\AttributeTok}[1]{\textcolor[rgb]{0.49,0.56,0.16}{{#1}}}
\newcommand{\InformationTok}[1]{\textcolor[rgb]{0.38,0.63,0.69}{\textbf{\textit{{#1}}}}}
\newcommand{\WarningTok}[1]{\textcolor[rgb]{0.38,0.63,0.69}{\textbf{\textit{{#1}}}}}
% Define a nice break command that doesn't care if a line doesn't already
% exist.
\def\br{\hspace*{\fill} \\* }
% Math Jax compatibility definitions
\def\gt{>}
\def\lt{<}
\let\Oldtex\TeX
\let\Oldlatex\LaTeX
\renewcommand{\TeX}{\textrm{\Oldtex}}
\renewcommand{\LaTeX}{\textrm{\Oldlatex}}
% Document parameters
% Document title
\title{PI}
% Pygments definitions
\makeatletter
\def\PY@reset{\let\PY@it=\relax \let\PY@bf=\relax%
\let\PY@ul=\relax \let\PY@tc=\relax%
\let\PY@bc=\relax \let\PY@ff=\relax}
\def\PY@tok#1{\csname PY@tok@#1\endcsname}
\def\PY@toks#1+{\ifx\relax#1\empty\else%
\PY@tok{#1}\expandafter\PY@toks\fi}
\def\PY@do#1{\PY@bc{\PY@tc{\PY@ul{%
\PY@it{\PY@bf{\PY@ff{#1}}}}}}}
\def\PY#1#2{\PY@reset\PY@toks#1+\relax+\PY@do{#2}}
\expandafter\def\csname PY@tok@w\endcsname{\def\PY@tc##1{\textcolor[rgb]{0.73,0.73,0.73}{##1}}}
\expandafter\def\csname PY@tok@c\endcsname{\let\PY@it=\textit\def\PY@tc##1{\textcolor[rgb]{0.25,0.50,0.50}{##1}}}
\expandafter\def\csname PY@tok@cp\endcsname{\def\PY@tc##1{\textcolor[rgb]{0.74,0.48,0.00}{##1}}}
\expandafter\def\csname PY@tok@k\endcsname{\let\PY@bf=\textbf\def\PY@tc##1{\textcolor[rgb]{0.00,0.50,0.00}{##1}}}
\expandafter\def\csname PY@tok@kp\endcsname{\def\PY@tc##1{\textcolor[rgb]{0.00,0.50,0.00}{##1}}}
\expandafter\def\csname PY@tok@kt\endcsname{\def\PY@tc##1{\textcolor[rgb]{0.69,0.00,0.25}{##1}}}
\expandafter\def\csname PY@tok@o\endcsname{\def\PY@tc##1{\textcolor[rgb]{0.40,0.40,0.40}{##1}}}
\expandafter\def\csname PY@tok@ow\endcsname{\let\PY@bf=\textbf\def\PY@tc##1{\textcolor[rgb]{0.67,0.13,1.00}{##1}}}
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\expandafter\def\csname PY@tok@ni\endcsname{\let\PY@bf=\textbf\def\PY@tc##1{\textcolor[rgb]{0.60,0.60,0.60}{##1}}}
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\expandafter\def\csname PY@tok@nt\endcsname{\let\PY@bf=\textbf\def\PY@tc##1{\textcolor[rgb]{0.00,0.50,0.00}{##1}}}
\expandafter\def\csname PY@tok@nd\endcsname{\def\PY@tc##1{\textcolor[rgb]{0.67,0.13,1.00}{##1}}}
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\expandafter\def\csname PY@tok@sd\endcsname{\let\PY@it=\textit\def\PY@tc##1{\textcolor[rgb]{0.73,0.13,0.13}{##1}}}
\expandafter\def\csname PY@tok@si\endcsname{\let\PY@bf=\textbf\def\PY@tc##1{\textcolor[rgb]{0.73,0.40,0.53}{##1}}}
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\expandafter\def\csname PY@tok@gr\endcsname{\def\PY@tc##1{\textcolor[rgb]{1.00,0.00,0.00}{##1}}}
\expandafter\def\csname PY@tok@ge\endcsname{\let\PY@it=\textit}
\expandafter\def\csname PY@tok@gs\endcsname{\let\PY@bf=\textbf}
\expandafter\def\csname PY@tok@gp\endcsname{\let\PY@bf=\textbf\def\PY@tc##1{\textcolor[rgb]{0.00,0.00,0.50}{##1}}}
\expandafter\def\csname PY@tok@go\endcsname{\def\PY@tc##1{\textcolor[rgb]{0.53,0.53,0.53}{##1}}}
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\begin{document}
\maketitle
\hypertarget{inf421-pi-2021-matching-under-constraints}{%
\section{INF421 PI 2021: Matching under
constraints}\label{inf421-pi-2021-matching-under-constraints}}
By : Taha Hammadia \& Styve Ngamou
For : M. Patrick Loiseau
\hypertarget{disclaimer}{%
\subsection{Disclaimer:}\label{disclaimer}}
In this assessment, we may use terms that are not politicaly correct. By
it, we mean no political statement and we are only discussing programing
aspects.
\hypertarget{task-1}{%
\subsection{Task 1:}\label{task-1}}
\hypertarget{feasibility}{%
\subsubsection{Feasibility:}\label{feasibility}}
The definition of stability is the same for both problems.
\hypertarget{fairness}{%
\subsubsection{Fairness:}\label{fairness}}
The definition of fairness is the same for both problems.
\$ \$
Since the two definitions coincide, it is sufficient to show that the
Gale and Shapley definition of stability implies non-wastefulness and
individual rationality:
\hypertarget{non-wastefulness}{%
\subsubsection{Non-wastefulness:}\label{non-wastefulness}}
Let \(\mu\) a matching. Suppose there is \((i, s) \in I \times S\) such
that \(s_{i} \gt _{i} \mu_{i}\) and \(|\mu_{s}| < q_{s}\).
We have: \(i >_{s} \emptyset\), therefore \(\mu\) is unstable.
\hypertarget{individual-rationality}{%
\subsubsection{Individual rationality:}\label{individual-rationality}}
???
\hypertarget{task-2}{%
\subsection{Task 2:}\label{task-2}}
\hypertarget{discussing-the-implementation}{%
\subsubsection{Discussing the
implementation:}\label{discussing-the-implementation}}
Here, we simply implement the algoithm of Gale and Shapley adapted to
many-to-matching problem where the ending condition must be ajusted.
\hypertarget{choice-of-data-structure}{%
\subsubsection{Choice of data
structure:}\label{choice-of-data-structure}}
We have opted to use class structures which helped us combine data over
students and schools in an efficient way.
In the \texttt{student} class, the list prefStud helped us to have the
rank of each school for the student. This choice of data structure makes
the access to the ranking easier.
In the \texttt{school} class, the list prefer containts the students
ordered acording to the preference of the school. This choice of data
structure makes the execution of the Gale and Shapley algorithm easier.
Indeed, we only have to take the students by order of prefer.
In the main function \texttt{mate}, the result is presented into the
shape of a list of lists. Each list within the great list contains the
students who has integrated the corresponding school. The last list
corresponds to students who do not have a school.
\begin{tcolorbox}[breakable, size=fbox, boxrule=1pt, pad at break*=1mm,colback=cellbackground, colframe=cellborder]
\prompt{In}{incolor}{1}{\boxspacing}
\begin{Verbatim}[commandchars=\\\{\}]
\PY{k}{class} \PY{n+nc}{student2}\PY{p}{:}
\PY{l+s+sd}{\PYZdq{}\PYZdq{}\PYZdq{}Definition of the class student\PYZdq{}\PYZdq{}\PYZdq{}}
\PY{k}{def} \PY{n+nf+fm}{\PYZus{}\PYZus{}init\PYZus{}\PYZus{}}\PY{p}{(}\PY{n+nb+bp}{self}\PY{p}{,} \PY{n}{prefStud}\PY{p}{)}\PY{p}{:}
\PY{l+s+sd}{\PYZdq{}\PYZdq{}\PYZdq{}}
\PY{l+s+sd}{ prefStud is a list that associates to each school its rank for the student.}
\PY{l+s+sd}{ A case is added for including the choice of not joining any school at all.}
\PY{l+s+sd}{ \PYZdq{}\PYZdq{}\PYZdq{}}
\PY{n+nb+bp}{self}\PY{o}{.}\PY{n}{prefStud} \PY{o}{=} \PY{n}{prefStud}
\PY{n+nb+bp}{self}\PY{o}{.}\PY{n}{schlStud} \PY{o}{=} \PY{n+nb}{len}\PY{p}{(}\PY{n}{prefStud}\PY{p}{)} \PY{o}{\PYZhy{}} \PY{l+m+mi}{1}
\PY{k}{class} \PY{n+nc}{school2}\PY{p}{:}
\PY{l+s+sd}{\PYZdq{}\PYZdq{}\PYZdq{}Definition of the school class\PYZdq{}\PYZdq{}\PYZdq{}}
\PY{n}{idx} \PY{o}{=} \PY{l+m+mi}{0}
\PY{n}{n} \PY{o}{=} \PY{l+m+mi}{0}
\PY{k}{def} \PY{n+nf+fm}{\PYZus{}\PYZus{}init\PYZus{}\PYZus{}}\PY{p}{(}\PY{n+nb+bp}{self}\PY{p}{,} \PY{n}{prefer}\PY{p}{,} \PY{n}{quota}\PY{p}{)}\PY{p}{:}
\PY{l+s+sd}{\PYZdq{}\PYZdq{}\PYZdq{}}
\PY{l+s+sd}{ prefer contains the students ordered according to the preference of the school.}
\PY{l+s+sd}{ \PYZdq{}\PYZdq{}\PYZdq{}}
\PY{n+nb+bp}{self}\PY{o}{.}\PY{n}{prefer} \PY{o}{=} \PY{n}{prefer}
\PY{n+nb+bp}{self}\PY{o}{.}\PY{n}{quota} \PY{o}{=} \PY{n}{quota}
\end{Verbatim}
\end{tcolorbox}
\begin{tcolorbox}[breakable, size=fbox, boxrule=1pt, pad at break*=1mm,colback=cellbackground, colframe=cellborder]
\prompt{In}{incolor}{2}{\boxspacing}
\begin{Verbatim}[commandchars=\\\{\}]
\PY{k}{def} \PY{n+nf}{mate2}\PY{p}{(}\PY{n}{schools}\PY{p}{,} \PY{n}{students}\PY{p}{)}\PY{p}{:}
\PY{n}{N} \PY{o}{=} \PY{n+nb}{len}\PY{p}{(}\PY{n}{students}\PY{p}{)}
\PY{n}{m} \PY{o}{=} \PY{n+nb}{len}\PY{p}{(}\PY{n}{schools}\PY{p}{)}
\PY{n}{i} \PY{o}{=} \PY{o}{\PYZhy{}}\PY{l+m+mi}{1}
\PY{c+c1}{\PYZsh{} since the algorithm goes through the preferences of the schools in decreasing order of}
\PY{c+c1}{\PYZsh{} preference, we have a stable matching when all quotas are met.}
\PY{k}{while} \PY{k+kc}{True}\PY{p}{:}
\PY{k}{for} \PY{n}{j} \PY{o+ow}{in} \PY{n+nb}{range}\PY{p}{(}\PY{n}{m}\PY{p}{)}\PY{p}{:}
\PY{k}{if} \PY{n}{schools}\PY{p}{[}\PY{n}{j}\PY{p}{]}\PY{o}{.}\PY{n}{n} \PY{o}{\PYZlt{}} \PY{n}{schools}\PY{p}{[}\PY{n}{j}\PY{p}{]}\PY{o}{.}\PY{n}{quota} \PY{o+ow}{and} \PY{n}{schools}\PY{p}{[}\PY{n}{i}\PY{p}{]}\PY{o}{.}\PY{n}{idx} \PY{o}{\PYZlt{}} \PY{n}{N}\PY{p}{:}
\PY{n}{i} \PY{o}{=} \PY{n}{j}
\PY{k}{break}
\PY{k}{else}\PY{p}{:}
\PY{k}{break}
\PY{k}{while} \PY{n}{schools}\PY{p}{[}\PY{n}{i}\PY{p}{]}\PY{o}{.}\PY{n}{n} \PY{o}{\PYZlt{}} \PY{n}{schools}\PY{p}{[}\PY{n}{i}\PY{p}{]}\PY{o}{.}\PY{n}{quota} \PY{o+ow}{and} \PY{n}{schools}\PY{p}{[}\PY{n}{i}\PY{p}{]}\PY{o}{.}\PY{n}{idx} \PY{o}{\PYZlt{}} \PY{n}{N}\PY{p}{:}
\PY{n}{stud} \PY{o}{=} \PY{n}{schools}\PY{p}{[}\PY{n}{i}\PY{p}{]}\PY{o}{.}\PY{n}{prefer}\PY{p}{[}\PY{n}{schools}\PY{p}{[}\PY{n}{i}\PY{p}{]}\PY{o}{.}\PY{n}{idx}\PY{p}{]}
\PY{k}{if} \PY{n}{stud}\PY{o}{.}\PY{n}{prefStud}\PY{p}{[}\PY{n}{i}\PY{p}{]} \PY{o}{\PYZlt{}} \PY{n}{stud}\PY{o}{.}\PY{n}{prefStud}\PY{p}{[}\PY{n}{stud}\PY{o}{.}\PY{n}{schlStud}\PY{p}{]}\PY{p}{:}
\PY{k}{if} \PY{n}{stud}\PY{o}{.}\PY{n}{schlStud} \PY{o}{\PYZlt{}} \PY{n}{m}\PY{p}{:}
\PY{n}{schools}\PY{p}{[}\PY{n}{stud}\PY{o}{.}\PY{n}{schlStud}\PY{p}{]}\PY{o}{.}\PY{n}{n} \PY{o}{\PYZhy{}}\PY{o}{=} \PY{l+m+mi}{1}
\PY{n}{schools}\PY{p}{[}\PY{n}{i}\PY{p}{]}\PY{o}{.}\PY{n}{n} \PY{o}{+}\PY{o}{=} \PY{l+m+mi}{1}
\PY{n}{stud}\PY{o}{.}\PY{n}{schlStud} \PY{o}{=} \PY{n}{i}
\PY{n}{schools}\PY{p}{[}\PY{n}{i}\PY{p}{]}\PY{o}{.}\PY{n}{idx} \PY{o}{+}\PY{o}{=} \PY{l+m+mi}{1}
\PY{n}{res} \PY{o}{=} \PY{p}{[}\PY{n+nb}{list}\PY{p}{(}\PY{p}{)} \PY{k}{for} \PY{n}{k} \PY{o+ow}{in} \PY{n+nb}{range}\PY{p}{(}\PY{n}{m} \PY{o}{+} \PY{l+m+mi}{1}\PY{p}{)}\PY{p}{]}
\PY{k}{for} \PY{n}{k} \PY{o+ow}{in} \PY{n+nb}{range}\PY{p}{(}\PY{n}{N}\PY{p}{)}\PY{p}{:}
\PY{n}{res}\PY{p}{[}\PY{n}{students}\PY{p}{[}\PY{n}{k}\PY{p}{]}\PY{o}{.}\PY{n}{schlStud}\PY{p}{]}\PY{o}{.}\PY{n}{append}\PY{p}{(}\PY{n}{k}\PY{p}{)}
\PY{k}{return} \PY{n}{res}
\end{Verbatim}
\end{tcolorbox}
\hypertarget{complexity}{%
\subsubsection{Complexity:}\label{complexity}}
\hypertarget{complexity-in-time}{%
\paragraph{Complexity in time:}\label{complexity-in-time}}
We count complexity in terms of how many times a ``school asks a
student''.
In the algorithm, each student is asked by a school at most once,
therefore the complexity is \(O(N m)\), where \(N\) is the number of
students and \(m\) the number of schools.
\hypertarget{complexity-in-memory}{%
\paragraph{Complexity in memory:}\label{complexity-in-memory}}
For each one of the \(N\) student, a list of \(\Theta(m)\) number
represent the ranking of schools. For each one of the \(m\) schools, a
list of \(N\) pointers to elements of the class \texttt{student}.
At the end, the space complexity is \(\Theta(N m)\).
\hypertarget{task-3}{%
\subsection{Task 3:}\label{task-3}}
\hypertarget{discussing-the-implementation}{%
\subsubsection{Discussing the
implementation:}\label{discussing-the-implementation}}
As before, we are adapting the Gale and Shapley algorithm to our
problem. The ending condition is not changed; whereas a school cannot
ask a person only if the number of waiting students of their group is
strictly less than the quota of the group.
\hypertarget{choice-of-data-structure}{%
\subsubsection{Choice of data
structure:}\label{choice-of-data-structure}}
We use the same structures as before. Moreover, in order to account for
group, we add an attribute group in the \texttt{student} class and a
list in the \texttt{school} class that correponds to the number of
people of each group.
\begin{tcolorbox}[breakable, size=fbox, boxrule=1pt, pad at break*=1mm,colback=cellbackground, colframe=cellborder]
\prompt{In}{incolor}{3}{\boxspacing}
\begin{Verbatim}[commandchars=\\\{\}]
\PY{k}{class} \PY{n+nc}{student3}\PY{p}{:}
\PY{l+s+sd}{\PYZdq{}\PYZdq{}\PYZdq{}Definition of the class student\PYZdq{}\PYZdq{}\PYZdq{}}
\PY{k}{def} \PY{n+nf+fm}{\PYZus{}\PYZus{}init\PYZus{}\PYZus{}}\PY{p}{(}\PY{n+nb+bp}{self}\PY{p}{,} \PY{n}{prefStud}\PY{p}{,} \PY{n}{grp}\PY{p}{)}\PY{p}{:}
\PY{l+s+sd}{\PYZdq{}\PYZdq{}\PYZdq{}}
\PY{l+s+sd}{ prefStud is a list that associates to each school its rank for the student.}
\PY{l+s+sd}{ A case is added for including the choice of not joining any school at all.}
\PY{l+s+sd}{ \PYZdq{}\PYZdq{}\PYZdq{}}
\PY{n+nb+bp}{self}\PY{o}{.}\PY{n}{prefStud} \PY{o}{=} \PY{n}{prefStud}
\PY{n+nb+bp}{self}\PY{o}{.}\PY{n}{schlStud} \PY{o}{=} \PY{n+nb}{len}\PY{p}{(}\PY{n}{prefStud}\PY{p}{)} \PY{o}{\PYZhy{}} \PY{l+m+mi}{1}
\PY{n+nb+bp}{self}\PY{o}{.}\PY{n}{grp} \PY{o}{=} \PY{n}{grp}
\PY{k}{class} \PY{n+nc}{school3}\PY{p}{:}
\PY{l+s+sd}{\PYZdq{}\PYZdq{}\PYZdq{}Definition of the school class\PYZdq{}\PYZdq{}\PYZdq{}}
\PY{n}{idx} \PY{o}{=} \PY{l+m+mi}{0}
\PY{n}{n} \PY{o}{=} \PY{l+m+mi}{0}
\PY{k}{def} \PY{n+nf+fm}{\PYZus{}\PYZus{}init\PYZus{}\PYZus{}}\PY{p}{(}\PY{n+nb+bp}{self}\PY{p}{,} \PY{n}{prefer}\PY{p}{,} \PY{n}{quota}\PY{p}{,} \PY{n}{quota\PYZus{}grp}\PY{p}{)}\PY{p}{:}
\PY{l+s+sd}{\PYZdq{}\PYZdq{}\PYZdq{}}
\PY{l+s+sd}{ prefer contains the students ordered according to the preference of the school.}
\PY{l+s+sd}{ \PYZdq{}\PYZdq{}\PYZdq{}}
\PY{n+nb+bp}{self}\PY{o}{.}\PY{n}{prefer} \PY{o}{=} \PY{n}{prefer}
\PY{n+nb+bp}{self}\PY{o}{.}\PY{n}{quota} \PY{o}{=} \PY{n}{quota}
\PY{n+nb+bp}{self}\PY{o}{.}\PY{n}{quota\PYZus{}grp} \PY{o}{=} \PY{n}{quota\PYZus{}grp}
\PY{n+nb+bp}{self}\PY{o}{.}\PY{n}{n\PYZus{}grp} \PY{o}{=} \PY{p}{[}\PY{l+m+mi}{0}\PY{p}{]} \PY{o}{*} \PY{n+nb}{len}\PY{p}{(}\PY{n}{quota\PYZus{}grp}\PY{p}{)}
\end{Verbatim}
\end{tcolorbox}
\begin{tcolorbox}[breakable, size=fbox, boxrule=1pt, pad at break*=1mm,colback=cellbackground, colframe=cellborder]
\prompt{In}{incolor}{4}{\boxspacing}
\begin{Verbatim}[commandchars=\\\{\}]
\PY{k}{def} \PY{n+nf}{mate3}\PY{p}{(}\PY{n}{schools}\PY{p}{,} \PY{n}{students}\PY{p}{)}\PY{p}{:}
\PY{n}{N} \PY{o}{=} \PY{n+nb}{len}\PY{p}{(}\PY{n}{students}\PY{p}{)}
\PY{n}{m} \PY{o}{=} \PY{n+nb}{len}\PY{p}{(}\PY{n}{schools}\PY{p}{)}
\PY{n}{i} \PY{o}{=} \PY{o}{\PYZhy{}}\PY{l+m+mi}{1}
\PY{c+c1}{\PYZsh{} since the algorithm goes through the preferences of the schools in decreasing order of}
\PY{c+c1}{\PYZsh{} preference, we have a stable matching when all quotas are met.}
\PY{k}{while} \PY{k+kc}{True}\PY{p}{:} \PY{c+c1}{\PYZsh{} stop condition unmodified}
\PY{k}{for} \PY{n}{j} \PY{o+ow}{in} \PY{n+nb}{range}\PY{p}{(}\PY{n}{m}\PY{p}{)}\PY{p}{:}
\PY{k}{if} \PY{n}{schools}\PY{p}{[}\PY{n}{j}\PY{p}{]}\PY{o}{.}\PY{n}{n} \PY{o}{\PYZlt{}} \PY{n}{schools}\PY{p}{[}\PY{n}{j}\PY{p}{]}\PY{o}{.}\PY{n}{quota} \PY{o+ow}{and} \PY{n}{schools}\PY{p}{[}\PY{n}{i}\PY{p}{]}\PY{o}{.}\PY{n}{idx} \PY{o}{\PYZlt{}} \PY{n}{N}\PY{p}{:}
\PY{n}{i} \PY{o}{=} \PY{n}{j}
\PY{k}{break}
\PY{k}{else}\PY{p}{:}
\PY{k}{break}
\PY{k}{while} \PY{n}{schools}\PY{p}{[}\PY{n}{i}\PY{p}{]}\PY{o}{.}\PY{n}{n} \PY{o}{\PYZlt{}} \PY{n}{schools}\PY{p}{[}\PY{n}{i}\PY{p}{]}\PY{o}{.}\PY{n}{quota} \PY{o+ow}{and} \PY{n}{schools}\PY{p}{[}\PY{n}{i}\PY{p}{]}\PY{o}{.}\PY{n}{idx} \PY{o}{\PYZlt{}} \PY{n}{N}\PY{p}{:}
\PY{n}{stud} \PY{o}{=} \PY{n}{schools}\PY{p}{[}\PY{n}{i}\PY{p}{]}\PY{o}{.}\PY{n}{prefer}\PY{p}{[}\PY{n}{schools}\PY{p}{[}\PY{n}{i}\PY{p}{]}\PY{o}{.}\PY{n}{idx}\PY{p}{]}
\PY{k}{if} \PY{n}{schools}\PY{p}{[}\PY{n}{i}\PY{p}{]}\PY{o}{.}\PY{n}{n\PYZus{}grp}\PY{p}{[}\PY{n}{stud}\PY{o}{.}\PY{n}{grp}\PY{p}{]} \PY{o}{\PYZlt{}} \PY{n}{schools}\PY{p}{[}\PY{n}{i}\PY{p}{]}\PY{o}{.}\PY{n}{quota\PYZus{}grp}\PY{p}{[}\PY{n}{stud}\PY{o}{.}\PY{n}{grp}\PY{p}{]}\PY{p}{:}
\PY{k}{if} \PY{n}{stud}\PY{o}{.}\PY{n}{prefStud}\PY{p}{[}\PY{n}{i}\PY{p}{]} \PY{o}{\PYZlt{}} \PY{n}{stud}\PY{o}{.}\PY{n}{prefStud}\PY{p}{[}\PY{n}{stud}\PY{o}{.}\PY{n}{schlStud}\PY{p}{]}\PY{p}{:}
\PY{k}{if} \PY{n}{stud}\PY{o}{.}\PY{n}{schlStud} \PY{o}{\PYZlt{}} \PY{n}{m}\PY{p}{:}
\PY{n}{schools}\PY{p}{[}\PY{n}{stud}\PY{o}{.}\PY{n}{schlStud}\PY{p}{]}\PY{o}{.}\PY{n}{n} \PY{o}{\PYZhy{}}\PY{o}{=} \PY{l+m+mi}{1}
\PY{n}{schools}\PY{p}{[}\PY{n}{i}\PY{p}{]}\PY{o}{.}\PY{n}{n\PYZus{}grp}\PY{p}{[}\PY{n}{stud}\PY{o}{.}\PY{n}{grp}\PY{p}{]} \PY{o}{\PYZhy{}}\PY{o}{=} \PY{l+m+mi}{1}
\PY{n}{schools}\PY{p}{[}\PY{n}{i}\PY{p}{]}\PY{o}{.}\PY{n}{n} \PY{o}{+}\PY{o}{=} \PY{l+m+mi}{1}
\PY{n}{stud}\PY{o}{.}\PY{n}{schlStud} \PY{o}{=} \PY{n}{i}
\PY{n}{schools}\PY{p}{[}\PY{n}{i}\PY{p}{]}\PY{o}{.}\PY{n}{idx} \PY{o}{+}\PY{o}{=} \PY{l+m+mi}{1}
\PY{n}{res} \PY{o}{=} \PY{p}{[}\PY{n+nb}{list}\PY{p}{(}\PY{p}{)} \PY{k}{for} \PY{n}{k} \PY{o+ow}{in} \PY{n+nb}{range}\PY{p}{(}\PY{n}{m} \PY{o}{+} \PY{l+m+mi}{1}\PY{p}{)}\PY{p}{]}
\PY{k}{for} \PY{n}{k} \PY{o+ow}{in} \PY{n+nb}{range}\PY{p}{(}\PY{n}{N}\PY{p}{)}\PY{p}{:}
\PY{n}{res}\PY{p}{[}\PY{n}{students}\PY{p}{[}\PY{n}{k}\PY{p}{]}\PY{o}{.}\PY{n}{schlStud}\PY{p}{]}\PY{o}{.}\PY{n}{append}\PY{p}{(}\PY{n}{k}\PY{p}{)}
\PY{k}{return} \PY{n}{res}
\end{Verbatim}
\end{tcolorbox}
\hypertarget{task-4}{%
\subsection{Task 4:}\label{task-4}}
\begin{tcolorbox}[breakable, size=fbox, boxrule=1pt, pad at break*=1mm,colback=cellbackground, colframe=cellborder]
\prompt{In}{incolor}{5}{\boxspacing}
\begin{Verbatim}[commandchars=\\\{\}]
\PY{k+kn}{import} \PY{n+nn}{matplotlib}\PY{n+nn}{.}\PY{n+nn}{pyplot} \PY{k}{as} \PY{n+nn}{plt}
\end{Verbatim}
\end{tcolorbox}
\begin{tcolorbox}[breakable, size=fbox, boxrule=1pt, pad at break*=1mm,colback=cellbackground, colframe=cellborder]
\prompt{In}{incolor}{6}{\boxspacing}
\begin{Verbatim}[commandchars=\\\{\}]
\PY{k}{def} \PY{n+nf}{transf}\PY{p}{(}\PY{n}{match}\PY{p}{)}\PY{p}{:}
\PY{n}{res} \PY{o}{=} \PY{p}{[}\PY{p}{]}\PY{p}{,} \PY{p}{[}\PY{p}{]}
\PY{k}{for} \PY{n}{i} \PY{o+ow}{in} \PY{n+nb}{range}\PY{p}{(}\PY{n+nb}{len}\PY{p}{(}\PY{n}{match}\PY{p}{)} \PY{o}{\PYZhy{}} \PY{l+m+mi}{1}\PY{p}{)}\PY{p}{:}
\PY{k}{for} \PY{n}{stud} \PY{o+ow}{in} \PY{n}{match}\PY{p}{[}\PY{n}{i}\PY{p}{]}\PY{p}{:}
\PY{n}{res}\PY{p}{[}\PY{l+m+mi}{0}\PY{p}{]}\PY{o}{.}\PY{n}{append}\PY{p}{(}\PY{n}{stud}\PY{p}{)}
\PY{n}{res}\PY{p}{[}\PY{l+m+mi}{1}\PY{p}{]}\PY{o}{.}\PY{n}{append}\PY{p}{(}\PY{n}{i}\PY{p}{)}
\PY{k}{return} \PY{n}{res}
\PY{k}{def} \PY{n+nf}{represent}\PY{p}{(}\PY{n}{match}\PY{p}{,} \PY{n}{col}\PY{p}{)}\PY{p}{:}
\PY{n}{res} \PY{o}{=} \PY{n}{transf}\PY{p}{(}\PY{n}{match}\PY{p}{)}
\PY{n}{plt}\PY{o}{.}\PY{n}{scatter}\PY{p}{(}\PY{n}{res}\PY{p}{[}\PY{l+m+mi}{0}\PY{p}{]}\PY{p}{,} \PY{n}{res}\PY{p}{[}\PY{l+m+mi}{1}\PY{p}{]}\PY{p}{,} \PY{n}{c} \PY{o}{=} \PY{n}{col}\PY{p}{)}
\PY{n}{plt}\PY{o}{.}\PY{n}{title}\PY{p}{(}\PY{l+s+s2}{\PYZdq{}}\PY{l+s+s2}{Matching of students to schools}\PY{l+s+s2}{\PYZdq{}}\PY{p}{)}
\PY{n}{plt}\PY{o}{.}\PY{n}{xlabel}\PY{p}{(}\PY{l+s+s2}{\PYZdq{}}\PY{l+s+s2}{Student}\PY{l+s+s2}{\PYZdq{}}\PY{p}{)}
\PY{n}{plt}\PY{o}{.}\PY{n}{ylabel}\PY{p}{(}\PY{l+s+s2}{\PYZdq{}}\PY{l+s+s2}{School}\PY{l+s+s2}{\PYZdq{}}\PY{p}{)}
\PY{n}{plt}\PY{o}{.}\PY{n}{show}\PY{p}{(}\PY{p}{)}
\PY{n}{plt}\PY{o}{.}\PY{n}{close}\PY{p}{(}\PY{p}{)}
\end{Verbatim}
\end{tcolorbox}
\hypertarget{test-instance-1}{%
\subsubsection{Test Instance 1:}\label{test-instance-1}}
\hypertarget{test-task-2}{%
\paragraph{Test Task 2:}\label{test-task-2}}
\begin{tcolorbox}[breakable, size=fbox, boxrule=1pt, pad at break*=1mm,colback=cellbackground, colframe=cellborder]
\prompt{In}{incolor}{7}{\boxspacing}
\begin{Verbatim}[commandchars=\\\{\}]
\PY{n}{stud1} \PY{o}{=} \PY{n}{student2}\PY{p}{(}\PY{p}{[}\PY{l+m+mi}{1}\PY{p}{,} \PY{l+m+mi}{2}\PY{p}{,} \PY{l+m+mi}{3}\PY{p}{]}\PY{p}{)}
\PY{n}{stud2} \PY{o}{=} \PY{n}{student2}\PY{p}{(}\PY{p}{[}\PY{l+m+mi}{2}\PY{p}{,} \PY{l+m+mi}{1}\PY{p}{,} \PY{l+m+mi}{3}\PY{p}{]}\PY{p}{)}
\PY{n}{stud3} \PY{o}{=} \PY{n}{student2}\PY{p}{(}\PY{p}{[}\PY{l+m+mi}{1}\PY{p}{,} \PY{l+m+mi}{3}\PY{p}{,} \PY{l+m+mi}{2}\PY{p}{]}\PY{p}{)}
\PY{n}{stud4} \PY{o}{=} \PY{n}{student2}\PY{p}{(}\PY{p}{[}\PY{l+m+mi}{3}\PY{p}{,} \PY{l+m+mi}{1}\PY{p}{,} \PY{l+m+mi}{2}\PY{p}{]}\PY{p}{)}
\end{Verbatim}
\end{tcolorbox}
\begin{tcolorbox}[breakable, size=fbox, boxrule=1pt, pad at break*=1mm,colback=cellbackground, colframe=cellborder]
\prompt{In}{incolor}{8}{\boxspacing}
\begin{Verbatim}[commandchars=\\\{\}]
\PY{n}{s1} \PY{o}{=} \PY{n}{school2}\PY{p}{(}\PY{p}{[}\PY{n}{stud4}\PY{p}{,} \PY{n}{stud3}\PY{p}{,} \PY{n}{stud2}\PY{p}{,} \PY{n}{stud1}\PY{p}{]}\PY{p}{,} \PY{l+m+mi}{2}\PY{p}{)}
\PY{n}{s2} \PY{o}{=} \PY{n}{school2}\PY{p}{(}\PY{p}{[}\PY{n}{stud4}\PY{p}{,} \PY{n}{stud3}\PY{p}{,} \PY{n}{stud2}\PY{p}{,} \PY{n}{stud1}\PY{p}{]}\PY{p}{,} \PY{l+m+mi}{2}\PY{p}{)}
\end{Verbatim}
\end{tcolorbox}
\begin{tcolorbox}[breakable, size=fbox, boxrule=1pt, pad at break*=1mm,colback=cellbackground, colframe=cellborder]
\prompt{In}{incolor}{9}{\boxspacing}
\begin{Verbatim}[commandchars=\\\{\}]
\PY{n}{represent}\PY{p}{(}\PY{n}{mate2}\PY{p}{(}\PY{p}{[}\PY{n}{s1}\PY{p}{,} \PY{n}{s2}\PY{p}{]}\PY{p}{,} \PY{p}{[}\PY{n}{stud1}\PY{p}{,} \PY{n}{stud2}\PY{p}{,} \PY{n}{stud3}\PY{p}{,} \PY{n}{stud4}\PY{p}{]}\PY{p}{)}\PY{p}{,} \PY{p}{[}\PY{l+s+s1}{\PYZsq{}}\PY{l+s+s1}{r}\PY{l+s+s1}{\PYZsq{}}\PY{p}{,} \PY{l+s+s1}{\PYZsq{}}\PY{l+s+s1}{b}\PY{l+s+s1}{\PYZsq{}}\PY{p}{,} \PY{l+s+s1}{\PYZsq{}}\PY{l+s+s1}{g}\PY{l+s+s1}{\PYZsq{}}\PY{p}{,} \PY{l+s+s1}{\PYZsq{}}\PY{l+s+s1}{y}\PY{l+s+s1}{\PYZsq{}}\PY{p}{]}\PY{p}{)}
\end{Verbatim}
\end{tcolorbox}
\begin{center}
\adjustimage{max size={0.9\linewidth}{0.9\paperheight}}{output_23_0.png}
\end{center}
{ \hspace*{\fill} \\}
\hypertarget{test-task-3}{%
\paragraph{Test Task 3:}\label{test-task-3}}
\begin{tcolorbox}[breakable, size=fbox, boxrule=1pt, pad at break*=1mm,colback=cellbackground, colframe=cellborder]
\prompt{In}{incolor}{10}{\boxspacing}
\begin{Verbatim}[commandchars=\\\{\}]
\PY{n}{stud1} \PY{o}{=} \PY{n}{student3}\PY{p}{(}\PY{p}{[}\PY{l+m+mi}{1}\PY{p}{,} \PY{l+m+mi}{2}\PY{p}{,} \PY{l+m+mi}{3}\PY{p}{]}\PY{p}{,} \PY{l+m+mi}{0}\PY{p}{)}
\PY{n}{stud2} \PY{o}{=} \PY{n}{student3}\PY{p}{(}\PY{p}{[}\PY{l+m+mi}{2}\PY{p}{,} \PY{l+m+mi}{1}\PY{p}{,} \PY{l+m+mi}{3}\PY{p}{]}\PY{p}{,} \PY{l+m+mi}{0}\PY{p}{)}
\PY{n}{stud3} \PY{o}{=} \PY{n}{student3}\PY{p}{(}\PY{p}{[}\PY{l+m+mi}{1}\PY{p}{,} \PY{l+m+mi}{3}\PY{p}{,} \PY{l+m+mi}{2}\PY{p}{]}\PY{p}{,} \PY{l+m+mi}{0}\PY{p}{)}
\PY{n}{stud4} \PY{o}{=} \PY{n}{student3}\PY{p}{(}\PY{p}{[}\PY{l+m+mi}{3}\PY{p}{,} \PY{l+m+mi}{1}\PY{p}{,} \PY{l+m+mi}{2}\PY{p}{]}\PY{p}{,} \PY{l+m+mi}{1}\PY{p}{)}
\end{Verbatim}
\end{tcolorbox}
\begin{tcolorbox}[breakable, size=fbox, boxrule=1pt, pad at break*=1mm,colback=cellbackground, colframe=cellborder]
\prompt{In}{incolor}{11}{\boxspacing}
\begin{Verbatim}[commandchars=\\\{\}]
\PY{n}{s1} \PY{o}{=} \PY{n}{school3}\PY{p}{(}\PY{p}{[}\PY{n}{stud4}\PY{p}{,} \PY{n}{stud3}\PY{p}{,} \PY{n}{stud2}\PY{p}{,} \PY{n}{stud1}\PY{p}{]}\PY{p}{,} \PY{l+m+mi}{2}\PY{p}{,} \PY{p}{[}\PY{l+m+mi}{2}\PY{p}{,} \PY{l+m+mi}{2}\PY{p}{]}\PY{p}{)}
\PY{n}{s2} \PY{o}{=} \PY{n}{school3}\PY{p}{(}\PY{p}{[}\PY{n}{stud4}\PY{p}{,} \PY{n}{stud3}\PY{p}{,} \PY{n}{stud2}\PY{p}{,} \PY{n}{stud1}\PY{p}{]}\PY{p}{,} \PY{l+m+mi}{2}\PY{p}{,} \PY{p}{[}\PY{l+m+mi}{2}\PY{p}{,} \PY{l+m+mi}{2}\PY{p}{]}\PY{p}{)}
\end{Verbatim}
\end{tcolorbox}
\begin{tcolorbox}[breakable, size=fbox, boxrule=1pt, pad at break*=1mm,colback=cellbackground, colframe=cellborder]
\prompt{In}{incolor}{12}{\boxspacing}
\begin{Verbatim}[commandchars=\\\{\}]
\PY{n}{represent}\PY{p}{(}\PY{n}{mate3}\PY{p}{(}\PY{p}{[}\PY{n}{s1}\PY{p}{,} \PY{n}{s2}\PY{p}{]}\PY{p}{,} \PY{p}{[}\PY{n}{stud1}\PY{p}{,} \PY{n}{stud2}\PY{p}{,} \PY{n}{stud3}\PY{p}{,} \PY{n}{stud4}\PY{p}{]}\PY{p}{)}\PY{p}{,} \PY{p}{[}\PY{l+s+s1}{\PYZsq{}}\PY{l+s+s1}{r}\PY{l+s+s1}{\PYZsq{}}\PY{p}{,} \PY{l+s+s1}{\PYZsq{}}\PY{l+s+s1}{b}\PY{l+s+s1}{\PYZsq{}}\PY{p}{,} \PY{l+s+s1}{\PYZsq{}}\PY{l+s+s1}{g}\PY{l+s+s1}{\PYZsq{}}\PY{p}{,} \PY{l+s+s1}{\PYZsq{}}\PY{l+s+s1}{y}\PY{l+s+s1}{\PYZsq{}}\PY{p}{]}\PY{p}{)}
\end{Verbatim}
\end{tcolorbox}
\begin{center}
\adjustimage{max size={0.9\linewidth}{0.9\paperheight}}{output_27_0.png}
\end{center}
{ \hspace*{\fill} \\}
\hypertarget{test-instance-2}{%
\subsubsection{Test instance 2:}\label{test-instance-2}}
\begin{tcolorbox}[breakable, size=fbox, boxrule=1pt, pad at break*=1mm,colback=cellbackground, colframe=cellborder]
\prompt{In}{incolor}{13}{\boxspacing}
\begin{Verbatim}[commandchars=\\\{\}]
\PY{k+kn}{from} \PY{n+nn}{random} \PY{k+kn}{import} \PY{n}{gauss}\PY{p}{,} \PY{n}{random}
\PY{k+kn}{from} \PY{n+nn}{math} \PY{k+kn}{import} \PY{n}{sqrt}
\end{Verbatim}
\end{tcolorbox}
\begin{tcolorbox}[breakable, size=fbox, boxrule=1pt, pad at break*=1mm,colback=cellbackground, colframe=cellborder]
\prompt{In}{incolor}{14}{\boxspacing}
\begin{Verbatim}[commandchars=\\\{\}]
\PY{k}{def} \PY{n+nf}{app}\PY{p}{(}\PY{n}{studs}\PY{p}{,} \PY{n}{grp}\PY{p}{)}\PY{p}{:}
\PY{n}{s} \PY{o}{=} \PY{n}{random}\PY{p}{(}\PY{p}{)}
\PY{k}{if} \PY{n}{s} \PY{o}{\PYZgt{}}\PY{o}{=} \PY{l+m+mf}{0.5} \PY{p}{:} \PY{n}{choice} \PY{o}{=} \PY{p}{[}\PY{l+m+mi}{1}\PY{p}{,} \PY{l+m+mi}{2}\PY{p}{]}
\PY{k}{else}\PY{p}{:} \PY{n}{choice} \PY{o}{=} \PY{p}{[}\PY{l+m+mi}{2}\PY{p}{,} \PY{l+m+mi}{1}\PY{p}{]}
\PY{n}{studs}\PY{o}{.}\PY{n}{append}\PY{p}{(}\PY{n}{student3}\PY{p}{(}\PY{n}{choice}\PY{p}{,} \PY{n}{grp}\PY{p}{)}\PY{p}{)}
\PY{k}{def} \PY{n+nf}{rand\PYZus{}instance2}\PY{p}{(}\PY{n}{n}\PY{p}{)}\PY{p}{:}
\PY{l+s+sd}{\PYZdq{}\PYZdq{}\PYZdq{}}
\PY{l+s+sd}{ mate2 works with arguments of the classes school3 and student3.}
\PY{l+s+sd}{ \PYZdq{}\PYZdq{}\PYZdq{}}
\PY{n}{studs} \PY{o}{=} \PY{p}{[}\PY{p}{]}
\PY{n}{ranQty} \PY{o}{=} \PY{p}{[}\PY{p}{]}
\PY{n}{m} \PY{o}{=} \PY{n+nb}{int}\PY{p}{(}\PY{l+m+mi}{9} \PY{o}{*} \PY{n}{n} \PY{o}{/} \PY{l+m+mi}{10}\PY{p}{)}
\PY{k}{for} \PY{n}{k} \PY{o+ow}{in} \PY{n+nb}{range}\PY{p}{(}\PY{n}{m} \PY{o}{+} \PY{l+m+mi}{1}\PY{p}{)}\PY{p}{:}
\PY{n}{app}\PY{p}{(}\PY{n}{studs}\PY{p}{,} \PY{l+m+mi}{0}\PY{p}{)}
\PY{n}{ranQty}\PY{o}{.}\PY{n}{append}\PY{p}{(}\PY{p}{(}\PY{n}{k}\PY{p}{,} \PY{n}{gauss}\PY{p}{(}\PY{l+m+mi}{0}\PY{p}{,} \PY{l+m+mi}{1}\PY{p}{)}\PY{p}{)}\PY{p}{)}
\PY{k}{for} \PY{n}{k} \PY{o+ow}{in} \PY{n+nb}{range}\PY{p}{(}\PY{n}{m} \PY{o}{+} \PY{l+m+mi}{1}\PY{p}{,} \PY{n}{n}\PY{p}{)}\PY{p}{:}
\PY{n}{app}\PY{p}{(}\PY{n}{studs}\PY{p}{,} \PY{l+m+mi}{1}\PY{p}{)}
\PY{n}{ranQty}\PY{o}{.}\PY{n}{append}\PY{p}{(}\PY{p}{(}\PY{n}{k}\PY{p}{,} \PY{n}{gauss}\PY{p}{(}\PY{l+m+mi}{0}\PY{p}{,} \PY{l+m+mi}{1}\PY{p}{)}\PY{p}{)}\PY{p}{)}
\PY{n}{ranQty1} \PY{o}{=} \PY{p}{[}\PY{p}{(}\PY{n}{k}\PY{p}{,} \PY{n}{ranQty}\PY{p}{[}\PY{n}{k}\PY{p}{]}\PY{p}{[}\PY{l+m+mi}{1}\PY{p}{]} \PY{o}{+} \PY{n}{gauss}\PY{p}{(}\PY{l+m+mi}{0}\PY{p}{,} \PY{l+m+mi}{1}\PY{p}{)}\PY{p}{)} \PY{k}{for} \PY{n}{k} \PY{o+ow}{in} \PY{n+nb}{range}\PY{p}{(}\PY{n}{n}\PY{p}{)}\PY{p}{]}
\PY{n}{ranQty2} \PY{o}{=} \PY{p}{[}\PY{p}{(}\PY{n}{k}\PY{p}{,} \PY{n}{ranQty}\PY{p}{[}\PY{n}{k}\PY{p}{]}\PY{p}{[}\PY{l+m+mi}{1}\PY{p}{]} \PY{o}{+} \PY{n}{gauss}\PY{p}{(}\PY{l+m+mi}{0}\PY{p}{,} \PY{l+m+mi}{1}\PY{p}{)}\PY{p}{)} \PY{k}{for} \PY{n}{k} \PY{o+ow}{in} \PY{n+nb}{range}\PY{p}{(}\PY{n}{n}\PY{p}{)}\PY{p}{]}
\PY{n}{ranQty1}\PY{o}{.}\PY{n}{sort}\PY{p}{(}\PY{n}{key} \PY{o}{=} \PY{k}{lambda} \PY{n}{x} \PY{p}{:} \PY{o}{\PYZhy{}}\PY{n}{x}\PY{p}{[}\PY{l+m+mi}{1}\PY{p}{]}\PY{p}{)} \PY{c+c1}{\PYZsh{} decreasing order of observed quality}
\PY{n}{ranQty2}\PY{o}{.}\PY{n}{sort}\PY{p}{(}\PY{n}{key} \PY{o}{=} \PY{k}{lambda} \PY{n}{x} \PY{p}{:} \PY{o}{\PYZhy{}}\PY{n}{x}\PY{p}{[}\PY{l+m+mi}{1}\PY{p}{]}\PY{p}{)}
\PY{n}{s1} \PY{o}{=} \PY{n}{school3}\PY{p}{(}\PY{p}{[}\PY{n}{studs}\PY{p}{[}\PY{n}{elt}\PY{p}{[}\PY{l+m+mi}{0}\PY{p}{]}\PY{p}{]} \PY{k}{for} \PY{n}{elt} \PY{o+ow}{in} \PY{n}{ranQty1}\PY{p}{]}\PY{p}{,} \PY{n}{n} \PY{o}{/}\PY{o}{/} \PY{l+m+mi}{4}\PY{p}{,} \PY{p}{[}\PY{n+nb}{int}\PY{p}{(}\PY{l+m+mf}{0.9} \PY{o}{*} \PY{n}{n} \PY{o}{/} \PY{l+m+mi}{4}\PY{p}{)}\PY{p}{,} \PY{n+nb}{int}\PY{p}{(}\PY{l+m+mf}{0.9} \PY{o}{*} \PY{n}{n} \PY{o}{/} \PY{l+m+mi}{4}\PY{p}{)}\PY{p}{]}\PY{p}{)}
\PY{n}{s2} \PY{o}{=} \PY{n}{school3}\PY{p}{(}\PY{p}{[}\PY{n}{studs}\PY{p}{[}\PY{n}{elt}\PY{p}{[}\PY{l+m+mi}{0}\PY{p}{]}\PY{p}{]} \PY{k}{for} \PY{n}{elt} \PY{o+ow}{in} \PY{n}{ranQty2}\PY{p}{]}\PY{p}{,} \PY{n}{n} \PY{o}{/}\PY{o}{/} \PY{l+m+mi}{4}\PY{p}{,} \PY{p}{[}\PY{n+nb}{int}\PY{p}{(}\PY{l+m+mf}{0.9} \PY{o}{*} \PY{n}{n} \PY{o}{/} \PY{l+m+mi}{4}\PY{p}{)}\PY{p}{,} \PY{n+nb}{int}\PY{p}{(}\PY{l+m+mf}{0.9} \PY{o}{*} \PY{n}{n} \PY{o}{/} \PY{l+m+mi}{4}\PY{p}{)}\PY{p}{]}\PY{p}{)}
\PY{k}{return} \PY{p}{[}\PY{n}{s1}\PY{p}{,} \PY{n}{s2}\PY{p}{]}\PY{p}{,} \PY{n}{studs}
\PY{k}{def} \PY{n+nf}{analyse2\PYZus{}2}\PY{p}{(}\PY{n}{n}\PY{p}{,} \PY{n}{rand\PYZus{}inst}\PY{p}{)}\PY{p}{:}
\PY{n}{res} \PY{o}{=} \PY{n}{mate2}\PY{p}{(}\PY{n}{rand\PYZus{}inst}\PY{p}{[}\PY{l+m+mi}{0}\PY{p}{]}\PY{p}{,} \PY{n}{rand\PYZus{}inst}\PY{p}{[}\PY{l+m+mi}{1}\PY{p}{]}\PY{p}{)}
\PY{n}{cptA}\PY{p}{,} \PY{n}{cptB} \PY{o}{=} \PY{l+m+mi}{0}\PY{p}{,} \PY{l+m+mi}{0}
\PY{k}{for} \PY{n}{s} \PY{o+ow}{in} \PY{p}{[}\PY{l+m+mi}{0}\PY{p}{,} \PY{l+m+mi}{1}\PY{p}{]}\PY{p}{:}
\PY{k}{for} \PY{n}{elt} \PY{o+ow}{in} \PY{n}{res}\PY{p}{[}\PY{n}{s}\PY{p}{]}\PY{p}{:}
\PY{k}{if} \PY{n}{rand\PYZus{}inst}\PY{p}{[}\PY{l+m+mi}{1}\PY{p}{]}\PY{p}{[}\PY{n}{elt}\PY{p}{]}\PY{o}{.}\PY{n}{prefStud}\PY{p}{[}\PY{n}{s}\PY{p}{]} \PY{o}{==} \PY{l+m+mi}{1}\PY{p}{:}
\PY{k}{if} \PY{n}{rand\PYZus{}inst}\PY{p}{[}\PY{l+m+mi}{1}\PY{p}{]}\PY{p}{[}\PY{n}{elt}\PY{p}{]}\PY{o}{.}\PY{n}{grp} \PY{o}{==} \PY{l+m+mi}{0}\PY{p}{:} \PY{n}{cptA} \PY{o}{+}\PY{o}{=} \PY{l+m+mi}{1}
\PY{k}{else}\PY{p}{:} \PY{n}{cptB} \PY{o}{+}\PY{o}{=} \PY{l+m+mi}{1}
\PY{k}{return} \PY{n}{n}\PY{p}{,} \PY{n}{cptA}\PY{p}{,} \PY{n}{cptB}
\PY{k}{def} \PY{n+nf}{analyse2\PYZus{}3}\PY{p}{(}\PY{n}{n}\PY{p}{,} \PY{n}{rand\PYZus{}inst}\PY{p}{)}\PY{p}{:}
\PY{n}{res} \PY{o}{=} \PY{n}{mate3}\PY{p}{(}\PY{n}{rand\PYZus{}inst}\PY{p}{[}\PY{l+m+mi}{0}\PY{p}{]}\PY{p}{,} \PY{n}{rand\PYZus{}inst}\PY{p}{[}\PY{l+m+mi}{1}\PY{p}{]}\PY{p}{)}
\PY{n}{cptA}\PY{p}{,} \PY{n}{cptB} \PY{o}{=} \PY{l+m+mi}{0}\PY{p}{,} \PY{l+m+mi}{0}
\PY{k}{for} \PY{n}{s} \PY{o+ow}{in} \PY{p}{[}\PY{l+m+mi}{0}\PY{p}{,} \PY{l+m+mi}{1}\PY{p}{]}\PY{p}{:}
\PY{k}{for} \PY{n}{elt} \PY{o+ow}{in} \PY{n}{res}\PY{p}{[}\PY{n}{s}\PY{p}{]}\PY{p}{:}
\PY{k}{if} \PY{n}{rand\PYZus{}inst}\PY{p}{[}\PY{l+m+mi}{1}\PY{p}{]}\PY{p}{[}\PY{n}{elt}\PY{p}{]}\PY{o}{.}\PY{n}{prefStud}\PY{p}{[}\PY{n}{s}\PY{p}{]} \PY{o}{==} \PY{l+m+mi}{1}\PY{p}{:}
\PY{k}{if} \PY{n}{rand\PYZus{}inst}\PY{p}{[}\PY{l+m+mi}{1}\PY{p}{]}\PY{p}{[}\PY{n}{elt}\PY{p}{]}\PY{o}{.}\PY{n}{grp} \PY{o}{==} \PY{l+m+mi}{0}\PY{p}{:} \PY{n}{cptA} \PY{o}{+}\PY{o}{=} \PY{l+m+mi}{1}
\PY{k}{else}\PY{p}{:} \PY{n}{cptB} \PY{o}{+}\PY{o}{=} \PY{l+m+mi}{1}
\PY{k}{return} \PY{n}{n}\PY{p}{,} \PY{n}{cptA}\PY{p}{,} \PY{n}{cptB}
\end{Verbatim}
\end{tcolorbox}
\hypertarget{test-task-2}{%
\paragraph{Test Task 2:}\label{test-task-2}}
\begin{tcolorbox}[breakable, size=fbox, boxrule=1pt, pad at break*=1mm,colback=cellbackground, colframe=cellborder]
\prompt{In}{incolor}{15}{\boxspacing}
\begin{Verbatim}[commandchars=\\\{\}]
\PY{k}{def} \PY{n+nf}{printAnalyse2\PYZus{}2}\PY{p}{(}\PY{n}{n}\PY{p}{)}\PY{p}{:}
\PY{n}{m} \PY{o}{=} \PY{n+nb}{int}\PY{p}{(}\PY{l+m+mi}{9} \PY{o}{*} \PY{n}{n} \PY{o}{/} \PY{l+m+mi}{10}\PY{p}{)}
\PY{n}{N} \PY{o}{=} \PY{l+m+mi}{200}
\PY{n}{cpta}\PY{p}{,} \PY{n}{cptb} \PY{o}{=} \PY{l+m+mi}{0}\PY{p}{,} \PY{l+m+mi}{0}
\PY{k}{for} \PY{n}{\PYZus{}} \PY{o+ow}{in} \PY{n+nb}{range}\PY{p}{(}\PY{n}{N}\PY{p}{)}\PY{p}{:}
\PY{n}{rand\PYZus{}inst} \PY{o}{=} \PY{n}{rand\PYZus{}instance2}\PY{p}{(}\PY{n}{n}\PY{p}{)}
\PY{n}{n}\PY{p}{,} \PY{n}{cptA}\PY{p}{,} \PY{n}{cptB} \PY{o}{=} \PY{n}{analyse2\PYZus{}2}\PY{p}{(}\PY{n}{n}\PY{p}{,} \PY{n}{rand\PYZus{}inst}\PY{p}{)}
\PY{n}{cpta} \PY{o}{+}\PY{o}{=} \PY{n}{cptA}
\PY{n}{cptb} \PY{o}{+}\PY{o}{=} \PY{n}{cptB}
\PY{n+nb}{print}\PY{p}{(}\PY{n}{n}\PY{p}{,} \PY{l+s+s2}{\PYZdq{}}\PY{l+s+s2}{:}\PY{l+s+s2}{\PYZdq{}}\PY{p}{)}
\PY{n+nb}{print}\PY{p}{(}\PY{l+s+s2}{\PYZdq{}}\PY{l+s+s2}{Average of cptA:}\PY{l+s+s2}{\PYZdq{}}\PY{p}{,} \PY{n}{cpta} \PY{o}{/} \PY{n}{N}\PY{p}{)}
\PY{n+nb}{print}\PY{p}{(}\PY{l+s+s2}{\PYZdq{}}\PY{l+s+s2}{Percentage of A}\PY{l+s+s2}{\PYZsq{}}\PY{l+s+s2}{s to get their first choice:}\PY{l+s+s2}{\PYZdq{}}\PY{p}{,} \PY{n+nb}{int}\PY{p}{(}\PY{n}{cpta} \PY{o}{/} \PY{n}{m} \PY{o}{/} \PY{n}{N} \PY{o}{*} \PY{l+m+mi}{10000}\PY{p}{)} \PY{o}{/} \PY{l+m+mi}{100}\PY{p}{,} \PY{l+s+s2}{\PYZdq{}}\PY{l+s+s2}{\PYZpc{}}\PY{l+s+s2}{\PYZdq{}}\PY{p}{)}
\PY{n+nb}{print}\PY{p}{(}\PY{l+s+s2}{\PYZdq{}}\PY{l+s+s2}{Average of cptB : }\PY{l+s+s2}{\PYZdq{}}\PY{p}{,} \PY{n}{cptb} \PY{o}{/} \PY{n}{N}\PY{p}{)}
\PY{n+nb}{print}\PY{p}{(}\PY{l+s+s2}{\PYZdq{}}\PY{l+s+s2}{Percentage of B}\PY{l+s+s2}{\PYZsq{}}\PY{l+s+s2}{s to get their first choice:}\PY{l+s+s2}{\PYZdq{}}\PY{p}{,} \PY{n+nb}{int}\PY{p}{(}\PY{n}{cptb} \PY{o}{/} \PY{p}{(}\PY{n}{n} \PY{o}{\PYZhy{}} \PY{n}{m}\PY{p}{)} \PY{o}{/} \PY{n}{N} \PY{o}{*} \PY{l+m+mi}{10000}\PY{p}{)} \PY{o}{/} \PY{l+m+mi}{100}\PY{p}{,} \PY{l+s+s2}{\PYZdq{}}\PY{l+s+s2}{\PYZpc{}}\PY{l+s+s2}{\PYZdq{}}\PY{p}{)}
\PY{n+nb}{print}\PY{p}{(}\PY{l+s+s2}{\PYZdq{}}\PY{l+s+s2}{\PYZus{}\PYZus{}\PYZus{}\PYZus{}\PYZus{}\PYZus{}\PYZus{}}\PY{l+s+s2}{\PYZdq{}}\PY{p}{)}
\end{Verbatim}
\end{tcolorbox}
\begin{tcolorbox}[breakable, size=fbox, boxrule=1pt, pad at break*=1mm,colback=cellbackground, colframe=cellborder]
\prompt{In}{incolor}{16}{\boxspacing}
\begin{Verbatim}[commandchars=\\\{\}]
\PY{n}{printAnalyse2\PYZus{}2}\PY{p}{(}\PY{l+m+mi}{1000}\PY{p}{)}
\PY{n}{printAnalyse2\PYZus{}2}\PY{p}{(}\PY{l+m+mi}{2000}\PY{p}{)}
\PY{n}{printAnalyse2\PYZus{}2}\PY{p}{(}\PY{l+m+mi}{3000}\PY{p}{)}
\end{Verbatim}
\end{tcolorbox}
\begin{Verbatim}[commandchars=\\\{\}]
1000 :
Average of cptA: 675.835
Percentage of A's to get their first choice: 75.09 \%
Average of cptB : 74.515
Percentage of B's to get their first choice: 74.51 \%
\_\_\_\_\_\_\_
2000 :
Average of cptA: 1350.55
Percentage of A's to get their first choice: 75.03 \%
Average of cptB : 148.81
Percentage of B's to get their first choice: 74.4 \%
\_\_\_\_\_\_\_
3000 :
Average of cptA: 2026.395
Percentage of A's to get their first choice: 75.05 \%
Average of cptB : 223.935
Percentage of B's to get their first choice: 74.64 \%
\_\_\_\_\_\_\_
\end{Verbatim}
The fraction of people getting their first choices seems independent
from the number n or their class. Nearly three fourths of students get
their first choice.
\hypertarget{test-task-3}{%
\paragraph{Test Task 3:}\label{test-task-3}}
\begin{tcolorbox}[breakable, size=fbox, boxrule=1pt, pad at break*=1mm,colback=cellbackground, colframe=cellborder]
\prompt{In}{incolor}{17}{\boxspacing}
\begin{Verbatim}[commandchars=\\\{\}]
\PY{k}{def} \PY{n+nf}{printAnalyse2\PYZus{}3}\PY{p}{(}\PY{n}{n}\PY{p}{)}\PY{p}{:}
\PY{n}{m} \PY{o}{=} \PY{n+nb}{int}\PY{p}{(}\PY{l+m+mi}{9} \PY{o}{*} \PY{n}{n} \PY{o}{/} \PY{l+m+mi}{10}\PY{p}{)}
\PY{n}{N} \PY{o}{=} \PY{l+m+mi}{200}
\PY{n}{cpta}\PY{p}{,} \PY{n}{cptb} \PY{o}{=} \PY{l+m+mi}{0}\PY{p}{,} \PY{l+m+mi}{0}
\PY{k}{for} \PY{n}{\PYZus{}} \PY{o+ow}{in} \PY{n+nb}{range}\PY{p}{(}\PY{n}{N}\PY{p}{)}\PY{p}{:}
\PY{n}{rand\PYZus{}inst} \PY{o}{=} \PY{n}{rand\PYZus{}instance2}\PY{p}{(}\PY{n}{n}\PY{p}{)}
\PY{n}{n}\PY{p}{,} \PY{n}{cptA}\PY{p}{,} \PY{n}{cptB} \PY{o}{=} \PY{n}{analyse2\PYZus{}3}\PY{p}{(}\PY{n}{n}\PY{p}{,} \PY{n}{rand\PYZus{}inst}\PY{p}{)}
\PY{n}{cpta} \PY{o}{+}\PY{o}{=} \PY{n}{cptA}
\PY{n}{cptb} \PY{o}{+}\PY{o}{=} \PY{n}{cptB}
\PY{n+nb}{print}\PY{p}{(}\PY{n}{n}\PY{p}{,} \PY{l+s+s2}{\PYZdq{}}\PY{l+s+s2}{:}\PY{l+s+s2}{\PYZdq{}}\PY{p}{)}
\PY{n+nb}{print}\PY{p}{(}\PY{l+s+s2}{\PYZdq{}}\PY{l+s+s2}{Average of cptA:}\PY{l+s+s2}{\PYZdq{}}\PY{p}{,} \PY{n}{cpta} \PY{o}{/} \PY{n}{N}\PY{p}{)}
\PY{n+nb}{print}\PY{p}{(}\PY{l+s+s2}{\PYZdq{}}\PY{l+s+s2}{Percentage of A}\PY{l+s+s2}{\PYZsq{}}\PY{l+s+s2}{s to get their first choice:}\PY{l+s+s2}{\PYZdq{}}\PY{p}{,} \PY{n+nb}{int}\PY{p}{(}\PY{n}{cpta} \PY{o}{/} \PY{n}{m} \PY{o}{/} \PY{n}{N} \PY{o}{*} \PY{l+m+mi}{10000}\PY{p}{)} \PY{o}{/} \PY{l+m+mi}{100}\PY{p}{,} \PY{l+s+s2}{\PYZdq{}}\PY{l+s+s2}{\PYZpc{}}\PY{l+s+s2}{\PYZdq{}}\PY{p}{)}
\PY{n+nb}{print}\PY{p}{(}\PY{l+s+s2}{\PYZdq{}}\PY{l+s+s2}{Average of cptB : }\PY{l+s+s2}{\PYZdq{}}\PY{p}{,} \PY{n}{cptb} \PY{o}{/} \PY{n}{N}\PY{p}{)}
\PY{n+nb}{print}\PY{p}{(}\PY{l+s+s2}{\PYZdq{}}\PY{l+s+s2}{Percentage of B}\PY{l+s+s2}{\PYZsq{}}\PY{l+s+s2}{s to get their first choice:}\PY{l+s+s2}{\PYZdq{}}\PY{p}{,} \PY{n+nb}{int}\PY{p}{(}\PY{n}{cptb} \PY{o}{/} \PY{p}{(}\PY{n}{n} \PY{o}{\PYZhy{}} \PY{n}{m}\PY{p}{)} \PY{o}{/} \PY{n}{N} \PY{o}{*} \PY{l+m+mi}{10000}\PY{p}{)} \PY{o}{/} \PY{l+m+mi}{100}\PY{p}{,} \PY{l+s+s2}{\PYZdq{}}\PY{l+s+s2}{\PYZpc{}}\PY{l+s+s2}{\PYZdq{}}\PY{p}{)}
\PY{n+nb}{print}\PY{p}{(}\PY{l+s+s2}{\PYZdq{}}\PY{l+s+s2}{\PYZus{}\PYZus{}\PYZus{}\PYZus{}\PYZus{}\PYZus{}\PYZus{}}\PY{l+s+s2}{\PYZdq{}}\PY{p}{)}
\end{Verbatim}
\end{tcolorbox}
\begin{tcolorbox}[breakable, size=fbox, boxrule=1pt, pad at break*=1mm,colback=cellbackground, colframe=cellborder]
\prompt{In}{incolor}{18}{\boxspacing}
\begin{Verbatim}[commandchars=\\\{\}]
\PY{n}{printAnalyse2\PYZus{}3}\PY{p}{(}\PY{l+m+mi}{1000}\PY{p}{)}
\PY{n}{printAnalyse2\PYZus{}3}\PY{p}{(}\PY{l+m+mi}{2000}\PY{p}{)}
\PY{n}{printAnalyse2\PYZus{}3}\PY{p}{(}\PY{l+m+mi}{3000}\PY{p}{)}
\end{Verbatim}
\end{tcolorbox}
\begin{Verbatim}[commandchars=\\\{\}]
1000 :
Average of cptA: 677.37
Percentage of A's to get their first choice: 75.26 \%
Average of cptB : 74.535
Percentage of B's to get their first choice: 74.53 \%
\_\_\_\_\_\_\_
2000 :
Average of cptA: 1349.425
Percentage of A's to get their first choice: 74.96 \%
Average of cptB : 149.915
Percentage of B's to get their first choice: 74.95 \%
\_\_\_\_\_\_\_
3000 :
Average of cptA: 2023.525
Percentage of A's to get their first choice: 74.94 \%
Average of cptB : 224.71
Percentage of B's to get their first choice: 74.9 \%
\_\_\_\_\_\_\_
\end{Verbatim}
The fraction of people getting their first choices seems independent
from the number n or their class. Nearly three fourths of students get
their first choice.
\hypertarget{test-instance-3}{%
\subsubsection{Test instance 3:}\label{test-instance-3}}
\begin{tcolorbox}[breakable, size=fbox, boxrule=1pt, pad at break*=1mm,colback=cellbackground, colframe=cellborder]
\prompt{In}{incolor}{19}{\boxspacing}
\begin{Verbatim}[commandchars=\\\{\}]
\PY{k}{def} \PY{n+nf}{app}\PY{p}{(}\PY{n}{studs}\PY{p}{,} \PY{n}{grp}\PY{p}{)}\PY{p}{:}
\PY{n}{s} \PY{o}{=} \PY{n}{random}\PY{p}{(}\PY{p}{)}
\PY{k}{if} \PY{n}{s} \PY{o}{\PYZgt{}}\PY{o}{=} \PY{l+m+mf}{0.5} \PY{p}{:} \PY{n}{choice} \PY{o}{=} \PY{p}{[}\PY{l+m+mi}{1}\PY{p}{,} \PY{l+m+mi}{2}\PY{p}{]}
\PY{k}{else}\PY{p}{:} \PY{n}{choice} \PY{o}{=} \PY{p}{[}\PY{l+m+mi}{2}\PY{p}{,} \PY{l+m+mi}{1}\PY{p}{]}
\PY{n}{studs}\PY{o}{.}\PY{n}{append}\PY{p}{(}\PY{n}{student3}\PY{p}{(}\PY{n}{choice}\PY{p}{,} \PY{n}{grp}\PY{p}{)}\PY{p}{)}
\PY{k}{def} \PY{n+nf}{rand\PYZus{}instance3}\PY{p}{(}\PY{n}{n}\PY{p}{)}\PY{p}{:}
\PY{l+s+sd}{\PYZdq{}\PYZdq{}\PYZdq{}}
\PY{l+s+sd}{ mate2 works with arguments of the classes school3 and student3.}
\PY{l+s+sd}{ \PYZdq{}\PYZdq{}\PYZdq{}}
\PY{n}{studs} \PY{o}{=} \PY{p}{[}\PY{p}{]}
\PY{n}{ranQty} \PY{o}{=} \PY{p}{[}\PY{p}{]}
\PY{n}{m1} \PY{o}{=} \PY{n}{n} \PY{o}{/}\PY{o}{/} \PY{l+m+mi}{2}
\PY{n}{m2} \PY{o}{=} \PY{n+nb}{int}\PY{p}{(}\PY{l+m+mi}{4} \PY{o}{*} \PY{n}{n} \PY{o}{/} \PY{l+m+mi}{5}\PY{p}{)}
\PY{n}{m3} \PY{o}{=} \PY{n+nb}{int}\PY{p}{(}\PY{l+m+mi}{19} \PY{o}{*} \PY{n}{n} \PY{o}{/} \PY{l+m+mi}{20}\PY{p}{)}
\PY{k}{for} \PY{n}{k} \PY{o+ow}{in} \PY{n+nb}{range}\PY{p}{(}\PY{n}{m1} \PY{o}{+} \PY{l+m+mi}{1}\PY{p}{)}\PY{p}{:}
\PY{n}{app}\PY{p}{(}\PY{n}{studs}\PY{p}{,} \PY{l+m+mi}{0}\PY{p}{)}
\PY{n}{ranQty}\PY{o}{.}\PY{n}{append}\PY{p}{(}\PY{p}{(}\PY{n}{k}\PY{p}{,} \PY{n}{gauss}\PY{p}{(}\PY{l+m+mi}{0}\PY{p}{,} \PY{l+m+mi}{1}\PY{p}{)}\PY{p}{)}\PY{p}{)}
\PY{k}{for} \PY{n}{k} \PY{o+ow}{in} \PY{n+nb}{range}\PY{p}{(}\PY{n}{m1} \PY{o}{+} \PY{l+m+mi}{1}\PY{p}{,} \PY{n}{m2} \PY{o}{+} \PY{l+m+mi}{1}\PY{p}{)}\PY{p}{:}
\PY{n}{app}\PY{p}{(}\PY{n}{studs}\PY{p}{,} \PY{l+m+mi}{1}\PY{p}{)}
\PY{n}{ranQty}\PY{o}{.}\PY{n}{append}\PY{p}{(}\PY{p}{(}\PY{n}{k}\PY{p}{,} \PY{n}{gauss}\PY{p}{(}\PY{l+m+mi}{0}\PY{p}{,} \PY{l+m+mi}{1}\PY{p}{)}\PY{p}{)}\PY{p}{)}
\PY{k}{for} \PY{n}{k} \PY{o+ow}{in} \PY{n+nb}{range}\PY{p}{(}\PY{n}{m2} \PY{o}{+} \PY{l+m+mi}{1}\PY{p}{,} \PY{n}{m3} \PY{o}{+} \PY{l+m+mi}{1}\PY{p}{)}\PY{p}{:}
\PY{n}{app}\PY{p}{(}\PY{n}{studs}\PY{p}{,} \PY{l+m+mi}{2}\PY{p}{)}
\PY{n}{ranQty}\PY{o}{.}\PY{n}{append}\PY{p}{(}\PY{p}{(}\PY{n}{k}\PY{p}{,} \PY{n}{gauss}\PY{p}{(}\PY{l+m+mi}{0}\PY{p}{,} \PY{l+m+mi}{1}\PY{p}{)}\PY{p}{)}\PY{p}{)}
\PY{k}{for} \PY{n}{k} \PY{o+ow}{in} \PY{n+nb}{range}\PY{p}{(}\PY{n}{m3} \PY{o}{+} \PY{l+m+mi}{1}\PY{p}{,} \PY{n}{n}\PY{p}{)}\PY{p}{:}
\PY{n}{app}\PY{p}{(}\PY{n}{studs}\PY{p}{,} \PY{l+m+mi}{3}\PY{p}{)}
\PY{n}{ranQty}\PY{o}{.}\PY{n}{append}\PY{p}{(}\PY{p}{(}\PY{n}{k}\PY{p}{,} \PY{n}{gauss}\PY{p}{(}\PY{l+m+mi}{0}\PY{p}{,} \PY{l+m+mi}{1}\PY{p}{)}\PY{p}{)}\PY{p}{)}
\PY{n}{ranQty1} \PY{o}{=} \PY{p}{[}\PY{p}{(}\PY{n}{k}\PY{p}{,} \PY{n}{ranQty}\PY{p}{[}\PY{n}{k}\PY{p}{]}\PY{p}{[}\PY{l+m+mi}{1}\PY{p}{]} \PY{o}{+} \PY{n}{gauss}\PY{p}{(}\PY{l+m+mi}{0}\PY{p}{,} \PY{l+m+mi}{1}\PY{p}{)}\PY{p}{)} \PY{k}{for} \PY{n}{k} \PY{o+ow}{in} \PY{n+nb}{range}\PY{p}{(}\PY{n}{n}\PY{p}{)}\PY{p}{]}
\PY{n}{ranQty2} \PY{o}{=} \PY{p}{[}\PY{p}{(}\PY{n}{k}\PY{p}{,} \PY{n}{ranQty}\PY{p}{[}\PY{n}{k}\PY{p}{]}\PY{p}{[}\PY{l+m+mi}{1}\PY{p}{]} \PY{o}{+} \PY{n}{gauss}\PY{p}{(}\PY{l+m+mi}{0}\PY{p}{,} \PY{l+m+mi}{1}\PY{p}{)}\PY{p}{)} \PY{k}{for} \PY{n}{k} \PY{o+ow}{in} \PY{n+nb}{range}\PY{p}{(}\PY{n}{n}\PY{p}{)}\PY{p}{]}
\PY{n}{ranQty1}\PY{o}{.}\PY{n}{sort}\PY{p}{(}\PY{n}{key} \PY{o}{=} \PY{k}{lambda} \PY{n}{x} \PY{p}{:} \PY{o}{\PYZhy{}}\PY{n}{x}\PY{p}{[}\PY{l+m+mi}{1}\PY{p}{]}\PY{p}{)} \PY{c+c1}{\PYZsh{} decreasing order of observed quality}
\PY{n}{ranQty2}\PY{o}{.}\PY{n}{sort}\PY{p}{(}\PY{n}{key} \PY{o}{=} \PY{k}{lambda} \PY{n}{x} \PY{p}{:} \PY{o}{\PYZhy{}}\PY{n}{x}\PY{p}{[}\PY{l+m+mi}{1}\PY{p}{]}\PY{p}{)}
\PY{n}{s1} \PY{o}{=} \PY{n}{school3}\PY{p}{(}\PY{p}{[}\PY{n}{studs}\PY{p}{[}\PY{n}{elt}\PY{p}{[}\PY{l+m+mi}{0}\PY{p}{]}\PY{p}{]} \PY{k}{for} \PY{n}{elt} \PY{o+ow}{in} \PY{n}{ranQty1}\PY{p}{]}\PY{p}{,} \PY{n}{n} \PY{o}{/}\PY{o}{/} \PY{l+m+mi}{4}\PY{p}{,} \PY{p}{[}\PY{n+nb}{int}\PY{p}{(}\PY{l+m+mf}{0.9} \PY{o}{*} \PY{n}{n} \PY{o}{/} \PY{l+m+mi}{4}\PY{p}{)}\PY{p}{,} \PY{n+nb}{int}\PY{p}{(}\PY{l+m+mf}{0.9} \PY{o}{*} \PY{n}{n} \PY{o}{/} \PY{l+m+mi}{4}\PY{p}{)}\PY{p}{,} \PY{n+nb}{int}\PY{p}{(}\PY{l+m+mf}{0.9} \PY{o}{*} \PY{n}{n} \PY{o}{/} \PY{l+m+mi}{4}\PY{p}{)}\PY{p}{,} \PY{n+nb}{int}\PY{p}{(}\PY{l+m+mf}{0.9} \PY{o}{*} \PY{n}{n} \PY{o}{/} \PY{l+m+mi}{4}\PY{p}{)}\PY{p}{]}\PY{p}{)}
\PY{n}{s2} \PY{o}{=} \PY{n}{school3}\PY{p}{(}\PY{p}{[}\PY{n}{studs}\PY{p}{[}\PY{n}{elt}\PY{p}{[}\PY{l+m+mi}{0}\PY{p}{]}\PY{p}{]} \PY{k}{for} \PY{n}{elt} \PY{o+ow}{in} \PY{n}{ranQty2}\PY{p}{]}\PY{p}{,} \PY{n}{n} \PY{o}{/}\PY{o}{/} \PY{l+m+mi}{4}\PY{p}{,} \PY{p}{[}\PY{n+nb}{int}\PY{p}{(}\PY{l+m+mf}{0.9} \PY{o}{*} \PY{n}{n} \PY{o}{/} \PY{l+m+mi}{4}\PY{p}{)}\PY{p}{,} \PY{n+nb}{int}\PY{p}{(}\PY{l+m+mf}{0.9} \PY{o}{*} \PY{n}{n} \PY{o}{/} \PY{l+m+mi}{4}\PY{p}{)}\PY{p}{,} \PY{n+nb}{int}\PY{p}{(}\PY{l+m+mf}{0.9} \PY{o}{*} \PY{n}{n} \PY{o}{/} \PY{l+m+mi}{4}\PY{p}{)}\PY{p}{,} \PY{n+nb}{int}\PY{p}{(}\PY{l+m+mf}{0.9} \PY{o}{*} \PY{n}{n} \PY{o}{/} \PY{l+m+mi}{4}\PY{p}{)}\PY{p}{]}\PY{p}{)}
\PY{k}{return} \PY{p}{[}\PY{n}{s1}\PY{p}{,} \PY{n}{s2}\PY{p}{]}\PY{p}{,} \PY{n}{studs}
\PY{k}{def} \PY{n+nf}{analyse3\PYZus{}2}\PY{p}{(}\PY{n}{n}\PY{p}{,} \PY{n}{rand\PYZus{}inst}\PY{p}{)}\PY{p}{:}
\PY{n}{res} \PY{o}{=} \PY{n}{mate2}\PY{p}{(}\PY{n}{rand\PYZus{}inst}\PY{p}{[}\PY{l+m+mi}{0}\PY{p}{]}\PY{p}{,} \PY{n}{rand\PYZus{}inst}\PY{p}{[}\PY{l+m+mi}{1}\PY{p}{]}\PY{p}{)}
\PY{n}{cptA}\PY{p}{,} \PY{n}{cptB}\PY{p}{,} \PY{n}{cptC}\PY{p}{,} \PY{n}{cptD} \PY{o}{=} \PY{l+m+mi}{0}\PY{p}{,} \PY{l+m+mi}{0}\PY{p}{,} \PY{l+m+mi}{0}\PY{p}{,} \PY{l+m+mi}{0}
\PY{k}{for} \PY{n}{s} \PY{o+ow}{in} \PY{p}{[}\PY{l+m+mi}{0}\PY{p}{,} \PY{l+m+mi}{1}\PY{p}{]}\PY{p}{:}
\PY{k}{for} \PY{n}{elt} \PY{o+ow}{in} \PY{n}{res}\PY{p}{[}\PY{n}{s}\PY{p}{]}\PY{p}{:}
\PY{k}{if} \PY{n}{rand\PYZus{}inst}\PY{p}{[}\PY{l+m+mi}{1}\PY{p}{]}\PY{p}{[}\PY{n}{elt}\PY{p}{]}\PY{o}{.}\PY{n}{prefStud}\PY{p}{[}\PY{n}{s}\PY{p}{]} \PY{o}{==} \PY{l+m+mi}{1}\PY{p}{:}
\PY{k}{if} \PY{n}{rand\PYZus{}inst}\PY{p}{[}\PY{l+m+mi}{1}\PY{p}{]}\PY{p}{[}\PY{n}{elt}\PY{p}{]}\PY{o}{.}\PY{n}{grp} \PY{o}{==} \PY{l+m+mi}{0}\PY{p}{:} \PY{n}{cptA} \PY{o}{+}\PY{o}{=} \PY{l+m+mi}{1}
\PY{k}{elif} \PY{n}{rand\PYZus{}inst}\PY{p}{[}\PY{l+m+mi}{1}\PY{p}{]}\PY{p}{[}\PY{n}{elt}\PY{p}{]}\PY{o}{.}\PY{n}{grp} \PY{o}{==} \PY{l+m+mi}{1}\PY{p}{:} \PY{n}{cptB} \PY{o}{+}\PY{o}{=} \PY{l+m+mi}{1}
\PY{k}{elif} \PY{n}{rand\PYZus{}inst}\PY{p}{[}\PY{l+m+mi}{1}\PY{p}{]}\PY{p}{[}\PY{n}{elt}\PY{p}{]}\PY{o}{.}\PY{n}{grp} \PY{o}{==} \PY{l+m+mi}{2}\PY{p}{:} \PY{n}{cptC} \PY{o}{+}\PY{o}{=} \PY{l+m+mi}{1}
\PY{k}{else}\PY{p}{:} \PY{n}{cptD} \PY{o}{+}\PY{o}{=} \PY{l+m+mi}{1}
\PY{k}{return} \PY{n}{n}\PY{p}{,} \PY{n}{cptA}\PY{p}{,} \PY{n}{cptB}\PY{p}{,} \PY{n}{cptC}\PY{p}{,} \PY{n}{cptD}
\PY{k}{def} \PY{n+nf}{analyse3\PYZus{}3}\PY{p}{(}\PY{n}{n}\PY{p}{,} \PY{n}{rand\PYZus{}inst}\PY{p}{)}\PY{p}{:}
\PY{n}{res} \PY{o}{=} \PY{n}{mate3}\PY{p}{(}\PY{n}{rand\PYZus{}inst}\PY{p}{[}\PY{l+m+mi}{0}\PY{p}{]}\PY{p}{,} \PY{n}{rand\PYZus{}inst}\PY{p}{[}\PY{l+m+mi}{1}\PY{p}{]}\PY{p}{)}
\PY{n}{cptA}\PY{p}{,} \PY{n}{cptB}\PY{p}{,} \PY{n}{cptC}\PY{p}{,} \PY{n}{cptD} \PY{o}{=} \PY{l+m+mi}{0}\PY{p}{,} \PY{l+m+mi}{0}\PY{p}{,} \PY{l+m+mi}{0}\PY{p}{,} \PY{l+m+mi}{0}
\PY{k}{for} \PY{n}{s} \PY{o+ow}{in} \PY{p}{[}\PY{l+m+mi}{0}\PY{p}{,} \PY{l+m+mi}{1}\PY{p}{]}\PY{p}{:}
\PY{k}{for} \PY{n}{elt} \PY{o+ow}{in} \PY{n}{res}\PY{p}{[}\PY{n}{s}\PY{p}{]}\PY{p}{:}
\PY{k}{if} \PY{n}{rand\PYZus{}inst}\PY{p}{[}\PY{l+m+mi}{1}\PY{p}{]}\PY{p}{[}\PY{n}{elt}\PY{p}{]}\PY{o}{.}\PY{n}{prefStud}\PY{p}{[}\PY{n}{s}\PY{p}{]} \PY{o}{==} \PY{l+m+mi}{1}\PY{p}{:}
\PY{k}{if} \PY{n}{rand\PYZus{}inst}\PY{p}{[}\PY{l+m+mi}{1}\PY{p}{]}\PY{p}{[}\PY{n}{elt}\PY{p}{]}\PY{o}{.}\PY{n}{grp} \PY{o}{==} \PY{l+m+mi}{0}\PY{p}{:} \PY{n}{cptA} \PY{o}{+}\PY{o}{=} \PY{l+m+mi}{1}
\PY{k}{elif} \PY{n}{rand\PYZus{}inst}\PY{p}{[}\PY{l+m+mi}{1}\PY{p}{]}\PY{p}{[}\PY{n}{elt}\PY{p}{]}\PY{o}{.}\PY{n}{grp} \PY{o}{==} \PY{l+m+mi}{1}\PY{p}{:} \PY{n}{cptB} \PY{o}{+}\PY{o}{=} \PY{l+m+mi}{1}
\PY{k}{elif} \PY{n}{rand\PYZus{}inst}\PY{p}{[}\PY{l+m+mi}{1}\PY{p}{]}\PY{p}{[}\PY{n}{elt}\PY{p}{]}\PY{o}{.}\PY{n}{grp} \PY{o}{==} \PY{l+m+mi}{2}\PY{p}{:} \PY{n}{cptC} \PY{o}{+}\PY{o}{=} \PY{l+m+mi}{1}
\PY{k}{else}\PY{p}{:} \PY{n}{cptD} \PY{o}{+}\PY{o}{=} \PY{l+m+mi}{1}
\PY{k}{return} \PY{n}{n}\PY{p}{,} \PY{n}{cptA}\PY{p}{,} \PY{n}{cptB}\PY{p}{,} \PY{n}{cptC}\PY{p}{,} \PY{n}{cptD}
\end{Verbatim}
\end{tcolorbox}
\hypertarget{test-task-2}{%
\paragraph{Test Task 2:}\label{test-task-2}}
\begin{tcolorbox}[breakable, size=fbox, boxrule=1pt, pad at break*=1mm,colback=cellbackground, colframe=cellborder]
\prompt{In}{incolor}{20}{\boxspacing}
\begin{Verbatim}[commandchars=\\\{\}]
\PY{k}{def} \PY{n+nf}{printAnalyse3\PYZus{}2}\PY{p}{(}\PY{n}{n}\PY{p}{)}\PY{p}{:}
\PY{n}{m1} \PY{o}{=} \PY{n}{n} \PY{o}{/}\PY{o}{/} \PY{l+m+mi}{2}
\PY{n}{m2} \PY{o}{=} \PY{n+nb}{int}\PY{p}{(}\PY{l+m+mi}{4} \PY{o}{*} \PY{n}{n} \PY{o}{/} \PY{l+m+mi}{5}\PY{p}{)}
\PY{n}{m3} \PY{o}{=} \PY{n+nb}{int}\PY{p}{(}\PY{l+m+mi}{19} \PY{o}{*} \PY{n}{n} \PY{o}{/} \PY{l+m+mi}{20}\PY{p}{)}
\PY{n}{N} \PY{o}{=} \PY{l+m+mi}{200}
\PY{n}{cpta}\PY{p}{,} \PY{n}{cptb}\PY{p}{,} \PY{n}{cptc}\PY{p}{,} \PY{n}{cptd} \PY{o}{=} \PY{l+m+mi}{0}\PY{p}{,} \PY{l+m+mi}{0}\PY{p}{,} \PY{l+m+mi}{0}\PY{p}{,} \PY{l+m+mi}{0}
\PY{k}{for} \PY{n}{\PYZus{}} \PY{o+ow}{in} \PY{n+nb}{range}\PY{p}{(}\PY{n}{N}\PY{p}{)}\PY{p}{:}
\PY{n}{rand\PYZus{}inst} \PY{o}{=} \PY{n}{rand\PYZus{}instance3}\PY{p}{(}\PY{n}{n}\PY{p}{)}
\PY{n}{n}\PY{p}{,} \PY{n}{cptA}\PY{p}{,} \PY{n}{cptB}\PY{p}{,} \PY{n}{cptC}\PY{p}{,} \PY{n}{cptD} \PY{o}{=} \PY{n}{analyse3\PYZus{}2}\PY{p}{(}\PY{n}{n}\PY{p}{,} \PY{n}{rand\PYZus{}inst}\PY{p}{)}
\PY{n}{cpta} \PY{o}{+}\PY{o}{=} \PY{n}{cptA}