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cost-allocation-breaks.tex
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cost-allocation-breaks.tex
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% Default to the notebook output style
% Inherit from the specified cell style.
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% Nicer default font than Computer Modern for most use cases
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% Basic figure setup, for now with no caption control since it's done
% automatically by Pandoc (which extracts  syntax from Markdown).
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% We will generate all images so they have a width \maxwidth. This means
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% are scaled down if they would overflow the margins.
\makeatletter
\def\maxwidth{\ifdim\Gin@nat@width>\linewidth\linewidth
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% Set max figure width to be 80% of text width, for now hardcoded.
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% 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}
\DeclareCaptionLabelFormat{nolabel}{}
\captionsetup{labelformat=nolabel}
\usepackage{adjustbox} % Used to constrain images to a maximum size
\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{%
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}
\usepackage{upquote} % Upright quotes for verbatim code
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\usepackage[mathletters]{ucs} % Extended unicode (utf-8) support
\usepackage[utf8x]{inputenc} % Allow utf-8 characters in the tex document
\usepackage{fancyvrb} % verbatim replacement that allows latex
\usepackage{grffile} % extends the file name processing of package graphics
% to support a larger range
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% Document parameters
\title{cost-allocation-breaks}
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% for compatibility with earlier versions
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linkcolor=linkcolor,
citecolor=citecolor,
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% Slightly bigger margins than the latex defaults
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\begin{document}
\maketitle
\section{Ride Share Allocation}
\subsection{Implementation of Gopalakrishnan et al.(2016) Algorithm}
This algorithm intake a list of passengers and their destination, and
designs a specific graph who's max flow min cost reduction is isomorphic
to the best route to take.
The biggest problem is that this algorithm assumes uncapacitated
vehicles, and takes the pickup order as inputted. For the first problem
we can assume now that this is a bus-sharing problem (think Google
sending shuttles to pick up their employees), and for the second we can
do some sorting on the input. To turn the algorithm into returning paths
for capacitated vehicles, we would need to modify the
max\_flow\_min\_cost algorithm for a graph, which is fairly
complicated/involved.
Code can be found \href{www.github.com/misingnoglic/cost-allocation}{on
Arya's Github}
Authors: Arya Boudaie; Kiranmayi P; Nidhi Rathi.
\subsubsection{Backend Logic Code:}
This part of the code deals with backend functions. Feel free to just
read the comments under the function headers, or just skip to Main
Algorithm
\begin{Verbatim}[commandchars=\\\{\}]
{\color{incolor}In [{\color{incolor}64}]:} \PY{n}{\PYZus{}\PYZus{}author\PYZus{}\PYZus{}} \PY{o}{=} \PY{l+s+s2}{\PYZdq{}}\PY{l+s+s2}{Arya}\PY{l+s+s2}{\PYZdq{}}
\PY{k+kn}{import} \PY{n+nn}{networkx} \PY{k+kn}{as} \PY{n+nn}{nx} \PY{c+c1}{\PYZsh{} Graph Library}
\PY{k+kn}{import} \PY{n+nn}{matplotlib.pyplot} \PY{k+kn}{as} \PY{n+nn}{plt} \PY{c+c1}{\PYZsh{} Plotting library}
\PY{k+kn}{from} \PY{n+nn}{matplotlib.colors} \PY{k+kn}{import} \PY{n}{cnames}
\PY{k+kn}{from} \PY{n+nn}{itertools} \PY{k+kn}{import} \PY{n}{cycle}
\PY{k+kn}{import} \PY{n+nn}{random}
\PY{o}{\PYZpc{}}\PY{k}{matplotlib} inline
\end{Verbatim}
\begin{Verbatim}[commandchars=\\\{\}]
{\color{incolor}In [{\color{incolor}65}]:} \PY{k}{def} \PY{n+nf}{new\PYZus{}round}\PY{p}{(}\PY{n}{num}\PY{p}{,} \PY{n}{digits}\PY{p}{)}\PY{p}{:}
\PY{l+s+sd}{\PYZsq{}\PYZsq{}\PYZsq{}}
\PY{l+s+sd}{ Because nx.nextwork\PYZus{}simplex doesnt work for }
\PY{l+s+sd}{ floats, this function just multiplies num}
\PY{l+s+sd}{ by digits and turns in into an integer}
\PY{l+s+sd}{ \PYZsq{}\PYZsq{}\PYZsq{}}
\PY{k}{return} \PY{n+nb}{int}\PY{p}{(}\PY{n}{num}\PY{o}{*}\PY{p}{(}\PY{l+m+mi}{10}\PY{o}{*}\PY{o}{*}\PY{n}{digits}\PY{p}{)}\PY{p}{)}
\PY{k}{def} \PY{n+nf}{node\PYZus{}numbers}\PY{p}{(}\PY{n}{node}\PY{p}{)}\PY{p}{:}
\PY{l+s+sd}{\PYZsq{}\PYZsq{}\PYZsq{}}
\PY{l+s+sd}{ Numbers the nodes, to allow for sorting }
\PY{l+s+sd}{ when printing them}
\PY{l+s+sd}{ \PYZsq{}\PYZsq{}\PYZsq{}}
\PY{k}{if} \PY{n}{node}\PY{p}{[}\PY{l+m+mi}{0}\PY{p}{]}\PY{o}{==}\PY{l+s+s1}{\PYZsq{}}\PY{l+s+s1}{source}\PY{l+s+s1}{\PYZsq{}}\PY{p}{:} \PY{k}{return} \PY{l+m+mi}{0}
\PY{k}{if} \PY{n}{node}\PY{p}{[}\PY{l+m+mi}{0}\PY{p}{]}\PY{o}{==}\PY{l+s+s1}{\PYZsq{}}\PY{l+s+s1}{sink}\PY{l+s+s1}{\PYZsq{}}\PY{p}{:} \PY{k}{return} \PY{l+m+mi}{1}
\PY{k}{if} \PY{n}{node}\PY{p}{[}\PY{l+m+mi}{0}\PY{p}{]}\PY{o}{==}\PY{l+s+s1}{\PYZsq{}}\PY{l+s+s1}{destination}\PY{l+s+s1}{\PYZsq{}}\PY{p}{:} \PY{k}{return} \PY{n+nb}{float}\PY{p}{(}\PY{l+s+s1}{\PYZsq{}}\PY{l+s+s1}{inf}\PY{l+s+s1}{\PYZsq{}}\PY{p}{)}
\PY{k}{else}\PY{p}{:}
\PY{k}{if} \PY{n}{node}\PY{p}{[}\PY{l+m+mi}{0}\PY{p}{]}\PY{p}{[}\PY{l+m+mi}{0}\PY{p}{]}\PY{o}{==}\PY{l+s+s2}{\PYZdq{}}\PY{l+s+s2}{d}\PY{l+s+s2}{\PYZdq{}}\PY{p}{:} \PY{k}{print} \PY{n}{node}
\PY{n}{cost}\PY{o}{=} \PY{l+m+mi}{2}\PY{o}{+}\PY{l+m+mi}{2}\PY{o}{*}\PY{n+nb}{int}\PY{p}{(}\PY{n}{node}\PY{p}{[}\PY{l+m+mi}{0}\PY{p}{]}\PY{p}{[}\PY{l+m+mi}{0}\PY{p}{]}\PY{p}{)}
\PY{k}{if} \PY{n}{node}\PY{p}{[}\PY{l+m+mi}{0}\PY{p}{]}\PY{p}{[}\PY{l+m+mi}{1}\PY{p}{:}\PY{p}{]}\PY{o}{==}\PY{l+s+s1}{\PYZsq{}}\PY{l+s+s1}{out}\PY{l+s+s1}{\PYZsq{}}\PY{p}{:} \PY{n}{cost}\PY{o}{+}\PY{o}{=}\PY{l+m+mi}{1}
\PY{k}{return} \PY{n}{cost}
\PY{k}{def} \PY{n+nf}{sort\PYZus{}nodes}\PY{p}{(}\PY{n}{nodes}\PY{p}{)}\PY{p}{:}
\PY{l+s+sd}{\PYZsq{}\PYZsq{}\PYZsq{}Sorts a list of nodes\PYZsq{}\PYZsq{}\PYZsq{}}
\PY{k}{return} \PY{n+nb}{sorted}\PY{p}{(}\PY{n}{nodes}\PY{p}{,}\PY{n}{key}\PY{o}{=}\PY{n}{node\PYZus{}numbers}\PY{p}{)}
\PY{k}{def} \PY{n+nf}{sort\PYZus{}edges}\PY{p}{(}\PY{n}{edges}\PY{p}{)}\PY{p}{:}
\PY{l+s+sd}{\PYZsq{}\PYZsq{}\PYZsq{}sorts a list of edges\PYZsq{}\PYZsq{}\PYZsq{}}
\PY{k}{return} \PY{n+nb}{sorted}\PY{p}{(}\PY{n}{edges}\PY{p}{,} \PY{n}{key} \PY{o}{=} \PY{k}{lambda} \PY{n}{x}\PY{p}{:}\PY{n}{node\PYZus{}numbers}\PY{p}{(}\PY{p}{[}\PY{n}{x}\PY{p}{[}\PY{l+m+mi}{0}\PY{p}{]}\PY{p}{]}\PY{p}{)}\PY{p}{)}
\end{Verbatim}
\subsubsection{Basic functions integral to the construction}
These are the building functions that are used in the algorithm. We can
replace them, as the main algorithm doesn't make any assumptions on
these. It might be interesting to use other distance functions for
instance.
\begin{Verbatim}[commandchars=\\\{\}]
{\color{incolor}In [{\color{incolor}66}]:} \PY{k+kn}{from} \PY{n+nn}{math} \PY{k+kn}{import} \PY{n}{sqrt}
\PY{k}{def} \PY{n+nf}{distance}\PY{p}{(}\PY{n}{node1}\PY{p}{,}\PY{n}{node2}\PY{p}{)}\PY{p}{:}
\PY{l+s+sd}{\PYZsq{}\PYZsq{}\PYZsq{}Simple euclidian distance formula between two points\PYZsq{}\PYZsq{}\PYZsq{}}
\PY{n}{x1} \PY{o}{=} \PY{n}{node1}\PY{p}{[}\PY{l+m+mi}{0}\PY{p}{]}
\PY{n}{x2} \PY{o}{=} \PY{n}{node2}\PY{p}{[}\PY{l+m+mi}{0}\PY{p}{]}
\PY{n}{y1} \PY{o}{=} \PY{n}{node1}\PY{p}{[}\PY{l+m+mi}{1}\PY{p}{]}
\PY{n}{y2} \PY{o}{=} \PY{n}{node2}\PY{p}{[}\PY{l+m+mi}{1}\PY{p}{]}
\PY{k}{return} \PY{n}{sqrt}\PY{p}{(}\PY{p}{(}\PY{n}{x2}\PY{o}{\PYZhy{}}\PY{n}{x1}\PY{p}{)}\PY{o}{*}\PY{o}{*}\PY{l+m+mi}{2} \PY{o}{+} \PY{p}{(}\PY{n}{y2}\PY{o}{\PYZhy{}}\PY{n}{y1}\PY{p}{)}\PY{o}{*}\PY{o}{*}\PY{l+m+mi}{2}\PY{p}{)}
\PY{k}{def} \PY{n+nf}{max\PYZus{}distance}\PY{p}{(}\PY{n}{points}\PY{p}{)}\PY{p}{:}
\PY{l+s+sd}{\PYZsq{}\PYZsq{}\PYZsq{}Gets the max distance between two points in the graph\PYZsq{}\PYZsq{}\PYZsq{}}
\PY{n}{current} \PY{o}{=} \PY{l+m+mi}{0}
\PY{k}{for} \PY{n}{node1} \PY{o+ow}{in} \PY{n}{points}\PY{p}{:}
\PY{k}{for} \PY{n}{node2} \PY{o+ow}{in} \PY{n}{points}\PY{p}{:}
\PY{n}{d} \PY{o}{=} \PY{n}{distance}\PY{p}{(}\PY{n}{node1}\PY{p}{,}\PY{n}{node2}\PY{p}{)} \PY{c+c1}{\PYZsh{} distance from n1 \PYZam{} n2}
\PY{k}{if} \PY{n}{d}\PY{o}{\PYZgt{}}\PY{n}{current}\PY{p}{:}
\PY{n}{current}\PY{o}{=}\PY{n}{d} \PY{c+c1}{\PYZsh{} store larger distance}
\PY{k}{return} \PY{n}{current}
\PY{c+c1}{\PYZsh{} Used in main algorithm to see which m\PYZsq{} has lowest cost}
\PY{k}{def} \PY{n+nf}{cost\PYZus{}of\PYZus{}min\PYZus{}cost}\PY{p}{(}\PY{n}{G}\PY{p}{)}\PY{p}{:}
\PY{l+s+sd}{\PYZsq{}\PYZsq{}\PYZsq{}}
\PY{l+s+sd}{ Given a graph, it gives the cost of the paths }
\PY{l+s+sd}{ in the max flow min cost over the graph.}
\PY{l+s+sd}{ It also returns the paths from source to }
\PY{l+s+sd}{ destination as a list, for convenience}
\PY{l+s+sd}{ \PYZsq{}\PYZsq{}\PYZsq{}}
\PY{c+c1}{\PYZsh{} Gets the max flow min cost path from source to sink}
\PY{n}{flow\PYZus{}path} \PY{o}{=} \PY{n}{nx}\PY{o}{.}\PY{n}{max\PYZus{}flow\PYZus{}min\PYZus{}cost}\PY{p}{(}\PY{n}{G}\PY{p}{,}\PY{l+s+s1}{\PYZsq{}}\PY{l+s+s1}{source}\PY{l+s+s1}{\PYZsq{}}\PY{p}{,}\PY{l+s+s1}{\PYZsq{}}\PY{l+s+s1}{sink}\PY{l+s+s1}{\PYZsq{}}\PY{p}{,}\PY{n}{weight}\PY{o}{=}\PY{l+s+s1}{\PYZsq{}}\PY{l+s+s1}{cost}\PY{l+s+s1}{\PYZsq{}}\PY{p}{)}
\PY{c+c1}{\PYZsh{} The beginning has the directions for where each cab starts}
\PY{n}{beginning} \PY{o}{=} \PY{n}{flow\PYZus{}path}\PY{p}{[}\PY{l+s+s1}{\PYZsq{}}\PY{l+s+s1}{source}\PY{l+s+s1}{\PYZsq{}}\PY{p}{]}
\PY{n}{total\PYZus{}cost} \PY{o}{=} \PY{l+m+mi}{0}
\PY{c+c1}{\PYZsh{} each node which the flow\PYZus{}path indicates to go}
\PY{n}{start\PYZus{}paths} \PY{o}{=} \PY{p}{[}\PY{n}{n}\PY{p}{[}\PY{l+m+mi}{0}\PY{p}{]} \PY{k}{for} \PY{n}{n} \PY{o+ow}{in} \PY{n}{beginning}\PY{o}{.}\PY{n}{items}\PY{p}{(}\PY{p}{)} \PY{k}{if} \PY{n}{n}\PY{p}{[}\PY{l+m+mi}{1}\PY{p}{]}\PY{o}{==}\PY{l+m+mi}{1}\PY{p}{]}
\PY{n}{paths} \PY{o}{=} \PY{p}{[}\PY{p}{]}
\PY{c+c1}{\PYZsh{} For each node that a cab goes to in order to start}
\PY{k}{for} \PY{n}{start\PYZus{}node} \PY{o+ow}{in} \PY{n}{start\PYZus{}paths}\PY{p}{:}
\PY{n}{path} \PY{o}{=} \PY{p}{[}\PY{n}{start\PYZus{}node}\PY{p}{]}
\PY{n}{curr\PYZus{}node}\PY{o}{=}\PY{n}{start\PYZus{}node}
\PY{n}{path\PYZus{}cost} \PY{o}{=} \PY{l+m+mi}{0}
\PY{c+c1}{\PYZsh{} Loop through the path until you reach the destination}
\PY{k}{while} \PY{n}{curr\PYZus{}node}\PY{o}{!=}\PY{l+s+s1}{\PYZsq{}}\PY{l+s+s1}{destination}\PY{l+s+s1}{\PYZsq{}}\PY{p}{:}
\PY{c+c1}{\PYZsh{} Node that current node travels to}
\PY{n}{next\PYZus{}node} \PY{o}{=} \PY{p}{[}\PY{n}{n} \PY{k}{for} \PY{n}{n} \PY{o+ow}{in}
\PY{n}{flow\PYZus{}path}\PY{p}{[}\PY{n}{curr\PYZus{}node}\PY{p}{]}\PY{o}{.}\PY{n}{items}\PY{p}{(}\PY{p}{)}
\PY{k}{if} \PY{n}{n}\PY{p}{[}\PY{l+m+mi}{1}\PY{p}{]}\PY{o}{==}\PY{l+m+mi}{1}\PY{p}{]}\PY{p}{[}\PY{l+m+mi}{0}\PY{p}{]}\PY{p}{[}\PY{l+m+mi}{0}\PY{p}{]}
\PY{c+c1}{\PYZsh{} Add the cost of the edge from the current node }
\PY{c+c1}{\PYZsh{}to the next node}
\PY{n}{path\PYZus{}cost} \PY{o}{+}\PY{o}{=} \PY{n}{G}\PY{o}{.}\PY{n}{edge}\PY{p}{[}\PY{n}{curr\PYZus{}node}\PY{p}{]}\PY{p}{[}\PY{n}{next\PYZus{}node}\PY{p}{]}\PY{p}{[}\PY{l+s+s1}{\PYZsq{}}\PY{l+s+s1}{cost}\PY{l+s+s1}{\PYZsq{}}\PY{p}{]}
\PY{c+c1}{\PYZsh{} Switch current node to be the node travelled to}
\PY{n}{curr\PYZus{}node} \PY{o}{=} \PY{n}{next\PYZus{}node}
\PY{k}{if} \PY{n}{G}\PY{o}{.}\PY{n}{node}\PY{p}{[}\PY{n}{curr\PYZus{}node}\PY{p}{]}\PY{p}{[}\PY{l+s+s1}{\PYZsq{}}\PY{l+s+s1}{type}\PY{l+s+s1}{\PYZsq{}}\PY{p}{]}\PY{o}{==}\PY{l+s+s1}{\PYZsq{}}\PY{l+s+s1}{passenger in}\PY{l+s+s1}{\PYZsq{}}\PY{p}{:}
\PY{n}{path}\PY{o}{.}\PY{n}{append}\PY{p}{(}\PY{n}{curr\PYZus{}node}\PY{p}{)}
\PY{c+c1}{\PYZsh{} When that node is finished, add it to the total cost }
\PY{c+c1}{\PYZsh{} and start over}
\PY{n}{total\PYZus{}cost} \PY{o}{+}\PY{o}{=} \PY{n}{path\PYZus{}cost}
\PY{c+c1}{\PYZsh{} add destination to end of path}
\PY{n}{paths}\PY{o}{.}\PY{n}{append}\PY{p}{(}\PY{n}{path}\PY{o}{+}\PY{p}{[}\PY{l+s+s1}{\PYZsq{}}\PY{l+s+s1}{destination}\PY{l+s+s1}{\PYZsq{}}\PY{p}{]}\PY{p}{)}
\PY{k}{return} \PY{n}{total\PYZus{}cost}\PY{p}{,} \PY{n}{paths}
\end{Verbatim}
\section{Main Algorithm}
In the code I include comments on what part of the paper is being
represented by the code
\begin{enumerate}[1.]
\item
The algorithm takes a list of passengers (x,y position tuples), and a
destination (x,y) tuple. Passengers are listed in pickup order
\item
From that, it creates a source node, a sink node, a destination node,
and two nodes for every passenger (an in and an out node)
\item
For each passenger it adds an edge from its in node to its out node
(cost 0, capacity 1)
\item
Add an edge from the source to every in node (cost 0, capacity 1)
\item
Calculate a value L, which is larger than the maximum distance between
two points multiplied by the number of passengers.
\item
Create an edge from each out node to the destination with the cost
being the distance between the passenger and the destination
\item
Add an edge from each out/in pair (where the index of the out is less
than the index of the in) where the cost is the distance between the
two passengers minus L.
\end{enumerate}
\begin{Verbatim}[commandchars=\\\{\}]
{\color{incolor}In [{\color{incolor}67}]:} \PY{k}{def} \PY{n+nf}{generate\PYZus{}graph}\PY{p}{(}\PY{n}{passengers}\PY{p}{,} \PY{n}{destination}\PY{p}{,}
\PY{n}{annotate}\PY{o}{=}\PY{n+nb+bp}{False}\PY{p}{,} \PY{n}{complete}\PY{o}{=}\PY{n+nb+bp}{False}\PY{p}{,} \PY{n}{mistake}\PY{o}{=}\PY{n+nb+bp}{False}\PY{p}{)}\PY{p}{:}
\PY{l+s+sd}{\PYZsq{}\PYZsq{}\PYZsq{}}
\PY{l+s+sd}{ This code generates the graph described by Raga in }
\PY{l+s+sd}{ section 7.3 steps 1 through 5.}
\PY{l+s+sd}{ The code is annotated with what it is doing in comments}
\PY{l+s+sd}{ passengers = List of x,y coordinates for each passenger}
\PY{l+s+sd}{ destination = x,y tuple with coord. of destination}
\PY{l+s+sd}{ annotate=True if you want print statements}
\PY{l+s+sd}{ complete=True if you want to do unordered paths}
\PY{l+s+sd}{ }
\PY{l+s+sd}{ \PYZsq{}\PYZsq{}\PYZsq{}}
\PY{n}{G}\PY{o}{=}\PY{n}{nx}\PY{o}{.}\PY{n}{DiGraph}\PY{p}{(}\PY{p}{)} \PY{c+c1}{\PYZsh{} Create a directed graph}
\PY{c+c1}{\PYZsh{} The rest of the algorithm closely follows Raga\PYZsq{}s paper}
\PY{c+c1}{\PYZsh{} I quote the steps directly}
\PY{c+c1}{\PYZsh{} Step 1}
\PY{c+c1}{\PYZsh{} \PYZdq{}s and t denote the source and sink vertices, respectively\PYZdq{}}
\PY{n}{G}\PY{o}{.}\PY{n}{add\PYZus{}node}\PY{p}{(}\PY{l+s+s2}{\PYZdq{}}\PY{l+s+s2}{source}\PY{l+s+s2}{\PYZdq{}}\PY{p}{,} \PY{n+nb}{type}\PY{o}{=}\PY{l+s+s2}{\PYZdq{}}\PY{l+s+s2}{meta}\PY{l+s+s2}{\PYZdq{}}\PY{p}{)}
\PY{n}{G}\PY{o}{.}\PY{n}{add\PYZus{}node}\PY{p}{(}\PY{l+s+s2}{\PYZdq{}}\PY{l+s+s2}{sink}\PY{l+s+s2}{\PYZdq{}}\PY{p}{,} \PY{n+nb}{type}\PY{o}{=}\PY{l+s+s2}{\PYZdq{}}\PY{l+s+s2}{meta}\PY{l+s+s2}{\PYZdq{}}\PY{p}{)}
\PY{c+c1}{\PYZsh{} Step 2}
\PY{c+c1}{\PYZsh{} For each passenger/pickup location u ∈ N , we create }
\PY{c+c1}{\PYZsh{} two vertices and an edge: an entry vertex u\PYZus{}in, an exit }
\PY{c+c1}{\PYZsh{} vertex u\PYZus{}out, and an edge of cost 0 and capacity 1 directed}
\PY{c+c1}{\PYZsh{} from u\PYZus{}in to u\PYZus{}out. We also create a vertex n+1 }
\PY{c+c1}{\PYZsh{} corresponding to the dropoff location.}
\PY{n}{i} \PY{o}{=} \PY{l+m+mi}{1}
\PY{k}{for} \PY{n}{passenger} \PY{o+ow}{in} \PY{n}{passengers}\PY{p}{:}
\PY{c+c1}{\PYZsh{} \PYZob{}\PYZcb{}in.format(x) will give the node the name xin, }
\PY{c+c1}{\PYZsh{} for any integer x}
\PY{n}{G}\PY{o}{.}\PY{n}{add\PYZus{}node}\PY{p}{(}\PY{l+s+s2}{\PYZdq{}}\PY{l+s+s2}{\PYZob{}\PYZcb{}in}\PY{l+s+s2}{\PYZdq{}}\PY{o}{.}\PY{n}{format}\PY{p}{(}\PY{n}{i}\PY{p}{)}\PY{p}{,} \PY{n}{number}\PY{o}{=}\PY{n}{i}\PY{p}{,} \PY{n}{x}\PY{o}{=}\PY{n}{passenger}\PY{p}{[}\PY{l+m+mi}{0}\PY{p}{]}\PY{p}{,}
\PY{n}{y}\PY{o}{=}\PY{n}{passenger}\PY{p}{[}\PY{l+m+mi}{1}\PY{p}{]}\PY{p}{,} \PY{n+nb}{type}\PY{o}{=}\PY{l+s+s2}{\PYZdq{}}\PY{l+s+s2}{passenger in}\PY{l+s+s2}{\PYZdq{}}\PY{p}{)}
\PY{n}{G}\PY{o}{.}\PY{n}{add\PYZus{}node}\PY{p}{(}\PY{l+s+s2}{\PYZdq{}}\PY{l+s+s2}{\PYZob{}\PYZcb{}out}\PY{l+s+s2}{\PYZdq{}}\PY{o}{.}\PY{n}{format}\PY{p}{(}\PY{n}{i}\PY{p}{)}\PY{p}{,}\PY{n}{number}\PY{o}{=}\PY{n}{i}\PY{p}{,} \PY{n}{x}\PY{o}{=}\PY{n}{passenger}\PY{p}{[}\PY{l+m+mi}{0}\PY{p}{]}\PY{p}{,}
\PY{n}{y}\PY{o}{=}\PY{n}{passenger}\PY{p}{[}\PY{l+m+mi}{1}\PY{p}{]}\PY{p}{,} \PY{n+nb}{type}\PY{o}{=}\PY{l+s+s2}{\PYZdq{}}\PY{l+s+s2}{passenger out}\PY{l+s+s2}{\PYZdq{}}\PY{p}{)}
\PY{n}{G}\PY{o}{.}\PY{n}{add\PYZus{}edge}\PY{p}{(}\PY{l+s+s2}{\PYZdq{}}\PY{l+s+s2}{\PYZob{}\PYZcb{}in}\PY{l+s+s2}{\PYZdq{}}\PY{o}{.}\PY{n}{format}\PY{p}{(}\PY{n}{i}\PY{p}{)}\PY{p}{,} \PY{l+s+s2}{\PYZdq{}}\PY{l+s+s2}{\PYZob{}\PYZcb{}out}\PY{l+s+s2}{\PYZdq{}}\PY{o}{.}\PY{n}{format}\PY{p}{(}\PY{n}{i}\PY{p}{)}\PY{p}{,} \PY{n}{cost}\PY{o}{=}\PY{l+m+mi}{0}\PY{p}{,}
\PY{n}{capacity}\PY{o}{=}\PY{l+m+mi}{1}\PY{p}{)}
\PY{n}{i}\PY{o}{+}\PY{o}{=}\PY{l+m+mi}{1}
\PY{n}{G}\PY{o}{.}\PY{n}{add\PYZus{}node}\PY{p}{(}\PY{l+s+s2}{\PYZdq{}}\PY{l+s+s2}{destination}\PY{l+s+s2}{\PYZdq{}}\PY{p}{,} \PY{n}{x}\PY{o}{=}\PY{n}{destination}\PY{p}{[}\PY{l+m+mi}{0}\PY{p}{]}\PY{p}{,} \PY{n}{y}\PY{o}{=}\PY{n}{destination}\PY{p}{[}\PY{l+m+mi}{1}\PY{p}{]}\PY{p}{,}
\PY{n+nb}{type}\PY{o}{=}\PY{l+s+s2}{\PYZdq{}}\PY{l+s+s2}{destination}\PY{l+s+s2}{\PYZdq{}}\PY{p}{)}
\PY{c+c1}{\PYZsh{} Step 3}
\PY{c+c1}{\PYZsh{} We create n edges, one each of cost 0 and capacity 1 from }
\PY{c+c1}{\PYZsh{} the source vertex s to each of the entry vertices u\PYZus{}in, }
\PY{c+c1}{\PYZsh{} u ∈ N.}
\PY{k}{for} \PY{n}{node} \PY{o+ow}{in} \PY{n}{G}\PY{o}{.}\PY{n}{nodes}\PY{p}{(}\PY{n}{data}\PY{o}{=}\PY{n+nb+bp}{True}\PY{p}{)}\PY{p}{:}
\PY{k}{if} \PY{n}{node}\PY{p}{[}\PY{l+m+mi}{1}\PY{p}{]}\PY{p}{[}\PY{l+s+s1}{\PYZsq{}}\PY{l+s+s1}{type}\PY{l+s+s1}{\PYZsq{}}\PY{p}{]}\PY{o}{==}\PY{l+s+s2}{\PYZdq{}}\PY{l+s+s2}{passenger in}\PY{l+s+s2}{\PYZdq{}}\PY{p}{:}
\PY{n}{G}\PY{o}{.}\PY{n}{add\PYZus{}edge}\PY{p}{(}\PY{l+s+s2}{\PYZdq{}}\PY{l+s+s2}{source}\PY{l+s+s2}{\PYZdq{}}\PY{p}{,} \PY{n}{node}\PY{p}{[}\PY{l+m+mi}{0}\PY{p}{]}\PY{p}{,} \PY{n}{cost}\PY{o}{=}\PY{l+m+mi}{0}\PY{p}{,} \PY{n}{capacity}\PY{o}{=}\PY{l+m+mi}{1}\PY{p}{)}
\PY{c+c1}{\PYZsh{} Calculating L}
\PY{c+c1}{\PYZsh{} passengers + [destination] is the concatonation of the }
\PY{c+c1}{\PYZsh{} two lists}
\PY{n}{L} \PY{o}{=} \PY{n+nb}{len}\PY{p}{(}\PY{n}{passengers}\PY{p}{)}\PY{o}{*}\PY{n}{max\PYZus{}distance}\PY{p}{(}\PY{n}{passengers} \PY{o}{+}
\PY{p}{[}\PY{n}{destination}\PY{p}{]}\PY{p}{)} \PY{o}{+} \PY{l+m+mi}{1}
\PY{k}{if} \PY{n}{annotate}\PY{p}{:} \PY{k}{print} \PY{l+s+s2}{\PYZdq{}}\PY{l+s+s2}{L = \PYZob{}\PYZcb{}}\PY{l+s+s2}{\PYZdq{}}\PY{o}{.}\PY{n}{format}\PY{p}{(}\PY{n}{L}\PY{p}{)}
\PY{c+c1}{\PYZsh{} Step 4}
\PY{c+c1}{\PYZsh{} We create n edges, one each of cost SuD and capacity 1 }
\PY{c+c1}{\PYZsh{} from each of the exit vertices uout, u ∈ N, to the }
\PY{c+c1}{\PYZsh{} dropoff vertex n + 1}
\PY{k}{for} \PY{n}{node} \PY{o+ow}{in} \PY{n}{G}\PY{o}{.}\PY{n}{nodes}\PY{p}{(}\PY{n}{data}\PY{o}{=}\PY{n+nb+bp}{True}\PY{p}{)}\PY{p}{:}
\PY{k}{if} \PY{n}{node}\PY{p}{[}\PY{l+m+mi}{1}\PY{p}{]}\PY{p}{[}\PY{l+s+s1}{\PYZsq{}}\PY{l+s+s1}{type}\PY{l+s+s1}{\PYZsq{}}\PY{p}{]}\PY{o}{==}\PY{l+s+s2}{\PYZdq{}}\PY{l+s+s2}{passenger out}\PY{l+s+s2}{\PYZdq{}}\PY{p}{:}
\PY{c+c1}{\PYZsh{} Calculating SuD}
\PY{n}{cost} \PY{o}{=} \PY{n}{distance}\PY{p}{(}\PY{n}{destination}\PY{p}{,} \PY{p}{(}\PY{n}{node}\PY{p}{[}\PY{l+m+mi}{1}\PY{p}{]}\PY{p}{[}\PY{l+s+s1}{\PYZsq{}}\PY{l+s+s1}{x}\PY{l+s+s1}{\PYZsq{}}\PY{p}{]}\PY{p}{,}
\PY{n}{node}\PY{p}{[}\PY{l+m+mi}{1}\PY{p}{]}\PY{p}{[}\PY{l+s+s1}{\PYZsq{}}\PY{l+s+s1}{y}\PY{l+s+s1}{\PYZsq{}}\PY{p}{]}\PY{p}{)}\PY{p}{)}\PY{o}{\PYZhy{}}\PY{n}{L}
\PY{k}{if} \PY{n}{mistake}\PY{p}{:} \PY{n}{cost}\PY{o}{+}\PY{o}{=}\PY{n}{L}
\PY{n}{cost} \PY{o}{=} \PY{n}{new\PYZus{}round}\PY{p}{(}\PY{n}{cost}\PY{p}{,} \PY{l+m+mi}{2}\PY{p}{)}
\PY{n}{node\PYZus{}name} \PY{o}{=} \PY{n}{node}\PY{p}{[}\PY{l+m+mi}{0}\PY{p}{]}
\PY{c+c1}{\PYZsh{} Creating the edge with that cost}
\PY{n}{G}\PY{o}{.}\PY{n}{add\PYZus{}edge}\PY{p}{(}\PY{n}{node\PYZus{}name}\PY{p}{,}\PY{l+s+s2}{\PYZdq{}}\PY{l+s+s2}{destination}\PY{l+s+s2}{\PYZdq{}}\PY{p}{,} \PY{n}{cost}\PY{o}{=}\PY{n}{cost}\PY{p}{,}
\PY{n}{capacity}\PY{o}{=}\PY{l+m+mi}{1}\PY{p}{)}
\PY{c+c1}{\PYZsh{} Step 5}
\PY{c+c1}{\PYZsh{} To encode the pickup order, for each 1 ≤ u \PYZlt{} v ≤ n we }
\PY{c+c1}{\PYZsh{} create an edge of cost (SuSv − L) and capacity 1 directed }
\PY{c+c1}{\PYZsh{} from u\PYZus{}out to v\PYZus{}in, where L is a sufficiently large number }
\PY{c+c1}{\PYZsh{} satisfying L \PYZgt{} 2*max [u,v∈N∪\PYZob{}n+1\PYZcb{} SuSv]. }
\PY{k}{for} \PY{n}{u} \PY{o+ow}{in} \PY{n+nb}{range}\PY{p}{(}\PY{l+m+mi}{1}\PY{p}{,}\PY{n+nb}{len}\PY{p}{(}\PY{n}{passengers}\PY{p}{)}\PY{o}{+}\PY{l+m+mi}{1}\PY{p}{)}\PY{p}{:} \PY{c+c1}{\PYZsh{} For each u}
\PY{c+c1}{\PYZsh{} If we want a complete graph we will do every v, }
\PY{c+c1}{\PYZsh{} otherwise just v\PYZgt{}u}
\PY{k}{if} \PY{n}{complete}\PY{p}{:}
\PY{n}{start} \PY{o}{=} \PY{l+m+mi}{1}
\PY{k}{else}\PY{p}{:}
\PY{n}{start} \PY{o}{=} \PY{n}{u}\PY{o}{+}\PY{l+m+mi}{1}
\PY{k}{for} \PY{n}{v} \PY{o+ow}{in} \PY{n+nb}{range}\PY{p}{(}\PY{n}{start}\PY{p}{,} \PY{n+nb}{len}\PY{p}{(}\PY{n}{passengers}\PY{p}{)}\PY{o}{+}\PY{l+m+mi}{1}\PY{p}{)}\PY{p}{:}
\PY{k}{if} \PY{n}{u}\PY{o}{!=}\PY{n}{v}\PY{p}{:}
\PY{n}{G}\PY{o}{.}\PY{n}{add\PYZus{}edge}\PY{p}{(}\PY{l+s+s2}{\PYZdq{}}\PY{l+s+s2}{\PYZob{}\PYZcb{}out}\PY{l+s+s2}{\PYZdq{}}\PY{o}{.}\PY{n}{format}\PY{p}{(}\PY{n}{u}\PY{p}{)}\PY{p}{,} \PY{l+s+s2}{\PYZdq{}}\PY{l+s+s2}{\PYZob{}\PYZcb{}in}\PY{l+s+s2}{\PYZdq{}}\PY{o}{.}\PY{n}{format}\PY{p}{(}\PY{n}{v}\PY{p}{)}\PY{p}{,}
\PY{n}{cost}\PY{o}{=}\PY{n}{new\PYZus{}round}\PY{p}{(}\PY{n}{distance}\PY{p}{(}
\PY{n}{passengers}\PY{p}{[}\PY{n}{u}\PY{o}{\PYZhy{}}\PY{l+m+mi}{1}\PY{p}{]}\PY{p}{,} \PY{n}{passengers}\PY{p}{[}\PY{n}{v}\PY{o}{\PYZhy{}}\PY{l+m+mi}{1}\PY{p}{]}\PY{p}{)}\PY{o}{\PYZhy{}}\PY{n}{L}\PY{p}{,}
\PY{l+m+mi}{2}\PY{p}{)}\PY{p}{,} \PY{n}{capacity}\PY{o}{=}\PY{l+m+mi}{1}\PY{p}{)}
\PY{c+c1}{\PYZsh{} pass this graph to step 6}
\PY{k}{return} \PY{n}{allocation}\PY{p}{(}\PY{n}{G}\PY{p}{,} \PY{n}{annotate}\PY{o}{=}\PY{n}{annotate}\PY{p}{)}
\end{Verbatim}
\subsubsection{Step 6:}
Finally, it creates a separate graph for each of n passengers, with each
one having an edge from destination to sink with cost 0 and capacity m'
(for each value m' from 1 to n).
After the graphs are generated, it iterates over each graph and
calculate the cost for n passengers pick up and returns the path with
minimum cost
\begin{Verbatim}[commandchars=\\\{\}]
{\color{incolor}In [{\color{incolor}68}]:} \PY{k}{def} \PY{n+nf}{allocation}\PY{p}{(}\PY{n}{G}\PY{p}{,} \PY{n}{annotate}\PY{o}{=}\PY{n+nb+bp}{False}\PY{p}{)}\PY{p}{:}
\PY{l+s+sd}{\PYZsq{}\PYZsq{}\PYZsq{}}
\PY{l+s+sd}{ Given the G in the generate\PYZus{}graph algo, it gives the graph w/}
\PY{l+s+sd}{ a node from destination to source m\PYZsq{} with the minimum cost}
\PY{l+s+sd}{ for traversing the max flow min cost graph.}
\PY{l+s+sd}{ annotate=True if you want print statements}
\PY{l+s+sd}{ \PYZsq{}\PYZsq{}\PYZsq{}}
\PY{c+c1}{\PYZsh{} Step 6}
\PY{c+c1}{\PYZsh{} We add a final edge of cost 0 and capacity m′ }
\PY{c+c1}{\PYZsh{} from the dropoff vertex n + 1 to the}
\PY{c+c1}{\PYZsh{} sink vertex t, thereby limiting the maximum flow }
\PY{c+c1}{\PYZsh{} in the network to m′ units.}
\PY{c+c1}{\PYZsh{} In this implementation, we make n copies of this graph, }
\PY{c+c1}{\PYZsh{} and give each one an m\PYZsq{} from i to n}
\PY{n}{passengers} \PY{o}{=} \PY{p}{[}\PY{n}{node} \PY{k}{for} \PY{n}{node} \PY{o+ow}{in} \PY{n}{G}\PY{o}{.}\PY{n}{nodes}\PY{p}{(}\PY{n}{data}\PY{o}{=}\PY{n+nb+bp}{True}\PY{p}{)}
\PY{k}{if} \PY{n}{node}\PY{p}{[}\PY{l+m+mi}{1}\PY{p}{]}\PY{p}{[}\PY{l+s+s1}{\PYZsq{}}\PY{l+s+s1}{type}\PY{l+s+s1}{\PYZsq{}}\PY{p}{]}\PY{o}{==}\PY{l+s+s1}{\PYZsq{}}\PY{l+s+s1}{passenger in}\PY{l+s+s1}{\PYZsq{}}\PY{p}{]}
\PY{n}{possible\PYZus{}graphs} \PY{o}{=} \PY{p}{[}\PY{n}{G}\PY{o}{.}\PY{n}{copy}\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}{passengers}\PY{p}{)}\PY{p}{)}\PY{p}{]}
\PY{k}{for} \PY{n}{i}\PY{p}{,} \PY{n}{pg} \PY{o+ow}{in} \PY{n+nb}{enumerate}\PY{p}{(}\PY{n}{possible\PYZus{}graphs}\PY{p}{,} \PY{l+m+mi}{1}\PY{p}{)}\PY{p}{:}
\PY{c+c1}{\PYZsh{} All the different m\PYZsq{} values}
\PY{n}{pg}\PY{o}{.}\PY{n}{add\PYZus{}edge}\PY{p}{(}\PY{l+s+s2}{\PYZdq{}}\PY{l+s+s2}{destination}\PY{l+s+s2}{\PYZdq{}}\PY{p}{,} \PY{l+s+s2}{\PYZdq{}}\PY{l+s+s2}{sink}\PY{l+s+s2}{\PYZdq{}}\PY{p}{,} \PY{n}{cost}\PY{o}{=}\PY{l+m+mi}{0}\PY{p}{,} \PY{n}{capacity}\PY{o}{=}\PY{n}{i}\PY{p}{)}
\PY{c+c1}{\PYZsh{} Now over these graphs we have to find the one }
\PY{c+c1}{\PYZsh{} with the lowest cost in the min cost max flow algorithm}
\PY{c+c1}{\PYZsh{} In other words which m\PYZsq{} gives the lowest cost}
\PY{n}{min\PYZus{}cost} \PY{o}{=} \PY{n+nb}{float}\PY{p}{(}\PY{l+s+s1}{\PYZsq{}}\PY{l+s+s1}{inf}\PY{l+s+s1}{\PYZsq{}}\PY{p}{)}
\PY{n}{min\PYZus{}flow\PYZus{}graph} \PY{o}{=} \PY{n+nb+bp}{None}
\PY{n}{costs} \PY{o}{=} \PY{p}{[}\PY{p}{]}
\PY{n}{paths} \PY{o}{=} \PY{p}{[}\PY{p}{]}
\PY{n}{optimal} \PY{o}{=} \PY{l+m+mi}{0}
\PY{k}{for} \PY{n}{index}\PY{p}{,} \PY{n}{pg} \PY{o+ow}{in} \PY{n+nb}{enumerate}\PY{p}{(}\PY{n}{possible\PYZus{}graphs}\PY{p}{)}\PY{p}{:}
\PY{k}{if} \PY{n}{annotate}\PY{p}{:} \PY{k}{print} \PY{l+s+s2}{\PYZdq{}}\PY{l+s+s2}{\PYZob{}\PYZcb{} cabs: }\PY{l+s+s2}{\PYZdq{}}\PY{o}{.}\PY{n}{format}\PY{p}{(}\PY{n}{index}\PY{o}{+}\PY{l+m+mi}{1}\PY{p}{)}\PY{p}{,}
\PY{c+c1}{\PYZsh{} For each possible graph, calculate the cost. }
\PY{n}{c}\PY{p}{,} \PY{n}{path} \PY{o}{=} \PY{n}{cost\PYZus{}of\PYZus{}min\PYZus{}cost}\PY{p}{(}\PY{n}{pg}\PY{p}{)}
\PY{n}{c} \PY{o}{=} \PY{n}{new\PYZus{}round}\PY{p}{(}\PY{n}{c}\PY{p}{,}\PY{l+m+mi}{2}\PY{p}{)}
\PY{n}{paths}\PY{o}{.}\PY{n}{append}\PY{p}{(}\PY{n}{path}\PY{p}{)}
\PY{n}{costs}\PY{o}{.}\PY{n}{append}\PY{p}{(}\PY{n}{c}\PY{p}{)}
\PY{k}{if} \PY{n}{annotate}\PY{p}{:}
\PY{k}{print} \PY{l+s+s2}{\PYZdq{}}\PY{l+s+s2}{path: }\PY{l+s+s2}{\PYZdq{}}\PY{p}{,}\PY{n}{path}\PY{p}{,}\PY{l+s+s2}{\PYZdq{}}\PY{l+s+s2}{\PYZhy{}\PYZhy{}}\PY{l+s+s2}{\PYZdq{}}\PY{p}{,}
\PY{k}{print} \PY{l+s+s2}{\PYZdq{}}\PY{l+s+s2}{cost: }\PY{l+s+s2}{\PYZdq{}}\PY{p}{,}\PY{n}{c}\PY{o}{/}\PY{l+m+mf}{10000.0}
\PY{c+c1}{\PYZsh{} If the cost is less than previous graphs, store it}
\PY{k}{if} \PY{n}{c}\PY{o}{\PYZlt{}}\PY{n}{min\PYZus{}cost}\PY{p}{:}
\PY{n}{min\PYZus{}cost}\PY{o}{=}\PY{n}{c}
\PY{n}{min\PYZus{}flow\PYZus{}graph}\PY{o}{=}\PY{n}{pg}
\PY{n}{optimal} \PY{o}{=} \PY{n}{index}
\PY{k}{if} \PY{n}{annotate}\PY{p}{:} \PY{k}{print} \PY{l+s+s2}{\PYZdq{}}\PY{l+s+s2}{Optimal path with \PYZob{}\PYZcb{} cabs}\PY{l+s+s2}{\PYZdq{}}\PY{o}{.}\PY{n}{format}\PY{p}{(}
\PY{n}{optimal}\PY{o}{+}\PY{l+m+mi}{1}\PY{p}{)}\PY{p}{,} \PY{n}{paths}\PY{p}{[}\PY{n}{optimal}\PY{p}{]}
\PY{k}{return} \PY{n}{min\PYZus{}flow\PYZus{}graph}\PY{p}{,}\PY{n}{paths}\PY{p}{[}\PY{n}{optimal}\PY{p}{]}
\end{Verbatim}
\subsubsection{Frontend Functions}
The following functions allow the user to interract with the algorithm
and visually inspect the output. (They are not commented, as they aren't
important to the algorithm.)
\begin{Verbatim}[commandchars=\\\{\}]
{\color{incolor}In [{\color{incolor}69}]:} \PY{k}{def} \PY{n+nf}{allocation\PYZus{}interractive}\PY{p}{(}\PY{o}{*}\PY{o}{*}\PY{n}{kwargs}\PY{p}{)}\PY{p}{:}
\PY{l+s+sd}{\PYZsq{}\PYZsq{}\PYZsq{}}
\PY{l+s+sd}{ Interractive version of the function}
\PY{l+s+sd}{ \PYZsq{}\PYZsq{}\PYZsq{}}
\PY{n}{players} \PY{o}{=} \PY{n+nb}{int}\PY{p}{(}\PY{n+nb}{raw\PYZus{}input}\PY{p}{(}\PY{l+s+s2}{\PYZdq{}}\PY{l+s+s2}{How many players are there: }\PY{l+s+s2}{\PYZdq{}}\PY{p}{)}\PY{p}{)}
\PY{n}{player\PYZus{}coordinates} \PY{o}{=} \PY{p}{[}\PY{p}{]}
\PY{k}{for} \PY{n}{i} \PY{o+ow}{in} \PY{n+nb}{range}\PY{p}{(}\PY{l+m+mi}{1}\PY{p}{,}\PY{n}{players}\PY{o}{+}\PY{l+m+mi}{1}\PY{p}{)}\PY{p}{:}
\PY{n}{coord} \PY{o}{=} \PY{n+nb}{raw\PYZus{}input}\PY{p}{(}\PY{l+s+s1}{\PYZsq{}\PYZsq{}\PYZsq{}}\PY{l+s+s1}{What are the x and y }
\PY{l+s+s1}{ coordinates for player \PYZob{}\PYZcb{}: }\PY{l+s+s1}{\PYZsq{}\PYZsq{}\PYZsq{}}\PY{o}{.}\PY{n}{format}\PY{p}{(}\PY{n}{i}\PY{p}{)}\PY{p}{)}
\PY{n}{coord} \PY{o}{=} \PY{p}{[}\PY{n+nb}{int}\PY{p}{(}\PY{n}{x}\PY{p}{)} \PY{k}{for} \PY{n}{x} \PY{o+ow}{in} \PY{n}{coord}\PY{o}{.}\PY{n}{split}\PY{p}{(}\PY{p}{)}\PY{p}{]}
\PY{n}{player\PYZus{}coordinates}\PY{o}{.}\PY{n}{append}\PY{p}{(}\PY{p}{(}\PY{n}{coord}\PY{p}{[}\PY{l+m+mi}{0}\PY{p}{]}\PY{p}{,}\PY{n}{coord}\PY{p}{[}\PY{l+m+mi}{1}\PY{p}{]}\PY{p}{)}\PY{p}{)}
\PY{n}{destination} \PY{o}{=} \PY{n+nb}{raw\PYZus{}input}\PY{p}{(}\PY{l+s+s1}{\PYZsq{}\PYZsq{}\PYZsq{}}\PY{l+s+s1}{What are the x and y coordinates }
\PY{l+s+s1}{ for the destination: }\PY{l+s+s1}{\PYZsq{}\PYZsq{}\PYZsq{}}\PY{p}{)}
\PY{n}{destination} \PY{o}{=} \PY{p}{[}\PY{n+nb}{int}\PY{p}{(}\PY{n}{x}\PY{p}{)} \PY{k}{for} \PY{n}{x} \PY{o+ow}{in} \PY{n}{destination}\PY{o}{.}\PY{n}{split}\PY{p}{(}\PY{p}{)}\PY{p}{]}
\PY{k}{return} \PY{n}{generate\PYZus{}graph}\PY{p}{(}\PY{n}{player\PYZus{}coordinates}\PY{p}{,}
\PY{p}{(}\PY{n}{destination}\PY{p}{[}\PY{l+m+mi}{0}\PY{p}{]}\PY{p}{,} \PY{n}{destination}\PY{p}{[}\PY{l+m+mi}{1}\PY{p}{]}\PY{p}{)}\PY{p}{,} \PY{o}{*}\PY{o}{*}\PY{n}{kwargs}\PY{p}{)}
\PY{k}{def} \PY{n+nf}{draw\PYZus{}paths}\PY{p}{(}\PY{n}{G}\PY{p}{,} \PY{n}{paths}\PY{o}{=}\PY{n+nb+bp}{None}\PY{p}{,} \PY{n}{title}\PY{o}{=}\PY{l+s+s2}{\PYZdq{}}\PY{l+s+s2}{Graph of optimal routes}\PY{l+s+s2}{\PYZdq{}}\PY{p}{)}\PY{p}{:}
\PY{l+s+sd}{\PYZsq{}\PYZsq{}\PYZsq{}}
\PY{l+s+sd}{ Given a graph and paths through the graph, it plots }
\PY{l+s+sd}{ them w/ arrows. If path not given, it gets the path with }
\PY{l+s+sd}{ min cost max flow.}
\PY{l+s+sd}{ \PYZsq{}\PYZsq{}\PYZsq{}}
\PY{k}{if} \PY{n}{paths} \PY{o}{==} \PY{n+nb+bp}{None}\PY{p}{:}
\PY{n}{paths} \PY{o}{=} \PY{n}{cost\PYZus{}of\PYZus{}min\PYZus{}cost}\PY{p}{(}\PY{n}{G}\PY{p}{)}\PY{p}{[}\PY{l+m+mi}{1}\PY{p}{]}
\PY{c+c1}{\PYZsh{} Way of getting random colors}
\PY{n}{color\PYZus{}names} \PY{o}{=} \PY{n+nb}{list}\PY{p}{(}\PY{l+s+s2}{\PYZdq{}}\PY{l+s+s2}{gbrcmyk}\PY{l+s+s2}{\PYZdq{}}\PY{p}{)}
\PY{n}{new\PYZus{}colors}\PY{o}{=}\PY{p}{[}\PY{p}{]}
\PY{k}{if} \PY{n+nb}{len}\PY{p}{(}\PY{n}{paths}\PY{p}{)} \PY{o}{\PYZgt{}} \PY{n+nb}{len}\PY{p}{(}\PY{n}{color\PYZus{}names}\PY{p}{)}\PY{p}{:}
\PY{n}{new\PYZus{}colors} \PY{o}{=} \PY{p}{[}\PY{n}{c} \PY{k}{for} \PY{n}{c} \PY{o+ow}{in} \PY{n}{cnames}\PY{o}{.}\PY{n}{keys}\PY{p}{(}\PY{p}{)}
\PY{k}{if} \PY{o+ow}{not}\PY{p}{(}\PY{l+s+s2}{\PYZdq{}}\PY{l+s+s2}{light}\PY{l+s+s2}{\PYZdq{}} \PY{o+ow}{in} \PY{n}{c}\PY{p}{)}\PY{p}{]}
\PY{n}{random}\PY{o}{.}\PY{n}{shuffle}\PY{p}{(}\PY{n}{new\PYZus{}colors}\PY{p}{)}
\PY{c+c1}{\PYZsh{}iterator through colors (so it will repeat)}
\PY{n}{colors} \PY{o}{=} \PY{n}{cycle}\PY{p}{(}\PY{n}{color\PYZus{}names}\PY{o}{+}\PY{n}{new\PYZus{}colors}\PY{p}{)}
\PY{c+c1}{\PYZsh{} plot labels}
\PY{n}{plt}\PY{o}{.}\PY{n}{xlabel}\PY{p}{(}\PY{l+s+s2}{\PYZdq{}}\PY{l+s+s2}{X}\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}{Y}\PY{l+s+s2}{\PYZdq{}}\PY{p}{)}
\PY{n}{plt}\PY{o}{.}\PY{n}{title}\PY{p}{(}\PY{n}{title}\PY{o}{+}\PY{l+s+s2}{\PYZdq{}}\PY{l+s+s2}{ \PYZhy{} \PYZob{}\PYZcb{} cabs}\PY{l+s+s2}{\PYZdq{}}\PY{o}{.}\PY{n}{format}\PY{p}{(}\PY{n+nb}{len}\PY{p}{(}\PY{n}{paths}\PY{p}{)}\PY{p}{)}\PY{p}{)}
\PY{c+c1}{\PYZsh{} add all the x,y points of the passengers}
\PY{n}{xs} \PY{o}{=} \PY{p}{[}\PY{p}{]}
\PY{n}{ys} \PY{o}{=} \PY{p}{[}\PY{p}{]}
\PY{k}{for} \PY{n}{n} \PY{o+ow}{in} \PY{n}{G}\PY{o}{.}\PY{n}{nodes}\PY{p}{(}\PY{n}{data}\PY{o}{=}\PY{n+nb+bp}{True}\PY{p}{)}\PY{p}{:}
\PY{k}{if} \PY{n}{n}\PY{p}{[}\PY{l+m+mi}{1}\PY{p}{]}\PY{p}{[}\PY{l+s+s1}{\PYZsq{}}\PY{l+s+s1}{type}\PY{l+s+s1}{\PYZsq{}}\PY{p}{]} \PY{o}{==} \PY{l+s+s1}{\PYZsq{}}\PY{l+s+s1}{passenger in}\PY{l+s+s1}{\PYZsq{}}\PY{p}{:}
\PY{n}{xs}\PY{o}{.}\PY{n}{append}\PY{p}{(}\PY{n}{n}\PY{p}{[}\PY{l+m+mi}{1}\PY{p}{]}\PY{p}{[}\PY{l+s+s1}{\PYZsq{}}\PY{l+s+s1}{x}\PY{l+s+s1}{\PYZsq{}}\PY{p}{]}\PY{p}{)}
\PY{n}{ys}\PY{o}{.}\PY{n}{append}\PY{p}{(}\PY{n}{n}\PY{p}{[}\PY{l+m+mi}{1}\PY{p}{]}\PY{p}{[}\PY{l+s+s1}{\PYZsq{}}\PY{l+s+s1}{y}\PY{l+s+s1}{\PYZsq{}}\PY{p}{]}\PY{p}{)}
\PY{n}{plt}\PY{o}{.}\PY{n}{plot}\PY{p}{(}\PY{n}{xs}\PY{p}{,}\PY{n}{ys}\PY{p}{,}\PY{l+s+s1}{\PYZsq{}}\PY{l+s+s1}{ro}\PY{l+s+s1}{\PYZsq{}}\PY{p}{)}
\PY{c+c1}{\PYZsh{} plot the destination}
\PY{n}{destination} \PY{o}{=} \PY{n}{G}\PY{o}{.}\PY{n}{node}\PY{p}{[}\PY{l+s+s1}{\PYZsq{}}\PY{l+s+s1}{destination}\PY{l+s+s1}{\PYZsq{}}\PY{p}{]}
\PY{n}{plt}\PY{o}{.}\PY{n}{plot}\PY{p}{(}\PY{n}{destination}\PY{p}{[}\PY{l+s+s1}{\PYZsq{}}\PY{l+s+s1}{x}\PY{l+s+s1}{\PYZsq{}}\PY{p}{]}\PY{p}{,} \PY{n}{destination}\PY{p}{[}\PY{l+s+s1}{\PYZsq{}}\PY{l+s+s1}{y}\PY{l+s+s1}{\PYZsq{}}\PY{p}{]}\PY{p}{,} \PY{l+s+s1}{\PYZsq{}}\PY{l+s+s1}{bo}\PY{l+s+s1}{\PYZsq{}}\PY{p}{)}
\PY{n}{xs}\PY{o}{.}\PY{n}{append}\PY{p}{(}\PY{n}{destination}\PY{p}{[}\PY{l+s+s1}{\PYZsq{}}\PY{l+s+s1}{x}\PY{l+s+s1}{\PYZsq{}}\PY{p}{]}\PY{p}{)}
\PY{n}{ys}\PY{o}{.}\PY{n}{append}\PY{p}{(}\PY{n}{destination}\PY{p}{[}\PY{l+s+s1}{\PYZsq{}}\PY{l+s+s1}{y}\PY{l+s+s1}{\PYZsq{}}\PY{p}{]}\PY{p}{)}
\PY{c+c1}{\PYZsh{} determine the axis}
\PY{n}{plt}\PY{o}{.}\PY{n}{axis}\PY{p}{(}\PY{p}{[}\PY{n+nb}{min}\PY{p}{(}\PY{n}{xs}\PY{p}{)}\PY{o}{\PYZhy{}}\PY{l+m+mi}{1}\PY{p}{,}\PY{n+nb}{max}\PY{p}{(}\PY{n}{xs}\PY{p}{)}\PY{o}{+}\PY{l+m+mi}{1}\PY{p}{,}\PY{n+nb}{min}\PY{p}{(}\PY{n}{ys}\PY{p}{)}\PY{o}{\PYZhy{}}\PY{l+m+mi}{1}\PY{p}{,}\PY{n+nb}{max}\PY{p}{(}\PY{n}{ys}\PY{p}{)}\PY{o}{+}\PY{l+m+mi}{1}\PY{p}{]}\PY{p}{)}
\PY{k}{for} \PY{n}{path} \PY{o+ow}{in} \PY{n}{paths}\PY{p}{:} \PY{c+c1}{\PYZsh{} for each path}
\PY{n}{c} \PY{o}{=} \PY{n}{colors}\PY{o}{.}\PY{n}{next}\PY{p}{(}\PY{p}{)} \PY{c+c1}{\PYZsh{} choose a random color}
\PY{k}{for} \PY{n}{i} \PY{o+ow}{in} \PY{n+nb}{range}\PY{p}{(}\PY{n+nb}{len}\PY{p}{(}\PY{n}{path}\PY{p}{)}\PY{o}{\PYZhy{}}\PY{l+m+mi}{1}\PY{p}{)}\PY{p}{:} \PY{c+c1}{\PYZsh{} for each pair in the path}
\PY{n}{from\PYZus{}node} \PY{o}{=} \PY{n}{G}\PY{o}{.}\PY{n}{node}\PY{p}{[}\PY{n}{path}\PY{p}{[}\PY{n}{i}\PY{p}{]}\PY{p}{]}
\PY{n}{to\PYZus{}node} \PY{o}{=} \PY{n}{G}\PY{o}{.}\PY{n}{node}\PY{p}{[}\PY{n}{path}\PY{p}{[}\PY{n}{i}\PY{o}{+}\PY{l+m+mi}{1}\PY{p}{]}\PY{p}{]}
\PY{k}{try}\PY{p}{:}
\PY{c+c1}{\PYZsh{} draw an arrow from from\PYZus{}node to to\PYZus{}node}
\PY{n}{plt}\PY{o}{.}\PY{n}{arrow}\PY{p}{(}\PY{n}{from\PYZus{}node}\PY{p}{[}\PY{l+s+s1}{\PYZsq{}}\PY{l+s+s1}{x}\PY{l+s+s1}{\PYZsq{}}\PY{p}{]}\PY{p}{,} \PY{n}{from\PYZus{}node}\PY{p}{[}\PY{l+s+s1}{\PYZsq{}}\PY{l+s+s1}{y}\PY{l+s+s1}{\PYZsq{}}\PY{p}{]}\PY{p}{,}
\PY{p}{(}\PY{n}{to\PYZus{}node}\PY{p}{[}\PY{l+s+s1}{\PYZsq{}}\PY{l+s+s1}{x}\PY{l+s+s1}{\PYZsq{}}\PY{p}{]}\PY{o}{\PYZhy{}}\PY{n}{from\PYZus{}node}\PY{p}{[}\PY{l+s+s1}{\PYZsq{}}\PY{l+s+s1}{x}\PY{l+s+s1}{\PYZsq{}}\PY{p}{]}\PY{p}{)}\PY{p}{,}
\PY{p}{(}\PY{n}{to\PYZus{}node}\PY{p}{[}\PY{l+s+s1}{\PYZsq{}}\PY{l+s+s1}{y}\PY{l+s+s1}{\PYZsq{}}\PY{p}{]}\PY{o}{\PYZhy{}}\PY{n}{from\PYZus{}node}\PY{p}{[}\PY{l+s+s1}{\PYZsq{}}\PY{l+s+s1}{y}\PY{l+s+s1}{\PYZsq{}}\PY{p}{]}\PY{p}{)}\PY{p}{,}
\PY{n}{head\PYZus{}width}\PY{o}{=}\PY{l+m+mf}{0.2}\PY{p}{,} \PY{n}{head\PYZus{}length}\PY{o}{=}\PY{l+m+mf}{0.05}\PY{p}{,}
\PY{n}{length\PYZus{}includes\PYZus{}head}\PY{o}{=}\PY{n+nb+bp}{True}\PY{p}{,}\PY{n}{fc}\PY{o}{=}\PY{n}{c}\PY{p}{,} \PY{n}{ec}\PY{o}{=}\PY{n}{c}\PY{p}{)}
\PY{k}{except}\PY{p}{:}
\PY{c+c1}{\PYZsh{} if error is thrown, it means that the point }
\PY{c+c1}{\PYZsh{} was repeated}
\PY{k}{continue}
\PY{n}{plt}\PY{o}{.}\PY{n}{show}\PY{p}{(}\PY{p}{)}
\end{Verbatim}
Running the cell below allows you to interractively give the program a
set of passengers and a destination. For example, try the simple case -
two passengers at (0,0) and (1,0) - destination (2,0)
\begin{Verbatim}[commandchars=\\\{\}]
{\color{incolor}In [{\color{incolor}70}]:} \PY{c+c1}{\PYZsh{} Run this cell to run the interractive simulation}
\PY{n}{a}\PY{p}{,} \PY{n}{paths} \PY{o}{=} \PY{n}{allocation\PYZus{}interractive}\PY{p}{(}\PY{n}{annotate}\PY{o}{=}\PY{n+nb+bp}{True}\PY{p}{)}
\PY{n}{draw\PYZus{}paths}\PY{p}{(}\PY{n}{a}\PY{p}{,}\PY{n}{paths}\PY{p}{)}
\end{Verbatim}
\begin{Verbatim}[commandchars=\\\{\}]
How many players are there: 2
What are the x and y coordinates for player 1: 0 0
What are the x and y coordinates for player 2: 1 0
What are the x and y coordinates for the destination: 2 0
L = 5.0
1 cabs: path: [['1in', '2in', 'destination']] -- cost: -8.0
2 cabs: path: [['2in', 'destination'], ['1in', 'destination']] -- cost: -7.0
Optimal path with 1 cabs [['1in', '2in', 'destination']]
\end{Verbatim}
\begin{center}
\adjustimage{max size={0.9\linewidth}{0.9\paperheight}}{cost-allocation-breaks_files/cost-allocation-breaks_14_1.png}
\end{center}
{ \hspace*{\fill} \\}
\paragraph{List of all the nodes}
\begin{Verbatim}[commandchars=\\\{\}]
{\color{incolor}In [{\color{incolor}71}]:} \PY{n}{sort\PYZus{}nodes}\PY{p}{(}\PY{n}{a}\PY{o}{.}\PY{n}{nodes}\PY{p}{(}\PY{n}{data}\PY{o}{=}\PY{n+nb+bp}{True}\PY{p}{)}\PY{p}{)}
\end{Verbatim}
\begin{Verbatim}[commandchars=\\\{\}]
{\color{outcolor}Out[{\color{outcolor}71}]:} [('source', \{'type': 'meta'\}),
('sink', \{'type': 'meta'\}),
('1in', \{'number': 1, 'type': 'passenger in', 'x': 0, 'y': 0\}),
('1out', \{'number': 1, 'type': 'passenger out', 'x': 0, 'y': 0\}),
('2in', \{'number': 2, 'type': 'passenger in', 'x': 1, 'y': 0\}),
('2out', \{'number': 2, 'type': 'passenger out', 'x': 1, 'y': 0\}),
('destination', \{'type': 'destination', 'x': 2, 'y': 0\})]
\end{Verbatim}
\paragraph{List of all the edges}
\begin{Verbatim}[commandchars=\\\{\}]
{\color{incolor}In [{\color{incolor}72}]:} \PY{c+c1}{\PYZsh{} All the edges in the graph}
\PY{n}{sort\PYZus{}edges}\PY{p}{(}\PY{n}{a}\PY{o}{.}\PY{n}{edges}\PY{p}{(}\PY{n}{data}\PY{o}{=}\PY{n+nb+bp}{True}\PY{p}{)}\PY{p}{)}
\end{Verbatim}
\begin{Verbatim}[commandchars=\\\{\}]
{\color{outcolor}Out[{\color{outcolor}72}]:} [('source', '2in', \{'capacity': 1, 'cost': 0\}),
('source', '1in', \{'capacity': 1, 'cost': 0\}),
('1in', '1out', \{'capacity': 1, 'cost': 0\}),
('1out', '2in', \{'capacity': 1, 'cost': -400\}),
('1out', 'destination', \{'capacity': 1, 'cost': -300\}),
('2in', '2out', \{'capacity': 1, 'cost': 0\}),
('2out', 'destination', \{'capacity': 1, 'cost': -400\}),
('destination', 'sink', \{'capacity': 1, 'cost': 0\})]
\end{Verbatim}
\paragraph{Drawing of the constructed graph}
This gets quite complicated with large number of players, but shows the
graph constructed by the algorithm, with the source, sink, destination,
and in/out nodes, as well as all the edges. Layout is random.
\begin{Verbatim}[commandchars=\\\{\}]
{\color{incolor}In [{\color{incolor}73}]:} \PY{n}{nx}\PY{o}{.}\PY{n}{draw}\PY{p}{(}\PY{n}{a}\PY{p}{,} \PY{n}{with\PYZus{}labels}\PY{o}{=}\PY{n+nb+bp}{True}\PY{p}{)}
\end{Verbatim}
\begin{center}
\adjustimage{max size={0.9\linewidth}{0.9\paperheight}}{cost-allocation-breaks_files/cost-allocation-breaks_20_0.png}
\end{center}
{ \hspace*{\fill} \\}
\begin{Verbatim}[commandchars=\\\{\}]
{\color{incolor}In [{\color{incolor}74}]:} \PY{c+c1}{\PYZsh{} The path that should be taken }
\PY{c+c1}{\PYZsh{} Starting from the source, a 1 in the node in the dictionary }
\PY{c+c1}{\PYZsh{} means send 1 cab there}
\PY{n}{nx}\PY{o}{.}\PY{n}{max\PYZus{}flow\PYZus{}min\PYZus{}cost}\PY{p}{(}\PY{n}{a}\PY{p}{,}\PY{l+s+s1}{\PYZsq{}}\PY{l+s+s1}{source}\PY{l+s+s1}{\PYZsq{}}\PY{p}{,}\PY{l+s+s1}{\PYZsq{}}\PY{l+s+s1}{sink}\PY{l+s+s1}{\PYZsq{}}\PY{p}{,}\PY{n}{weight}\PY{o}{=}\PY{l+s+s1}{\PYZsq{}}\PY{l+s+s1}{cost}\PY{l+s+s1}{\PYZsq{}}\PY{p}{)}
\end{Verbatim}
\begin{Verbatim}[commandchars=\\\{\}]
{\color{outcolor}Out[{\color{outcolor}74}]:} \{'1in': \{'1out': 1\},
'1out': \{'2in': 1, 'destination': 0\},
'2in': \{'2out': 1\},
'2out': \{'destination': 1\},
'destination': \{'sink': 1\},
'sink': \{\},
'source': \{'1in': 1, '2in': 0\}\}
\end{Verbatim}
\subsection{More examples}
\begin{Verbatim}[commandchars=\\\{\}]
{\color{incolor}In [{\color{incolor}75}]:} \PY{n}{passengers} \PY{o}{=} \PY{p}{[}\PY{p}{(}\PY{l+m+mi}{0}\PY{p}{,}\PY{l+m+mi}{0}\PY{p}{)}\PY{p}{,} \PY{p}{(}\PY{l+m+mi}{1}\PY{p}{,}\PY{l+m+mi}{0}\PY{p}{)}\PY{p}{]}
\PY{n}{destination} \PY{o}{=} \PY{p}{(}\PY{l+m+mi}{2}\PY{p}{,}\PY{l+m+mi}{0}\PY{p}{)}
\PY{n}{a}\PY{p}{,} \PY{n}{paths} \PY{o}{=} \PY{n}{generate\PYZus{}graph}\PY{p}{(}\PY{n}{passengers}\PY{p}{,} \PY{n}{destination}\PY{p}{)}
\PY{n}{draw\PYZus{}paths}\PY{p}{(}\PY{n}{a}\PY{p}{,}\PY{n}{paths}\PY{p}{,} \PY{n}{title}\PY{o}{=}\PY{l+s+s2}{\PYZdq{}}\PY{l+s+s2}{Simple Example}\PY{l+s+s2}{\PYZdq{}}\PY{p}{)}
\end{Verbatim}
\begin{center}
\adjustimage{max size={0.9\linewidth}{0.9\paperheight}}{cost-allocation-breaks_files/cost-allocation-breaks_23_0.png}
\end{center}
{ \hspace*{\fill} \\}
\begin{Verbatim}[commandchars=\\\{\}]
{\color{incolor}In [{\color{incolor}76}]:} \PY{n}{passengers} \PY{o}{=} \PY{p}{[}\PY{p}{(}\PY{l+m+mi}{2}\PY{p}{,}\PY{l+m+mi}{0}\PY{p}{)}\PY{p}{,} \PY{p}{(}\PY{l+m+mi}{1}\PY{p}{,}\PY{l+m+mi}{0}\PY{p}{)}\PY{p}{,} \PY{p}{(}\PY{l+m+mi}{0}\PY{p}{,}\PY{l+m+mi}{2}\PY{p}{)}\PY{p}{,} \PY{p}{(}\PY{l+m+mi}{0}\PY{p}{,}\PY{l+m+mi}{1}\PY{p}{)}\PY{p}{,} \PY{p}{(}\PY{l+m+mi}{0}\PY{p}{,}\PY{o}{\PYZhy{}}\PY{l+m+mi}{2}\PY{p}{)}\PY{p}{,} \PY{p}{(}\PY{l+m+mi}{0}\PY{p}{,}\PY{o}{\PYZhy{}}\PY{l+m+mi}{1}\PY{p}{)}\PY{p}{]}
\PY{n}{destination} \PY{o}{=} \PY{p}{(}\PY{l+m+mi}{0}\PY{p}{,}\PY{l+m+mi}{0}\PY{p}{)}
\PY{n}{a}\PY{p}{,} \PY{n}{paths} \PY{o}{=} \PY{n}{generate\PYZus{}graph}\PY{p}{(}\PY{n}{passengers}\PY{p}{,} \PY{n}{destination}\PY{p}{)}
\PY{n}{draw\PYZus{}paths}\PY{p}{(}\PY{n}{a}\PY{p}{,}\PY{n}{paths}\PY{p}{,} \PY{n}{title}\PY{o}{=}\PY{l+s+s2}{\PYZdq{}}\PY{l+s+s2}{Three cabs needed}\PY{l+s+s2}{\PYZdq{}}\PY{p}{)}
\end{Verbatim}
\begin{center}
\adjustimage{max size={0.9\linewidth}{0.9\paperheight}}{cost-allocation-breaks_files/cost-allocation-breaks_24_0.png}
\end{center}
{ \hspace*{\fill} \\}
\begin{Verbatim}[commandchars=\\\{\}]
{\color{incolor}In [{\color{incolor}77}]:} \PY{n}{passengers} \PY{o}{=} \PY{p}{[}\PY{p}{(}\PY{l+m+mi}{1}\PY{p}{,}\PY{l+m+mi}{1}\PY{p}{)}\PY{p}{,} \PY{p}{(}\PY{o}{\PYZhy{}}\PY{l+m+mi}{1}\PY{p}{,}\PY{l+m+mi}{1}\PY{p}{)}\PY{p}{,} \PY{p}{(}\PY{o}{\PYZhy{}}\PY{l+m+mi}{1}\PY{p}{,}\PY{o}{\PYZhy{}}\PY{l+m+mi}{1}\PY{p}{)}\PY{p}{,} \PY{p}{(}\PY{l+m+mi}{1}\PY{p}{,}\PY{o}{\PYZhy{}}\PY{l+m+mi}{1}\PY{p}{)}\PY{p}{]}
\PY{n}{destination} \PY{o}{=} \PY{p}{(}\PY{l+m+mi}{0}\PY{p}{,}\PY{l+m+mi}{0}\PY{p}{)}
\PY{n}{a}\PY{p}{,} \PY{n}{paths} \PY{o}{=} \PY{n}{generate\PYZus{}graph}\PY{p}{(}\PY{n}{passengers}\PY{p}{,} \PY{n}{destination}\PY{p}{)}
\PY{n}{draw\PYZus{}paths}\PY{p}{(}\PY{n}{a}\PY{p}{,}\PY{n}{paths}\PY{p}{,} \PY{n}{title}\PY{o}{=}\PY{l+s+s2}{\PYZdq{}}\PY{l+s+s2}{Points in Square}\PY{l+s+s2}{\PYZdq{}}\PY{p}{)}
\end{Verbatim}
\begin{center}
\adjustimage{max size={0.9\linewidth}{0.9\paperheight}}{cost-allocation-breaks_files/cost-allocation-breaks_25_0.png}
\end{center}
{ \hspace*{\fill} \\}
\begin{Verbatim}[commandchars=\\\{\}]
{\color{incolor}In [{\color{incolor}78}]:} \PY{n}{passengers} \PY{o}{=} \PY{p}{[}\PY{p}{(}\PY{l+m+mi}{2}\PY{p}{,}\PY{l+m+mi}{2}\PY{p}{)}\PY{p}{,} \PY{p}{(}\PY{l+m+mi}{1}\PY{p}{,}\PY{l+m+mi}{1}\PY{p}{)}\PY{p}{,} \PY{p}{(}\PY{l+m+mi}{1}\PY{p}{,}\PY{l+m+mi}{0}\PY{p}{)}\PY{p}{,} \PY{p}{(}\PY{l+m+mi}{1}\PY{p}{,}\PY{o}{\PYZhy{}}\PY{l+m+mi}{1}\PY{p}{)}\PY{p}{,} \PY{p}{(}\PY{l+m+mi}{3}\PY{p}{,}\PY{l+m+mi}{1}\PY{p}{)}\PY{p}{,} \PY{p}{(}\PY{l+m+mi}{3}\PY{p}{,}\PY{l+m+mi}{0}\PY{p}{)}\PY{p}{,} \PY{p}{(}\PY{l+m+mi}{3}\PY{p}{,}\PY{o}{\PYZhy{}}\PY{l+m+mi}{1}\PY{p}{)}\PY{p}{]}
\PY{n}{destination} \PY{o}{=} \PY{p}{(}\PY{l+m+mi}{2}\PY{p}{,}\PY{o}{\PYZhy{}}\PY{l+m+mi}{2}\PY{p}{)}
\PY{n}{a}\PY{p}{,} \PY{n}{paths} \PY{o}{=} \PY{n}{generate\PYZus{}graph}\PY{p}{(}\PY{n}{passengers}\PY{p}{,} \PY{n}{destination}\PY{p}{)}
\PY{n}{draw\PYZus{}paths}\PY{p}{(}\PY{n}{a}\PY{p}{,}\PY{n}{paths}\PY{p}{,} \PY{n}{title}\PY{o}{=}\PY{l+s+s2}{\PYZdq{}}\PY{l+s+s2}{Points in hexagon}\PY{l+s+s2}{\PYZdq{}}\PY{p}{)}
\end{Verbatim}
\begin{center}
\adjustimage{max size={0.9\linewidth}{0.9\paperheight}}{cost-allocation-breaks_files/cost-allocation-breaks_26_0.png}
\end{center}
{ \hspace*{\fill} \\}
\begin{Verbatim}[commandchars=\\\{\}]
{\color{incolor}In [{\color{incolor}79}]:} \PY{n}{passengers} \PY{o}{=} \PY{p}{[}\PY{p}{(}\PY{l+m+mi}{2}\PY{p}{,}\PY{l+m+mi}{2}\PY{p}{)}\PY{p}{,} \PY{p}{(}\PY{l+m+mi}{1}\PY{p}{,}\PY{l+m+mi}{1}\PY{p}{)}\PY{p}{,} \PY{p}{(}\PY{l+m+mi}{1}\PY{p}{,}\PY{l+m+mi}{0}\PY{p}{)}\PY{p}{,} \PY{p}{(}\PY{l+m+mi}{1}\PY{p}{,}\PY{o}{\PYZhy{}}\PY{l+m+mi}{1}\PY{p}{)}\PY{p}{,} \PY{p}{(}\PY{l+m+mi}{3}\PY{p}{,}\PY{l+m+mi}{1}\PY{p}{)}\PY{p}{,} \PY{p}{(}\PY{l+m+mi}{3}\PY{p}{,}\PY{l+m+mi}{0}\PY{p}{)}\PY{p}{,} \PY{p}{(}\PY{l+m+mi}{3}\PY{p}{,}\PY{o}{\PYZhy{}}\PY{l+m+mi}{1}\PY{p}{)}\PY{p}{]}
\PY{n}{destination} \PY{o}{=} \PY{p}{(}\PY{l+m+mi}{2}\PY{p}{,}\PY{l+m+mi}{0}\PY{p}{)}
\PY{n}{a}\PY{p}{,} \PY{n}{paths} \PY{o}{=} \PY{n}{generate\PYZus{}graph}\PY{p}{(}\PY{n}{passengers}\PY{p}{,} \PY{n}{destination}\PY{p}{)}
\PY{n}{draw\PYZus{}paths}\PY{p}{(}\PY{n}{a}\PY{p}{,}\PY{n}{paths}\PY{p}{,} \PY{n}{title}\PY{o}{=}\PY{l+s+s2}{\PYZdq{}}\PY{l+s+s2}{hexagon with destination in middle}\PY{l+s+s2}{\PYZdq{}}\PY{p}{)}
\end{Verbatim}
\begin{center}
\adjustimage{max size={0.9\linewidth}{0.9\paperheight}}{cost-allocation-breaks_files/cost-allocation-breaks_27_0.png}
\end{center}
{ \hspace*{\fill} \\}
The below graphs are completely random (except for destination being
0,0) - feel free to change the variables in the below cell to get
different graphs. The first random graph will have non optimal paths, as
the ordering is set, but the second graph shows the paths when we order
the pickups by distance from destination (largest first).
\begin{Verbatim}[commandchars=\\\{\}]
{\color{incolor}In [{\color{incolor}80}]:} \PY{c+c1}{\PYZsh{} Constraints for random graphs}
\PY{n}{n} \PY{o}{=} \PY{l+m+mi}{50} \PY{c+c1}{\PYZsh{} number of passengers}
\PY{n}{min\PYZus{}x} \PY{o}{=} \PY{o}{\PYZhy{}}\PY{l+m+mi}{30}
\PY{n}{max\PYZus{}x} \PY{o}{=} \PY{l+m+mi}{30}
\PY{n}{min\PYZus{}y} \PY{o}{=} \PY{o}{\PYZhy{}}\PY{l+m+mi}{30}
\PY{n}{max\PYZus{}y} \PY{o}{=} \PY{l+m+mi}{30}
\end{Verbatim}
\begin{Verbatim}[commandchars=\\\{\}]
{\color{incolor}In [{\color{incolor}81}]:} \PY{n}{passengers} \PY{o}{=} \PY{p}{[}\PY{p}{(}\PY{n}{random}\PY{o}{.}\PY{n}{randint}\PY{p}{(}\PY{n}{min\PYZus{}x}\PY{p}{,}\PY{n}{max\PYZus{}x}\PY{p}{)}\PY{p}{,}
\PY{n}{random}\PY{o}{.}\PY{n}{randint}\PY{p}{(}\PY{n}{min\PYZus{}y}\PY{p}{,}\PY{n}{max\PYZus{}y}\PY{p}{)}\PY{p}{)} \PY{k}{for} \PY{n}{i} \PY{o+ow}{in} \PY{n+nb}{range}\PY{p}{(}\PY{n}{n}\PY{p}{)}\PY{p}{]}
\PY{n}{destination} \PY{o}{=} \PY{p}{(}\PY{l+m+mi}{0}\PY{p}{,}\PY{l+m+mi}{0}\PY{p}{)}
\PY{n}{a}\PY{p}{,} \PY{n}{paths} \PY{o}{=} \PY{n}{generate\PYZus{}graph}\PY{p}{(}\PY{n}{passengers}\PY{p}{,} \PY{n}{destination}\PY{p}{)}
\PY{n}{draw\PYZus{}paths}\PY{p}{(}\PY{n}{a}\PY{p}{,}\PY{n}{paths}\PY{p}{,} \PY{n}{title}\PY{o}{=}\PY{l+s+s1}{\PYZsq{}\PYZsq{}\PYZsq{}}\PY{l+s+s1}{Completely random graph }
\PY{l+s+s1}{ of \PYZob{}\PYZcb{} passengers}\PY{l+s+s1}{\PYZsq{}\PYZsq{}\PYZsq{}}\PY{o}{.}\PY{n}{format}\PY{p}{(}\PY{n}{n}\PY{p}{)}\PY{p}{)}
\end{Verbatim}
\begin{center}
\adjustimage{max size={0.9\linewidth}{0.9\paperheight}}{cost-allocation-breaks_files/cost-allocation-breaks_30_0.png}
\end{center}
{ \hspace*{\fill} \\}
Here we order the passengers and the paths taken are much more optimal.
\begin{Verbatim}[commandchars=\\\{\}]
{\color{incolor}In [{\color{incolor}82}]:} \PY{n}{passengers} \PY{o}{=} \PY{n+nb}{sorted}\PY{p}{(}\PY{n}{passengers}\PY{p}{,} \PY{n}{key} \PY{o}{=}
\PY{k}{lambda} \PY{n}{x} \PY{p}{:} \PY{n}{distance}\PY{p}{(}\PY{n}{x}\PY{p}{,} \PY{n}{destination}\PY{p}{)}\PY{p}{,}
\PY{n}{reverse}\PY{o}{=}\PY{n+nb+bp}{True}\PY{p}{)}
\PY{n}{a}\PY{p}{,} \PY{n}{paths} \PY{o}{=} \PY{n}{generate\PYZus{}graph}\PY{p}{(}\PY{n}{passengers}\PY{p}{,} \PY{n}{destination}\PY{p}{)}
\PY{n}{draw\PYZus{}paths}\PY{p}{(}\PY{n}{a}\PY{p}{,}\PY{n}{paths}\PY{p}{,} \PY{n}{title}\PY{o}{=}\PY{l+s+s2}{\PYZdq{}}\PY{l+s+s2}{Same graph with sorted passengers}\PY{l+s+s2}{\PYZdq{}}\PY{p}{)}
\end{Verbatim}
\begin{center}
\adjustimage{max size={0.9\linewidth}{0.9\paperheight}}{cost-allocation-breaks_files/cost-allocation-breaks_32_0.png}
\end{center}
{ \hspace*{\fill} \\}
\subsection{The bug we found in the paper}
As described in the presentation, the original cost of the edge from a
passenger's out node to the destination was just the distance between
the passenger and the destination. This subtle bug caused the optimal
number of cabs to always be 1, and we fixed it by subtracting the same L
from the cost.
This is not a formal proof, but since the problem was only found in the
implementation, we have recreated the results of the mistake below with
the mistake=True flag - on graphs which we have already seen the best
path for.
\begin{Verbatim}[commandchars=\\\{\}]
{\color{incolor}In [{\color{incolor}83}]:} \PY{n}{passengers} \PY{o}{=} \PY{p}{[}\PY{p}{(}\PY{l+m+mi}{1}\PY{p}{,}\PY{l+m+mi}{1}\PY{p}{)}\PY{p}{,} \PY{p}{(}\PY{o}{\PYZhy{}}\PY{l+m+mi}{1}\PY{p}{,}\PY{l+m+mi}{1}\PY{p}{)}\PY{p}{,} \PY{p}{(}\PY{o}{\PYZhy{}}\PY{l+m+mi}{1}\PY{p}{,}\PY{o}{\PYZhy{}}\PY{l+m+mi}{1}\PY{p}{)}\PY{p}{,} \PY{p}{(}\PY{l+m+mi}{1}\PY{p}{,}\PY{o}{\PYZhy{}}\PY{l+m+mi}{1}\PY{p}{)}\PY{p}{]}
\PY{n}{destination} \PY{o}{=} \PY{p}{(}\PY{l+m+mi}{0}\PY{p}{,}\PY{l+m+mi}{0}\PY{p}{)}
\PY{n}{a}\PY{p}{,} \PY{n}{paths} \PY{o}{=} \PY{n}{generate\PYZus{}graph}\PY{p}{(}\PY{n}{passengers}\PY{p}{,} \PY{n}{destination}\PY{p}{,} \PY{n}{mistake}\PY{o}{=}\PY{n+nb+bp}{True}\PY{p}{)}
\PY{n}{draw\PYZus{}paths}\PY{p}{(}\PY{n}{a}\PY{p}{,}\PY{n}{paths}\PY{p}{,} \PY{n}{title}\PY{o}{=}\PY{l+s+s2}{\PYZdq{}}\PY{l+s+s2}{Square Mistake}\PY{l+s+s2}{\PYZdq{}}\PY{p}{)}
\end{Verbatim}
\begin{center}
\adjustimage{max size={0.9\linewidth}{0.9\paperheight}}{cost-allocation-breaks_files/cost-allocation-breaks_34_0.png}
\end{center}
{ \hspace*{\fill} \\}
\begin{Verbatim}[commandchars=\\\{\}]
{\color{incolor}In [{\color{incolor}84}]:} \PY{n}{passengers} \PY{o}{=} \PY{p}{[}\PY{p}{(}\PY{n}{random}\PY{o}{.}\PY{n}{randint}\PY{p}{(}\PY{n}{min\PYZus{}x}\PY{p}{,}\PY{n}{max\PYZus{}x}\PY{p}{)}\PY{p}{,}
\PY{n}{random}\PY{o}{.}\PY{n}{randint}\PY{p}{(}\PY{n}{min\PYZus{}y}\PY{p}{,}\PY{n}{max\PYZus{}y}\PY{p}{)}\PY{p}{)} \PY{k}{for} \PY{n}{i} \PY{o+ow}{in} \PY{n+nb}{range}\PY{p}{(}\PY{n}{n}\PY{p}{)}\PY{p}{]}
\PY{n}{destination} \PY{o}{=} \PY{p}{(}\PY{l+m+mi}{0}\PY{p}{,}\PY{l+m+mi}{0}\PY{p}{)}
\PY{n}{a}\PY{p}{,} \PY{n}{paths} \PY{o}{=} \PY{n}{generate\PYZus{}graph}\PY{p}{(}\PY{n}{passengers}\PY{p}{,} \PY{n}{destination}\PY{p}{,} \PY{n}{mistake}\PY{o}{=}\PY{n+nb+bp}{True}\PY{p}{)}
\PY{n}{draw\PYZus{}paths}\PY{p}{(}\PY{n}{a}\PY{p}{,}\PY{n}{paths}\PY{p}{,} \PY{n}{title}\PY{o}{=}\PY{l+s+s2}{\PYZdq{}}\PY{l+s+s2}{Random graph with mistake}\PY{l+s+s2}{\PYZdq{}}\PY{o}{.}\PY{n}{format}\PY{p}{(}\PY{n}{n}\PY{p}{)}\PY{p}{)}
\end{Verbatim}
\begin{center}
\adjustimage{max size={0.9\linewidth}{0.9\paperheight}}{cost-allocation-breaks_files/cost-allocation-breaks_35_0.png}
\end{center}
{ \hspace*{\fill} \\}
\subsection{Unused work}
We tried to make progress in calculating costs, but as the min cost max
flow algorithm is not based on individual rationality, the output of
optimal path need not satisy SIR. However, the following part of code
can be used to verify if the resultant path is SIR compliant or not.
\begin{Verbatim}[commandchars=\\\{\}]
{\color{incolor}In [{\color{incolor}85}]:} \PY{k}{def} \PY{n+nf}{operational\PYZus{}cost}\PY{p}{(}\PY{n}{G}\PY{p}{,} \PY{n}{route}\PY{p}{,} \PY{n}{meter\PYZus{}fare} \PY{o}{=} \PY{l+m+mi}{8}\PY{p}{)}\PY{p}{:}
\PY{n}{oc} \PY{o}{=} \PY{l+m+mi}{0}
\PY{n}{oc\PYZus{}values} \PY{o}{=} \PY{p}{[}\PY{p}{]}
\PY{n}{players\PYZus{}on\PYZus{}route} \PY{o}{=} \PY{l+m+mi}{1}
\PY{k}{for} \PY{n}{i} \PY{o+ow}{in} \PY{n+nb}{range}\PY{p}{(}\PY{n+nb}{len}\PY{p}{(}\PY{n}{route}\PY{p}{)}\PY{o}{\PYZhy{}}\PY{l+m+mi}{1}\PY{p}{)}\PY{p}{:}
\PY{n}{first} \PY{o}{=} \PY{n}{G}\PY{o}{.}\PY{n}{node}\PY{p}{[}\PY{n}{route}\PY{p}{[}\PY{n}{i}\PY{p}{]}\PY{p}{]}
\PY{n}{second} \PY{o}{=} \PY{n}{G}\PY{o}{.}\PY{n}{node}\PY{p}{[}\PY{n}{route}\PY{p}{[}\PY{n}{i}\PY{o}{+}\PY{l+m+mi}{1}\PY{p}{]}\PY{p}{]}
\PY{n}{oc} \PY{o}{+}\PY{o}{=} \PY{n}{meter\PYZus{}fare}\PY{o}{*}\PY{p}{(}\PY{n}{distance}\PY{p}{(} \PY{p}{(}\PY{n}{first}\PY{p}{[}\PY{l+s+s1}{\PYZsq{}}\PY{l+s+s1}{x}\PY{l+s+s1}{\PYZsq{}}\PY{p}{]}\PY{p}{,} \PY{n}{first}\PY{p}{[}\PY{l+s+s1}{\PYZsq{}}\PY{l+s+s1}{y}\PY{l+s+s1}{\PYZsq{}}\PY{p}{]}\PY{p}{)}\PY{p}{,}
\PY{p}{(}\PY{n}{second}\PY{p}{[}\PY{l+s+s1}{\PYZsq{}}\PY{l+s+s1}{x}\PY{l+s+s1}{\PYZsq{}}\PY{p}{]}\PY{p}{,} \PY{n}{second}\PY{p}{[}\PY{l+s+s1}{\PYZsq{}}\PY{l+s+s1}{y}\PY{l+s+s1}{\PYZsq{}}\PY{p}{]}\PY{p}{)} \PY{p}{)} \PY{o}{/} \PY{n+nb}{float}\PY{p}{(}\PY{n}{players\PYZus{}on\PYZus{}route}\PY{p}{)} \PY{p}{)}
\PY{n}{oc\PYZus{}values}\PY{o}{.}\PY{n}{append}\PY{p}{(}\PY{p}{(}\PY{n}{distance}\PY{p}{(} \PY{p}{(}\PY{n}{first}\PY{p}{[}\PY{l+s+s1}{\PYZsq{}}\PY{l+s+s1}{x}\PY{l+s+s1}{\PYZsq{}}\PY{p}{]}\PY{p}{,} \PY{n}{first}\PY{p}{[}\PY{l+s+s1}{\PYZsq{}}\PY{l+s+s1}{y}\PY{l+s+s1}{\PYZsq{}}\PY{p}{]}\PY{p}{)}\PY{p}{,}
\PY{p}{(}\PY{n}{second}\PY{p}{[}\PY{l+s+s1}{\PYZsq{}}\PY{l+s+s1}{x}\PY{l+s+s1}{\PYZsq{}}\PY{p}{]}\PY{p}{,} \PY{n}{second}\PY{p}{[}\PY{l+s+s1}{\PYZsq{}}\PY{l+s+s1}{y}\PY{l+s+s1}{\PYZsq{}}\PY{p}{]}\PY{p}{)} \PY{p}{)} \PY{o}{/} \PY{n+nb}{float}\PY{p}{(}\PY{n}{players\PYZus{}on\PYZus{}route}\PY{p}{)} \PY{p}{)}\PY{p}{)}
\PY{n}{players\PYZus{}on\PYZus{}route} \PY{o}{+}\PY{o}{=}\PY{l+m+mi}{1}
\PY{k}{return} \PY{n}{oc}
\PY{k}{def} \PY{n+nf}{shared\PYZus{}cost\PYZus{}per\PYZus{}player}\PY{p}{(}\PY{n}{G}\PY{p}{,} \PY{n}{route}\PY{p}{,} \PY{n}{meter\PYZus{}fare}\PY{o}{=}\PY{l+m+mi}{8}\PY{p}{)}\PY{p}{:}
\PY{n}{operational\PYZus{}costs} \PY{o}{=} \PY{p}{[}\PY{p}{]}
\PY{n}{inconvenience\PYZus{}costs} \PY{o}{=} \PY{p}{[}\PY{l+m+mi}{0}\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}{route}\PY{p}{)}\PY{o}{\PYZhy{}}\PY{l+m+mi}{1}\PY{p}{)}\PY{p}{:}
\PY{n}{operational\PYZus{}costs}\PY{o}{.}\PY{n}{append}\PY{p}{(}\PY{n}{operational\PYZus{}cost}\PY{p}{(}\PY{n}{G}\PY{p}{,} \PY{n}{route}\PY{p}{[}\PY{n}{i}\PY{p}{:}\PY{p}{]}\PY{p}{,}
\PY{n}{meter\PYZus{}fare}\PY{p}{)}\PY{p}{)}
\PY{c+c1}{\PYZsh{}IC(i) is the inconvenience cost caused due to }
\PY{c+c1}{\PYZsh{} player(i) to other players}
\PY{k}{for} \PY{n}{i} \PY{o+ow}{in} \PY{n+nb}{range}\PY{p}{(}\PY{l+m+mi}{1}\PY{p}{,}\PY{n+nb}{len}\PY{p}{(}\PY{n}{route}\PY{p}{)}\PY{o}{\PYZhy{}}\PY{l+m+mi}{1}\PY{p}{)}\PY{p}{:}
\PY{n}{first} \PY{o}{=} \PY{n}{G}\PY{o}{.}\PY{n}{node}\PY{p}{[}\PY{n}{route}\PY{p}{[}\PY{n}{i}\PY{o}{\PYZhy{}}\PY{l+m+mi}{1}\PY{p}{]}\PY{p}{]}
\PY{n}{second} \PY{o}{=} \PY{n}{G}\PY{o}{.}\PY{n}{node}\PY{p}{[}\PY{n}{route}\PY{p}{[}\PY{n}{i}\PY{p}{]}\PY{p}{]}
\PY{n}{destination} \PY{o}{=} \PY{n}{G}\PY{o}{.}\PY{n}{node}\PY{p}{[}\PY{l+s+s2}{\PYZdq{}}\PY{l+s+s2}{destination}\PY{l+s+s2}{\PYZdq{}}\PY{p}{]}
\PY{n}{new\PYZus{}distance} \PY{o}{=} \PY{p}{(}\PY{n}{distance}\PY{p}{(} \PY{p}{(}\PY{n}{first}\PY{p}{[}\PY{l+s+s1}{\PYZsq{}}\PY{l+s+s1}{x}\PY{l+s+s1}{\PYZsq{}}\PY{p}{]}\PY{p}{,} \PY{n}{first}\PY{p}{[}\PY{l+s+s1}{\PYZsq{}}\PY{l+s+s1}{y}\PY{l+s+s1}{\PYZsq{}}\PY{p}{]}\PY{p}{)}\PY{p}{,}
\PY{p}{(}\PY{n}{second}\PY{p}{[}\PY{l+s+s1}{\PYZsq{}}\PY{l+s+s1}{x}\PY{l+s+s1}{\PYZsq{}}\PY{p}{]}\PY{p}{,} \PY{n}{second}\PY{p}{[}\PY{l+s+s1}{\PYZsq{}}\PY{l+s+s1}{y}\PY{l+s+s1}{\PYZsq{}}\PY{p}{]}\PY{p}{)} \PY{p}{)}\PY{o}{+}
\PY{n}{distance}\PY{p}{(} \PY{p}{(}\PY{n}{second}\PY{p}{[}\PY{l+s+s1}{\PYZsq{}}\PY{l+s+s1}{x}\PY{l+s+s1}{\PYZsq{}}\PY{p}{]}\PY{p}{,} \PY{n}{second}\PY{p}{[}\PY{l+s+s1}{\PYZsq{}}\PY{l+s+s1}{y}\PY{l+s+s1}{\PYZsq{}}\PY{p}{]}\PY{p}{)}\PY{p}{,}
\PY{p}{(}\PY{n}{destination}\PY{p}{[}\PY{l+s+s1}{\PYZsq{}}\PY{l+s+s1}{x}\PY{l+s+s1}{\PYZsq{}}\PY{p}{]}\PY{p}{,}
\PY{n}{destination}\PY{p}{[}\PY{l+s+s1}{\PYZsq{}}\PY{l+s+s1}{y}\PY{l+s+s1}{\PYZsq{}}\PY{p}{]}\PY{p}{)} \PY{p}{)}\PY{p}{)}
\PY{n}{original\PYZus{}distance} \PY{o}{=} \PY{n}{distance}\PY{p}{(} \PY{p}{(}\PY{n}{first}\PY{p}{[}\PY{l+s+s1}{\PYZsq{}}\PY{l+s+s1}{x}\PY{l+s+s1}{\PYZsq{}}\PY{p}{]}\PY{p}{,} \PY{n}{first}\PY{p}{[}\PY{l+s+s1}{\PYZsq{}}\PY{l+s+s1}{y}\PY{l+s+s1}{\PYZsq{}}\PY{p}{]}\PY{p}{)}\PY{p}{,}
\PY{p}{(}\PY{n}{destination}\PY{p}{[}\PY{l+s+s1}{\PYZsq{}}\PY{l+s+s1}{x}\PY{l+s+s1}{\PYZsq{}}\PY{p}{]}\PY{p}{,}
\PY{n}{destination}\PY{p}{[}\PY{l+s+s1}{\PYZsq{}}\PY{l+s+s1}{y}\PY{l+s+s1}{\PYZsq{}}\PY{p}{]}\PY{p}{)} \PY{p}{)}
\PY{n}{inconvenience\PYZus{}costs}\PY{o}{.}\PY{n}{append}\PY{p}{(}
\PY{p}{(}\PY{n}{new\PYZus{}distance}\PY{o}{\PYZhy{}}\PY{n}{original\PYZus{}distance}\PY{p}{)}\PY{o}{*}\PY{n}{meter\PYZus{}fare}\PY{p}{)}
\PY{n}{total\PYZus{}cost} \PY{o}{=} \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}{route}\PY{p}{)}\PY{o}{\PYZhy{}}\PY{l+m+mi}{1}\PY{p}{)}\PY{p}{:}
\PY{n}{total\PYZus{}cost}\PY{o}{.}\PY{n}{append}\PY{p}{(}
\PY{n}{operational\PYZus{}costs}\PY{p}{[}\PY{n}{i}\PY{p}{]}\PY{o}{+}\PY{n}{i}\PY{o}{*}\PY{n}{inconvenience\PYZus{}costs}\PY{p}{[}\PY{n}{i}\PY{p}{]}\PY{p}{)}
\PY{k}{return} \PY{n}{total\PYZus{}cost}
\PY{k}{def} \PY{n+nf}{original\PYZus{}cost\PYZus{}per\PYZus{}player}\PY{p}{(}\PY{n}{G}\PY{p}{,} \PY{n}{meter\PYZus{}fare}\PY{o}{=}\PY{l+m+mi}{8}\PY{p}{)}\PY{p}{:}
\PY{n}{costs} \PY{o}{=} \PY{p}{[}\PY{p}{]}
\PY{n}{players} \PY{o}{=} \PY{p}{[}\PY{n}{n} \PY{k}{for} \PY{n}{n} \PY{o+ow}{in} \PY{n}{G}\PY{o}{.}\PY{n}{nodes}\PY{p}{(}\PY{n}{data}\PY{o}{=}\PY{n+nb+bp}{True}\PY{p}{)}
\PY{k}{if} \PY{n}{n}\PY{p}{[}\PY{l+m+mi}{1}\PY{p}{]}\PY{p}{[}\PY{l+s+s1}{\PYZsq{}}\PY{l+s+s1}{type}\PY{l+s+s1}{\PYZsq{}}\PY{p}{]}\PY{o}{==}\PY{l+s+s1}{\PYZsq{}}\PY{l+s+s1}{passenger in}\PY{l+s+s1}{\PYZsq{}}\PY{p}{]}
\PY{n}{destination} \PY{o}{=} \PY{n}{G}\PY{o}{.}\PY{n}{node}\PY{p}{[}\PY{l+s+s2}{\PYZdq{}}\PY{l+s+s2}{destination}\PY{l+s+s2}{\PYZdq{}}\PY{p}{]}
\PY{k}{for} \PY{n}{player} \PY{o+ow}{in} \PY{n}{players}\PY{p}{:}
\PY{n}{costs}\PY{o}{.}\PY{n}{append}\PY{p}{(}
\PY{n}{meter\PYZus{}fare}\PY{o}{*}\PY{p}{(}\PY{n}{distance}\PY{p}{(}\PY{p}{(}\PY{n}{player}\PY{p}{[}\PY{l+m+mi}{1}\PY{p}{]}\PY{p}{[}\PY{l+s+s1}{\PYZsq{}}\PY{l+s+s1}{x}\PY{l+s+s1}{\PYZsq{}}\PY{p}{]}\PY{p}{,} \PY{n}{player}\PY{p}{[}\PY{l+m+mi}{1}\PY{p}{]}\PY{p}{[}\PY{l+s+s1}{\PYZsq{}}\PY{l+s+s1}{y}\PY{l+s+s1}{\PYZsq{}}\PY{p}{]}\PY{p}{)}\PY{p}{,}
\PY{p}{(}\PY{n}{destination}\PY{p}{[}\PY{l+s+s1}{\PYZsq{}}\PY{l+s+s1}{x}\PY{l+s+s1}{\PYZsq{}}\PY{p}{]}\PY{p}{,} \PY{n}{destination}\PY{p}{[}\PY{l+s+s1}{\PYZsq{}}\PY{l+s+s1}{y}\PY{l+s+s1}{\PYZsq{}}\PY{p}{]}\PY{p}{)} \PY{p}{)}\PY{p}{)}\PY{p}{)}
\PY{k}{return} \PY{n}{costs}