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% Most of the following are not required for the majority
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% Document Info
\author{Phoenix (cddc)}
\pdfinfo{
  /Title (cpsc221mt.pdf)
  /Creator (Cheatography)
  /Author (Phoenix (cddc))
  /Subject (CPSC221MT Cheat Sheet)
}

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% Header and Footer
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\fancyhead{} % Set header to blank
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\fancyhead[L]{
\noindent
\begin{multicols}{3}
\begin{tabulary}{5.8cm}{C}
    \SetRowColor{DarkBackground}
    \vspace{-7pt}
    {\parbox{\dimexpr\textwidth-2\fboxsep\relax}{\noindent
        \hspace*{-6pt}\includegraphics[width=5.8cm]{/web/www.cheatography.com/public/images/cheatography_logo.pdf}}
    }
\end{tabulary}
\columnbreak
\begin{tabulary}{11cm}{L}
    \vspace{-2pt}\large{\bf{\textcolor{DarkBackground}{\textrm{CPSC221MT Cheat Sheet}}}} \\
    \normalsize{by \textcolor{DarkBackground}{Phoenix (cddc)} via \textcolor{DarkBackground}{\uline{cheatography.com/26246/cs/7303/}}}
\end{tabulary}
\end{multicols}}

\fancyfoot[L]{ \footnotesize
\noindent
\begin{multicols}{3}
\begin{tabulary}{5.8cm}{LL}
  \SetRowColor{FootBackground}
  \mymulticolumn{2}{p{5.377cm}}{\bf\textcolor{white}{Cheatographer}}  \\
  \vspace{-2pt}Phoenix (cddc) \\
  \uline{cheatography.com/cddc} \\
        \uline{\seqsplit{codenut}.weebly.com}
  \end{tabulary}
\vfill
\columnbreak
\begin{tabulary}{5.8cm}{L}
  \SetRowColor{FootBackground}
  \mymulticolumn{1}{p{5.377cm}}{\bf\textcolor{white}{Cheat Sheet}}  \\
   \vspace{-2pt}Published 23rd February, 2016.\\
   Updated 12th May, 2016.\\
   Page {\thepage} of \pageref{LastPage}.
\end{tabulary}
\vfill
\columnbreak
\begin{tabulary}{5.8cm}{L}
  \SetRowColor{FootBackground}
  \mymulticolumn{1}{p{5.377cm}}{\bf\textcolor{white}{Sponsor}}  \\
  \SetRowColor{white}
  \vspace{-5pt}
  %\includegraphics[width=48px,height=48px]{dave.jpeg}
  Measure your website readability!\\
  www.readability-score.com
\end{tabulary}
\end{multicols}}




\begin{document}
\raggedright
\raggedcolumns

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% www.emerson.emory.edu/services/latex/latex_169.html
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\begin{multicols*}{3}

\begin{tabularx}{5.377cm}{X}
\SetRowColor{DarkBackground}
\mymulticolumn{1}{x{5.377cm}}{\bf\textcolor{white}{Logarithm Rules and Properties}}  \tn
\SetRowColor{LightBackground}
\mymulticolumn{1}{p{5.377cm}}{\vspace{1px}\centerline{\includegraphics[width=5.1cm]{/web/www.cheatography.com/public/uploads/cddc_1456260419_clip_image001[3].png}}} \tn 
\hhline{>{\arrayrulecolor{DarkBackground}}-}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{5.377cm}{X}
\SetRowColor{DarkBackground}
\mymulticolumn{1}{x{5.377cm}}{\bf\textcolor{white}{Summations for Printing}}  \tn
\SetRowColor{LightBackground}
\mymulticolumn{1}{p{5.377cm}}{\vspace{1px}\centerline{\includegraphics[width=5.1cm]{/web/www.cheatography.com/public/uploads/cddc_1456272498_summation_formulae.png}}} \tn 
\hhline{>{\arrayrulecolor{DarkBackground}}-}
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\par\addvspace{1.3em}

\begin{tabularx}{5.377cm}{X}
\SetRowColor{DarkBackground}
\mymulticolumn{1}{x{5.377cm}}{\bf\textcolor{white}{Summation Rules}}  \tn
\SetRowColor{LightBackground}
\mymulticolumn{1}{p{5.377cm}}{\vspace{1px}\centerline{\includegraphics[width=5.1cm]{/web/www.cheatography.com/public/uploads/cddc_1456260571_clip_image004[5].png}}} \tn 
\hhline{>{\arrayrulecolor{DarkBackground}}-}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{5.377cm}{X}
\SetRowColor{DarkBackground}
\mymulticolumn{1}{x{5.377cm}}{\bf\textcolor{white}{Don`t panic}}  \tn
\SetRowColor{LightBackground}
\mymulticolumn{1}{p{5.377cm}}{\vspace{1px}\centerline{\includegraphics[width=5.1cm]{/web/www.cheatography.com/public/uploads/cddc_1456271055_Don't Panic HHG-500x500.png}}} \tn 
\hhline{>{\arrayrulecolor{DarkBackground}}-}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{5.377cm}{x{2.38896 cm} x{2.58804 cm} }
\SetRowColor{DarkBackground}
\mymulticolumn{2}{x{5.377cm}}{\bf\textcolor{white}{Insertion Sort - Algorithm}}  \tn
% Row 0
\SetRowColor{LightBackground}
1  3 7 2 0 & {[}1 3 7{]}2 0 \tn 
% Row Count 1 (+ 1)
% Row 1
\SetRowColor{white}
{[}1{]}3 7 2 0 & {[}1 2 3 7{]} 0 \tn 
% Row Count 2 (+ 1)
% Row 2
\SetRowColor{LightBackground}
{[}1 3{]}7 2 0 & {[}0 1 2 3 7{]} \tn 
% Row Count 3 (+ 1)
\hhline{>{\arrayrulecolor{DarkBackground}}--}
\SetRowColor{LightBackground}
\mymulticolumn{2}{x{5.377cm}}{Starting from one end, ensure the sub-array sorted in each pivot}  \tn 
\hhline{>{\arrayrulecolor{DarkBackground}}--}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{5.377cm}{x{2.38896 cm} x{2.58804 cm} }
\SetRowColor{DarkBackground}
\mymulticolumn{2}{x{5.377cm}}{\bf\textcolor{white}{Selection sort - Algorithm}}  \tn
% Row 0
\SetRowColor{LightBackground}
1  3 7 2 0 & {[}0 1 2{]}3 7 \tn 
% Row Count 1 (+ 1)
% Row 1
\SetRowColor{white}
{[}0{]}1 3 7 2 & {[}0 1 2 3{]}7 \tn 
% Row Count 2 (+ 1)
% Row 2
\SetRowColor{LightBackground}
{[}0 1{]}3 7 2 & {[}0 1 2 3 7{]} \tn 
% Row Count 3 (+ 1)
\hhline{>{\arrayrulecolor{DarkBackground}}--}
\SetRowColor{LightBackground}
\mymulticolumn{2}{x{5.377cm}}{Starting from one end, ensure the sub-array sorted in each pivot}  \tn 
\hhline{>{\arrayrulecolor{DarkBackground}}--}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{5.377cm}{x{2.4885 cm} x{2.4885 cm} }
\SetRowColor{DarkBackground}
\mymulticolumn{2}{x{5.377cm}}{\bf\textcolor{white}{Bubble sort - Algorithm}}  \tn
% Row 0
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1 3 7 2 0 & 1 2(0 3)7 \tn 
% Row Count 1 (+ 1)
% Row 1
\SetRowColor{white}
1 3(2 7)0 & 1(0 2)3 7 \tn 
% Row Count 2 (+ 1)
% Row 2
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1 3 2(0 7) & (0 1)2 3 7 \tn 
% Row Count 3 (+ 1)
% Row 3
\SetRowColor{white}
1(2 3)0 7 & 0 1 2 3 7 \tn 
% Row Count 4 (+ 1)
\hhline{>{\arrayrulecolor{DarkBackground}}--}
\SetRowColor{LightBackground}
\mymulticolumn{2}{x{5.377cm}}{Binary comparison and swap, shift each pivot by at most 1 position in an iteration}  \tn 
\hhline{>{\arrayrulecolor{DarkBackground}}--}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{5.377cm}{x{1.33664 cm} x{1.00248 cm} x{1.00248 cm} x{0.8354 cm} }
\SetRowColor{DarkBackground}
\mymulticolumn{4}{x{5.377cm}}{\bf\textcolor{white}{Sorting Complexities}}  \tn
% Row 0
\SetRowColor{LightBackground}
Algorithm & {\bf{Time}} &  & {\bf{Space}} \tn 
% Row Count 2 (+ 2)
% Row 1
\SetRowColor{white}
 & {\bf{Best}} & {\bf{Worst}} & {\bf{Worst}} \tn 
% Row Count 4 (+ 2)
% Row 2
\SetRowColor{LightBackground}
Bubble Sort & O(n) & O(n\textasciicircum{}2) & O(1) \tn 
% Row Count 5 (+ 1)
% Row 3
\SetRowColor{white}
Insertion Sort & O(n) & O(n\textasciicircum{}2) & O(1) \tn 
% Row Count 7 (+ 2)
% Row 4
\SetRowColor{LightBackground}
Selection Sort & O(n\textasciicircum{}2) & O(n\textasciicircum{}2) & O(1) \tn 
% Row Count 9 (+ 2)
% Row 5
\SetRowColor{white}
Quicksort & O(n log(n)) & O(n\textasciicircum{}2) & \seqsplit{O(log(n))} \tn 
% Row Count 11 (+ 2)
% Row 6
\SetRowColor{LightBackground}
Heapsort & O(n log(n)) & O(n log(n)) & O(1) \tn 
% Row Count 13 (+ 2)
% Row 7
\SetRowColor{white}
Mergesort & O(n log(n)) & O(n log(n)) & O(n) \tn 
% Row Count 15 (+ 2)
\hhline{>{\arrayrulecolor{DarkBackground}}----}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{5.377cm}{X}
\SetRowColor{DarkBackground}
\mymulticolumn{1}{x{5.377cm}}{\bf\textcolor{white}{Asymptotic Notations}}  \tn
% Row 0
\SetRowColor{LightBackground}
\mymulticolumn{1}{x{5.377cm}}{Big-O} \tn 
\mymulticolumn{1}{x{5.377cm}}{\hspace*{6 px}\rule{2px}{6px}\hspace*{6 px}T(n) ∈ O(f(n)) if there are constants c \textgreater{} 0 and n0such that T(n) ≤c f(n) for all n ≥n0} \tn 
% Row Count 3 (+ 3)
% Row 1
\SetRowColor{white}
\mymulticolumn{1}{x{5.377cm}}{Big-Omega} \tn 
\mymulticolumn{1}{x{5.377cm}}{\hspace*{6 px}\rule{2px}{6px}\hspace*{6 px}T(n) ∈ Ω(f(n)) if f(n) ∈O(T(n))} \tn 
% Row Count 5 (+ 2)
% Row 2
\SetRowColor{LightBackground}
\mymulticolumn{1}{x{5.377cm}}{Big-Theta} \tn 
\mymulticolumn{1}{x{5.377cm}}{\hspace*{6 px}\rule{2px}{6px}\hspace*{6 px}T(n) ∈ Θ(f(n)) if T(n) ∈O(f(n)) and T(n) ∈ Ω(f(n))} \tn 
% Row Count 8 (+ 3)
\hhline{>{\arrayrulecolor{DarkBackground}}-}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{5.377cm}{X}
\SetRowColor{DarkBackground}
\mymulticolumn{1}{x{5.377cm}}{\bf\textcolor{white}{Big-O Complexity Chart}}  \tn
\SetRowColor{LightBackground}
\mymulticolumn{1}{p{5.377cm}}{\vspace{1px}\centerline{\includegraphics[width=5.1cm]{/web/www.cheatography.com/public/uploads/cddc_1456261420_big-o-complexity.png}}} \tn 
\hhline{>{\arrayrulecolor{DarkBackground}}-}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{5.377cm}{X}
\SetRowColor{DarkBackground}
\mymulticolumn{1}{x{5.377cm}}{\bf\textcolor{white}{Queue ADT}}  \tn
% Row 0
\SetRowColor{LightBackground}
\mymulticolumn{1}{x{5.377cm}}{Queue property} \tn 
\mymulticolumn{1}{x{5.377cm}}{\hspace*{6 px}\rule{2px}{6px}\hspace*{6 px}FIFO: First In First Out} \tn 
% Row Count 2 (+ 2)
% Row 1
\SetRowColor{white}
\mymulticolumn{1}{x{5.377cm}}{Core operations} \tn 
\mymulticolumn{1}{x{5.377cm}}{\hspace*{6 px}\rule{2px}{6px}\hspace*{6 px}- enqueue – dequeue – is\_empty} \tn 
% Row Count 4 (+ 2)
\hhline{>{\arrayrulecolor{DarkBackground}}-}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{5.377cm}{X}
\SetRowColor{DarkBackground}
\mymulticolumn{1}{x{5.377cm}}{\bf\textcolor{white}{Priority Queue ADT}}  \tn
% Row 0
\SetRowColor{LightBackground}
\mymulticolumn{1}{x{5.377cm}}{Queue property} \tn 
\mymulticolumn{1}{x{5.377cm}}{\hspace*{6 px}\rule{2px}{6px}\hspace*{6 px}Lower Priority Value Out  First} \tn 
% Row Count 2 (+ 2)
% Row 1
\SetRowColor{white}
\mymulticolumn{1}{x{5.377cm}}{Core operations} \tn 
\mymulticolumn{1}{x{5.377cm}}{\hspace*{6 px}\rule{2px}{6px}\hspace*{6 px}- insert –deleteMin –isEmpty} \tn 
% Row Count 4 (+ 2)
\hhline{>{\arrayrulecolor{DarkBackground}}-}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{5.377cm}{X}
\SetRowColor{DarkBackground}
\mymulticolumn{1}{x{5.377cm}}{\bf\textcolor{white}{Stack ADT}}  \tn
% Row 0
\SetRowColor{LightBackground}
\mymulticolumn{1}{x{5.377cm}}{Stack property} \tn 
\mymulticolumn{1}{x{5.377cm}}{\hspace*{6 px}\rule{2px}{6px}\hspace*{6 px}LIFO: Last In First Out} \tn 
% Row Count 2 (+ 2)
% Row 1
\SetRowColor{white}
\mymulticolumn{1}{x{5.377cm}}{Core operations} \tn 
\mymulticolumn{1}{x{5.377cm}}{\hspace*{6 px}\rule{2px}{6px}\hspace*{6 px}– push – pop – top – is\_empty} \tn 
% Row Count 4 (+ 2)
\hhline{>{\arrayrulecolor{DarkBackground}}-}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{5.377cm}{x{1.64241 cm} x{3.33459 cm} }
\SetRowColor{DarkBackground}
\mymulticolumn{2}{x{5.377cm}}{\bf\textcolor{white}{d-Heap ADT}}  \tn
% Row 0
\SetRowColor{LightBackground}
child & (i-1)*d+2 \textasciitilde{} i*d+1 \tn 
% Row Count 1 (+ 1)
% Row 1
\SetRowColor{white}
parent & ⌊(i-2)/d⌋+1 \tn 
% Row Count 2 (+ 1)
% Row 2
\SetRowColor{LightBackground}
root & 1 \tn 
% Row Count 3 (+ 1)
% Row 3
\SetRowColor{white}
next free & size+1 \tn 
% Row Count 4 (+ 1)
\hhline{>{\arrayrulecolor{DarkBackground}}--}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{5.377cm}{X}
\SetRowColor{DarkBackground}
\mymulticolumn{1}{x{5.377cm}}{\bf\textcolor{white}{(Min) Heap Tree - ADT}}  \tn
% Row 0
\SetRowColor{LightBackground}
\mymulticolumn{1}{x{5.377cm}}{Heap-order property} \tn 
\mymulticolumn{1}{x{5.377cm}}{\hspace*{6 px}\rule{2px}{6px}\hspace*{6 px}parent's key \textless{}= children's keys} \tn 
% Row Count 2 (+ 2)
% Row 1
\SetRowColor{white}
\mymulticolumn{1}{x{5.377cm}}{Structure property} \tn 
\mymulticolumn{1}{x{5.377cm}}{\hspace*{6 px}\rule{2px}{6px}\hspace*{6 px}nearly complete tree} \tn 
% Row Count 4 (+ 2)
% Row 2
\SetRowColor{LightBackground}
\mymulticolumn{1}{x{5.377cm}}{Heapify algorithm} \tn 
\mymulticolumn{1}{x{5.377cm}}{\hspace*{6 px}\rule{2px}{6px}\hspace*{6 px}Heapify the tree from bottom up, percolate DOWN a node as deep as needed for each node.} \tn 
% Row Count 7 (+ 3)
\hhline{>{\arrayrulecolor{DarkBackground}}-}
\SetRowColor{LightBackground}
\mymulticolumn{1}{x{5.377cm}}{Binary Heap Operations Complexity: \newline Heapify - O(n) \newline Find Min - O(1) \newline Insert - O(log(n)) \newline Delete - O(log(n))}  \tn 
\hhline{>{\arrayrulecolor{DarkBackground}}-}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{5.377cm}{X}
\SetRowColor{DarkBackground}
\mymulticolumn{1}{x{5.377cm}}{\bf\textcolor{white}{Loop -\textgreater{} (Tail) Recursion}}  \tn
\SetRowColor{LightBackground}
\mymulticolumn{1}{x{5.377cm}}{//Loop \newline int i = 0; \newline while (i \textless{} n) \newline   doFoo(i); \newline   i++; \newline  \newline //Recursion \newline void recDoFoo(int i, int n)\{ \newline   if (i \textless{} n) \{ \newline     doFoo(i); \newline     recDoFoo(i + 1, n);\} \newline \} \newline recDoFoo(0, n);} \tn 
\hhline{>{\arrayrulecolor{DarkBackground}}-}
\SetRowColor{LightBackground}
\mymulticolumn{1}{x{5.377cm}}{Equivalent for loop: \newline for (int i=0; i\textless{}n; i++) doFoo(i);}  \tn 
\hhline{>{\arrayrulecolor{DarkBackground}}-}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{5.377cm}{X}
\SetRowColor{DarkBackground}
\mymulticolumn{1}{x{5.377cm}}{\bf\textcolor{white}{Tail Recursion -\textgreater{} Iteration}}  \tn
\SetRowColor{LightBackground}
\mymulticolumn{1}{x{5.377cm}}{//Tail Recursion \newline int fact\_acc (int n, int acc) \{ \newline   if (n) \newline     return fact\_acc(n –1, acc {\emph{ n); \newline   return acc; \newline \} \newline //Iteration \newline int fact\_acc (int n, int acc) \{ \newline   for(;n;n-{}-) \newline     acc = acc }} n; \newline return acc; \newline \}} \tn 
\hhline{>{\arrayrulecolor{DarkBackground}}-}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{5.377cm}{X}
\SetRowColor{DarkBackground}
\mymulticolumn{1}{x{5.377cm}}{\bf\textcolor{white}{Recurrence Simplification Strategy}}  \tn
% Row 0
\SetRowColor{LightBackground}
\mymulticolumn{1}{x{5.377cm}}{1) Find T(n) for the base cases.} \tn 
% Row Count 1 (+ 1)
% Row 1
\SetRowColor{white}
\mymulticolumn{1}{x{5.377cm}}{2) Expand T(n) for the general patterns.} \tn 
% Row Count 2 (+ 1)
% Row 2
\SetRowColor{LightBackground}
\mymulticolumn{1}{x{5.377cm}}{3) Drive the recursive term to the base case.} \tn 
% Row Count 3 (+ 1)
% Row 3
\SetRowColor{white}
\mymulticolumn{1}{x{5.377cm}}{4) Solve and represent k with n by inequality(equality).} \tn 
% Row Count 5 (+ 2)
% Row 4
\SetRowColor{LightBackground}
\mymulticolumn{1}{x{5.377cm}}{5) Substitute back to T(n).} \tn 
% Row Count 6 (+ 1)
\hhline{>{\arrayrulecolor{DarkBackground}}-}
\SetRowColor{LightBackground}
\mymulticolumn{1}{x{5.377cm}}{e.g. \newline T{[}n \textless{}= n0{]} = C1 \newline T{[}n{]}= T{[}n/2{]}+C2 \newline -\textgreater{} T{[}n{]} = T{[}n*(1/2){]}+C2  \newline -\textgreater{} T{[}n{]} = T{[}n*(1/2)\textasciicircum{}k{]} + k*C2 \newline  \newline Let n*(1/2)\textasciicircum{}k \textless{}= n0  -\textgreater{} T(n*(1/2)\textasciicircum{}k) = C1 \newline  \newline -\textgreater{} n*(1/2)\textasciicircum{}k * (2)\textasciicircum{}k \textless{}= n0* (2)\textasciicircum{}k \newline -\textgreater{} n/n0 \textless{}= (2)\textasciicircum{}k \newline -\textgreater{} log2(n/n0) \textless{}= log2((2)\textasciicircum{}k) \newline -\textgreater{} log2(n/n0) \textless{}= k \newline  \newline -\textgreater{} T{[}n{]} = C1+k*C2 = C1 + C2*log2(n/n0)}  \tn 
\hhline{>{\arrayrulecolor{DarkBackground}}-}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{5.377cm}{X}
\SetRowColor{DarkBackground}
\mymulticolumn{1}{x{5.377cm}}{\bf\textcolor{white}{Summation Fomulas}}  \tn
\SetRowColor{LightBackground}
\mymulticolumn{1}{p{5.377cm}}{\vspace{1px}\centerline{\includegraphics[width=5.1cm]{/web/www.cheatography.com/public/uploads/cddc_1456260490_image[8].png}}} \tn 
\hhline{>{\arrayrulecolor{DarkBackground}}-}
\end{tabularx}
\par\addvspace{1.3em}


% That's all folks
\end{multicols*}

\end{document}