From 8d2d0e49dad935f7902fa5dd46fd6e62deb2ba15 Mon Sep 17 00:00:00 2001 From: pezy Date: Wed, 13 May 2015 20:12:10 +0800 Subject: [PATCH] finished 2.2-2 --- Foundations/overview/README.md | 8 +- Foundations/overview/selection-sort.pdf | Bin 73794 -> 153511 bytes Foundations/overview/selection-sort.tex | 139 +++++++++++++++++++++--- 3 files changed, 132 insertions(+), 15 deletions(-) diff --git a/Foundations/overview/README.md b/Foundations/overview/README.md index 4f86c83..9852935 100644 --- a/Foundations/overview/README.md +++ b/Foundations/overview/README.md @@ -131,5 +131,11 @@ appear in A. ### 2.2-2 > Consider sorting `n` numbers stored in array `A` by first finding the smallest element of `A` and exchanging it with the element in `A[1]`. Then find the second smallest element of `A`, and exchange it with `A[2]`􏰃. Continue in this manner for the first `n-1` elements of `A`. Write pseudocode for this algorithm, which is known as **selection sort**. What loop invariant does this algorithm maintain? Why does it need to run for only the first `n-1` elements, rather than for all `n` elements? Give the best-case and worst-case running times of selection sort in O-notation. -[pseudocode](selection-sort.pdf) | [code](insertion_sort.py) | +**loop invariant**: At the start of each iteration `for` loop, the subarray `A[1..j-1]` consists of the smallest elements of the array, and in sorted order. + +Because we always find the smallest element of `A`, when the first `n-1` elements in sorted order. The last element must bigger than before. Thus, we don't need to run for all `n` elements. + +[O-notation and pseudocode](selection-sort.pdf) | [code](insertion_sort.py) | [unittest](insertion_sort_unittest.py) + +Selection sort is always a ![image](https://cloud.githubusercontent.com/assets/1147451/7592456/dd19381e-f905-11e4-8089-8ec444fca706.png) algorithm. diff --git a/Foundations/overview/selection-sort.pdf b/Foundations/overview/selection-sort.pdf index f106e8fd2b1b079ffdaa5f6bfaba8cb2bfa98fc9..8fed232b014e33f4ef286e6be137fb853efb6e3e 100644 GIT binary patch delta 128591 zcmbTdbwE_n_b*CG2_jt*(j7A}Fr;)N9nzhGbO{F(BqXH-Bt%p|kPwiNZlt>r6e&UJ zhIeMrum0Zs-QT$0YpIoVV)x%M_Ow;$LjTuC4 z_4fzu&St{tvL5b=C*q8RLUFo(i4f`#n@6H4-I=uZO@62dU!QtLh>>tV7gAK!%O6N1Lz)NrCgD{7sHgK$RgGq_X_oy7(S#~x2LTD_>!d5_$0WE z=v}qDigoY3+*{W*WZk|R8mrnSZo7*Ne0X|%erL^dO;Rw1x6<96Wfa(^POg|mbGnIR z>8ZBM(h?f{9iO(>^)m$no5EtSy~64|M4zz3#L|DmRs!{N_LLpb{ujEWTT|O*|XfdGl>_^tw>LW8FfH?>&5At0UcMm9B}QS30Y{t9EhA@Ije0OS#z-+fJj>J`xE=2Gd(>YqA)# 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b/Foundations/overview/selection-sort.tex @@ -1,20 +1,131 @@ -\documentclass{article} +\documentclass[twoside]{article} +\setlength{\oddsidemargin}{0.25 in} +\setlength{\evensidemargin}{-0.25 in} +\setlength{\topmargin}{-0.6 in} +\setlength{\textwidth}{6.5 in} +\setlength{\textheight}{8.5 in} +\setlength{\headsep}{0.75 in} +\setlength{\parindent}{0 in} +\setlength{\parskip}{0.1 in} + \usepackage{clrscode3e} +\usepackage{multicol} +\usepackage{amsmath} + \begin{document} \title{2.2-2} \author{pezy} \maketitle - \begin{codebox} - \Procname{$\proc{Selection-Sort}(A)$} - \li \For $j \gets 1$ \To $\attrib{A}{length} - 1$ - \li \Do $min \gets j$ - \li \For $i \gets j + 1$ \To $\attrib{A}{length}$ - \li \Do \If $A[i] < A[min]$ - \li \Do $min \gets i$ - \End - \End - \li \func{swap}($A[min], A[j]$) - \End - \li \Return $A$ - \end{codebox} + \begin{multicols}{3} + \begin{codebox} + \Procname{$\proc{Selection-Sort}(A)$} + \li \For $j \gets 1$ \To $\attrib{A}{length} - 1$ + \li \Do $min \gets j$ + \li \For $i \gets j + 1$ \To $\attrib{A}{length}$ + \li \Do \If $A[i] < A[min]$ + \li \Do $min \gets i$ + \End + \End + \li \func{swap}($A[min], A[j]$) + \End + \end{codebox} + + \columnbreak + + \begin{center} % column 2 holds cost of each statement + \textbf{\large Cost} + \\*$c_{1}$ + \\*$c_{2}$ + \\*$c_{3}$ + \\*$c_{4}$ + \\*$c_{5}$ + \\*$c_{6}$ + \end{center} + + \columnbreak + + \begin{flushleft} % displays run time of each statement + \textbf{\large Time} + \\* $n$ + \\* $n-1$ + \\* $\sum _{ j=1 }^{ n-1 } t_j $ + \\* $\sum _{ j=1 }^{ n-1 } (t_j - 1) $ + \\* $0 \To \sum _{ j=1 }^{ n-1 } (t_j - 1)$ + \\* $n-1$ + \end{flushleft} + \end{multicols} + + We will now calculate the running time, $T(n)$, of \proc{Selection-Sort}: + + \begin{align} + \notag + T(n) &= c_1 n + c_2 (n-1) + c_3 \sum _{j=1}^{n-1} t_j + c_4 \sum _{j=1}^{n-1} ( t_j -1 ) + k c_5 \sum _{j=1}^{n-1} (t_j -1) + c_6 (n-1),\\ + \notag + &= c_1 n+ (c_2 + c_6)(n-1) + c_3 \sum _{j=1}^{n-1} t_j + (c_4 + k c_5) \sum _{j=1}^{n-1} (t_j -1),\\ + \notag + &= c_1 n+ (c_2 + c_6)(n-1) + (c_3 + c_4 + k c_5 ) \sum _{j=1}^{n-1} t_j - (c_4 + k c_5) (n-1),\\ + \label{eq:Tn} + &= c_1 n +(c_2 + c_6 - c_4 - k c_5) (n-1) + (c_3 + c_4 + k c_5 ) \sum _{j=1}^{n-1} t_j. + \end{align} + + \section{Best case running time} + In the {\bf BEST CASE} running time, the list of input will already be sorted. Thus, the body of \If is never step in, and $k = 0$. we obtain that $t_j = j+1$, for every choice of j. Thus, + + \[ + \sum_{j=1}^{n-1} t_j = \frac{1}{2} n(n+1) - 1 = (\frac{n}{2}+1)(n-1) + \] + + Substituting this into the last term of Eqn.~(\ref{eq:Tn}) yields, + \begin{align} + T(n) &= c_1 n +(c_2 + c_6 - c_4)(n-1) + (c_3 + c_4)(\frac{n}{2}+1)(n-1) \\ + &= \frac{c_3+c_4}{2}n^2 + (c_1+c_2+\frac{c_3}{2}-\frac{c_4}{2}+c_6)n - (c_2+c_3+c_6) + \end{align} + + which can be simplified to the linear equation $T(n) = An^2+Bn+C$ where + \begin{align*} + A &= \frac{c_3+c_4}{2} > 0,\\ + B &= c_1+c_2+\frac{c_3}{2}-\frac{c_4}{2}+c_6, \quad\text{and,}\\ + C &= -c_2-c_3-c_6 < 0. + \end{align*} + + Therefore, the {\bf BEST CASE} running time of the \proc{Selection-Sort} Algorithm equals: + \begin{center} + $\boldsymbol{T(n) = An^2+Bn+C}$ + \end{center} + + % + %Begin section for Worst Case Running Time + % + % + \section{Worst case running time} + We will now look at the {\bf WORST CASE} for \proc{Selection-Sort}: + \begin{itemize} + \item In the worst case, the \If statement is invoked on every occasion. + \item This means $ k = 1$ + \end{itemize} + + Substituting $t_{ j }$ with $j$ into the last summation in Eqn.~(\ref{eq:Tn}) yields, + + \[ + \sum_{j=1}^{n-1} t_j = \frac{1}{2} n(n+1) - 1 = (\frac{n}{2}+1)(n-1) + \] + + Thus, Eqn.~(\ref{eq:Tn}) becomes, + \begin{align*} + T(n) &= c_1 n +(c_2+c_6-c_4-c_5)(n-1) + (c_3+c_4+c_5)(\frac{n}{2}+1)(n-1),\\ + &= \frac{c_3+c_4+c_5}{2} n^2 + (c_1+c_2+\frac{c_3}{2}-\frac{c_4}{2}-\frac{c_5}{2}+c_6) n - (c_2+c_3+c_6) + \end{align*} + a \emph{quadratic function} of $n$, the input sequence length, where, + \begin{align*} + A' &= \frac{c_3+c_4+c_5}{2} > 0,\\ + B' &= c_1+c_2+\frac{c_3}{2}-\frac{c_4}{2}-\frac{c_5}{2}+c_6, \quad\text{and,}\\ + C' &= -c_{2}-c_{3}-c_{6} < 0. + \end{align*} + + + Therefore, the {\bf WORST CASE} running time of the \proc{Selection-Sort} Algorithm also equals: + \begin{center} + $\boldsymbol{T(n) = An^2+Bn+C}$ + \end{center} + \end{document} \ No newline at end of file