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0000000..a8d329d --- /dev/null +++ b/SPE/OPT/TP 3/Questions.tex @@ -0,0 +1,35 @@ +\documentclass[a4paper,10pt]{article} +%\documentclass[a4paper,10pt]{scrartcl} +\usepackage{mathtools} + +\usepackage[utf8]{inputenc} +\renewcommand{\thesection}{\Roman{section}} +\newcommand{\sectioni}[2]{\setcounter{section}{#1}\addtocounter{section}{-1}\section{#2}} +\newcommand{\subsectioni}[2]{\setcounter{subsection}{#1}\addtocounter{subsection}{-1}\subsection{#2}} +\DeclarePairedDelimiter\ceil{\lceil}{\rceil} +\DeclarePairedDelimiter\floor{\lfloor}{\rfloor} +\title{Questions du TP 3} +\author{} +\date{} + +\pdfinfo{% + /Title (Questions du TP 3) + /Author () + /Creator () + /Producer () + /Subject () + /Keywords () +} + +\begin{document} + \maketitle + \sectioni{3}{Arbres bouc-émissaires} + \begin{enumerate} + \setcounter{enumi}{11} + \item En notant $h$ la hauteur de l'arbre traité, s'il n'y a pas reconstruction, on effectue une opération en $O(1)$ et un appel récursif sur un abre de hauteur $h-1$. Ainsi, en notant $c_h$ la complexité de l'appel insere\_be sur un arbre de hauteur $h$ dans le pire des cas, c'est à dire lorsque l'appel parcourt la branche la plus longue (de longueur $h$), on a : $$\left\{\begin{array}{ll} c_{-1}= O(1) \\ c_h = O(1) + c_{h-1} \end{array}\right.$$ Donc $c_h = O(h)$. Or $h\leq \floor{\log_{\frac{1}{\alpha}}(n)} + 1$ donc $c_h = O(lg(n))$ + $$ $$ + S'il y a reconstruction, dans le pire des cas, cette reconstruction s'effectue sur l'arbre entier en $O(n)$ puis puisque la reconstruction crée un arbre dont tous les noeuds sont équilibrés, il n'y aura donc pas d'autre reconstruction. Ainsi en notant $C_h$ la complexité d'un appel entrainant une reconstruction, on a : $$C_h = O(n) + c_h = O(n) + O(lg(n)) = O(n)$$ + CQFD (ça va, c'est pas si pénible que ça les maths (de ce niveau là en tout cas)) + \end{enumerate} + +\end{document} diff --git a/SPE/OPT/TP 3/questions.kilepr b/SPE/OPT/TP 3/questions.kilepr new file mode 100644 index 0000000..9f5da29 --- /dev/null +++ b/SPE/OPT/TP 3/questions.kilepr @@ -0,0 +1,31 @@ +[General] +bib_extensions=.bib +bibliographyBackendAutoDetected= +bibliographyBackendUserOverride= +def_graphic_ext=eps +img_extIsRegExp=false +img_extensions=.eps .jpg .jpeg .png .pdf .ps .fig .gif +kileprversion=3 +kileversion=2.9.93 +masterDocument= +name=Questions +pkg_extIsRegExp=false +pkg_extensions=.cls .sty .bbx .cbx .lbx +src_extIsRegExp=false +src_extensions=.tex .ltx .latex .dtx .ins + +[Tools] +MakeIndex= +QuickBuild= + +[item:Questions.tex] +archive=true +encoding= +highlight= +mode= + +[item:questions.kilepr] +archive=true +encoding= +highlight= +mode=