17. Notions, Concepts

Stratégies

Il n’existe pas toujours de solution parfaite ou de solution parfaite qui puisse être mise en oeuvre en un temps raisonnable. Une solution est souvent un assemblage de briques. Certains reviennent plus souvent que d’autres comme trier les éléments, procéder par récurrence (ajouter de nouveaux éléments un par un en adaptant la meilleure solution).

On a utilise parfois un langage guerrier quand on parle d’un problème. Comment l’attaquer ? Il faut une stratégie pour le résoudre, un plan de bataille. Il faut le couper en petits bouts et chercher une solution pour chacun de ses bouts.

Les informaticiens aiment aussi beaucoup le terme heuristique. C’est une sorte de règle dont on sait qu’elle marche bien pour un problème donné.

Propriété locale - globale

C’est une façon de découper une problème. Une solution à un problème n’est pas toujours facile à comprendre dans son ensemble mais elle est plus facile à interpréter localement.

Lorsqu’un ensemble d’objets est trié, cela veut dire que si on prendre deux éléments rangés côte à côte, ils sont toujours dans le bon ordre. Une idée pour trier des éléments est de faire respecter cette propriété locale : et d’inverser deux éléments proches qui sont mal rangés. C’est l’idée du tri à bulles qui permet effectivement d’obtenir un tableau trié.

Algorithme optimal

On parle d”algorithme optimal pour un problème données lorsque celui-ci produit une solution et qu’il n’est pas possible de produire cette même solution plus rapidement.

On parle de solution optimale à un problème lorsqu’il n’existe pas de meilleure solution à ce problème.

Pour la grande majorité de problème, on sait écrire un algorithme qui calcule la solution optimale. Mais ce n’est pas forcément l’algorithme optimal. On ne sait pas toujours dire si un algorithme est optimal. Dans la grande majorité, on fabrique des algorithmes rapides qui produisent des solutions approchées mais tout-à-fait satsifaisante.

Concept clés d’un algorithme

  • La séquence - on effectue des tâches les unes à la suite des autres.
  • Le choix - on fait telle ou telle chose selon une certaine condition.
  • La répétition - on recommence jusqu’à ce qu’on ait fini (mais entre-temps le résultat final se construit petit à petit). On parle aussi de boucles.
  • Variables - pour mémoriser l’état de l’algorithme.
  • Données - les données que manipule l’algorithme (les nombres qu’il trie, …)
  • Résultats - quelque chose à partir des données.
  • Coût - ou sa vitesse, le coût d’un algorithme est le nombre d’opérations élémentaires nécessaires pour qu’il se termine (une opération numérique, la consultation d’une variable, un test…) On le calcule rarement précisément mais on veur toujours avoir un ordre de grandeur : si mon problème est dix fois plus grand, l’algorithme sera combien sera combien de fois plus lent ?

Par exemple, le tri le plus rapide est en

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est le nombre d’éléments à trier. La fonction

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ou logarithme détermine le nombre de fois qu’il a fallu multiplier 10 pour obtenir

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. Par exemple,

System Message: WARNING/2 (1000=10*10*10)

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donc

System Message: WARNING/2 (\log(1000)=3)

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.

Donc, pour trier

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éléments, il faut environ

System Message: WARNING/2 (100 \log(100) = 100 * 2 = 200)

latex exited with error [stdout] This is pdfTeX, Version 3.14159265-2.6-1.40.17 (TeX Live 2016/Debian) (preloaded format=latex) restricted \write18 enabled. entering extended mode (./math.tex LaTeX2e <2017/01/01> patch level 3 Babel <3.9r> and hyphenation patterns for 32 language(s) loaded. (/usr/share/texlive/texmf-dist/tex/latex/base/article.cls Document Class: article 2014/09/29 v1.4h Standard LaTeX document class (/usr/share/texlive/texmf-dist/tex/latex/base/size12.clo)) (/usr/share/texlive/texmf-dist/tex/latex/base/inputenc.sty (/usr/share/texlive/texmf-dist/tex/latex/ucs/utf8x.def)) (/usr/share/texlive/texmf-dist/tex/latex/ucs/ucs.sty (/usr/share/texlive/texmf-dist/tex/latex/ucs/data/uni-global.def)) (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsmath.sty For additional information on amsmath, use the `?' option. (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amstext.sty (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsgen.sty)) (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsbsy.sty) (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsopn.sty)) (/usr/share/texlive/texmf-dist/tex/latex/amscls/amsthm.sty) (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/amssymb.sty (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/amsfonts.sty)) (/usr/share/texlive/texmf-dist/tex/latex/anyfontsize/anyfontsize.sty) (/usr/share/texlive/texmf-dist/tex/latex/tools/bm.sty) (/usr/share/texlive/texmf-dist/tex/latex/eepic/epic.sty Enhancements to Picture Environment. Version 1.2 - Released June 1, 1986 ) (/usr/share/texlive/texmf-dist/tex/generic/xypic/xy.sty (/usr/share/texlive/texmf-dist/tex/generic/xypic/xy.tex Bootstrap'ing: catcodes, docmode, (/usr/share/texlive/texmf-dist/tex/generic/xypic/xyrecat.tex ) (/usr/share/texlive/texmf-dist/tex/generic/xypic/xyidioms.tex) Xy-pic version 3.8.9 <2013/10/06> Copyright (c) 1991-2013 by Kristoffer H. Rose <krisrose@tug.org> and others Xy-pic is free software: see the User's Guide for details. Loading kernel: messages; fonts; allocations: state, direction, utility macros; pictures: \xy, positions, objects, decorations; kernel objects: directionals, circles, text; options; algorithms: directions, edges, connections; Xy-pic loaded) (/usr/share/texlive/texmf-dist/tex/generic/oberdiek/ifpdf.sty) (/usr/share/texlive/texmf-dist/tex/generic/xypic/xyall.tex Xy-pic option: All features v.3.8 (/usr/share/texlive/texmf-dist/tex/generic/xypic/xycurve.tex Xy-pic option: Curve and Spline extension v.3.12 curve, circles, loaded) (/usr/share/texlive/texmf-dist/tex/generic/xypic/xyframe.tex Xy-pic option: Frame and Bracket extension v.3.14 loaded) (/usr/share/texlive/texmf-dist/tex/generic/xypic/xycmtip.tex Xy-pic option: Computer Modern tip extension v.3.7 (/usr/share/texlive/texmf-dist/tex/generic/xypic/xytips.tex Xy-pic option: More Tips extension v.3.11 loaded) loaded) (/usr/share/texlive/texmf-dist/tex/generic/xypic/xyline.tex Xy-pic option: Line styles extension v.3.10 loaded) (/usr/share/texlive/texmf-dist/tex/generic/xypic/xyrotate.tex Xy-pic option: Rotate and Scale extension v.3.8 loaded) (/usr/share/texlive/texmf-dist/tex/generic/xypic/xycolor.tex Xy-pic option: Colour extension v.3.11 loaded) (/usr/share/texlive/texmf-dist/tex/generic/xypic/xymatrix.tex Xy-pic option: Matrix feature v.3.14 loaded) (/usr/share/texlive/texmf-dist/tex/generic/xypic/xyarrow.tex Xy-pic option: Arrow and Path feature v.3.9 path, \ar, loaded) (/usr/share/texlive/texmf-dist/tex/generic/xypic/xygraph.tex Xy-pic option: Graph feature v.3.11 loaded) loaded)) (/usr/share/texlive/texmf-dist/tex/latex/movie15/movie15.sty (/usr/share/texlive/texmf-dist/tex/latex/graphics/keyval.sty) (/usr/share/texlive/texmf-dist/tex/latex/base/ifthen.sty) (/usr/share/texlive/texmf-dist/tex/latex/oberdiek/ifdraft.sty) (/usr/share/texlive/texmf-dist/tex/latex/ms/everyshi.sty)) (./math.aux) (/usr/share/texlive/texmf-dist/tex/latex/ucs/ucsencs.def) ! Package movie15 Error: Package `hyperref' has not been loaded yet . See the movie15 package documentation for explanation. Type H <return> for immediate help. ... l.44 \begin{document} ! Undefined control sequence. \@begindocumenthook ...hyperref}}\ifHy@colorlinks \gdef \@MXV@pdfborder {/Bo... l.44 \begin{document} ! Extra \else. \@begindocumenthook ...der {/Border [0 0 0]}\else \gdef \@MXV@pdfborder {}\f... l.44 \begin{document} ! Extra \fi. \@begindocumenthook ...gdef \@MXV@pdfborder {}\fi \@EveryShipout@Init l.44 \begin{document} ABD: EveryShipout initializing macros (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/umsa.fd) (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/umsb.fd) Package movie15 Warning: @@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ (movie15) @@ Package `movie15' is obsolete and @@ (movie15) @@ superseded by `media9'. @@ (movie15) @@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@. [1] (./math.aux) ) (see the transcript file for additional information) Output written on math.dvi (1 page, 292 bytes). Transcript written on math.log.
. Pour trier 10 fois plus d’éléments, il faut :

System Message: WARNING/2 (1000 \log(1000) = 3000)

latex exited with error [stdout] This is pdfTeX, Version 3.14159265-2.6-1.40.17 (TeX Live 2016/Debian) (preloaded format=latex) restricted \write18 enabled. entering extended mode (./math.tex LaTeX2e <2017/01/01> patch level 3 Babel <3.9r> and hyphenation patterns for 32 language(s) loaded. (/usr/share/texlive/texmf-dist/tex/latex/base/article.cls Document Class: article 2014/09/29 v1.4h Standard LaTeX document class (/usr/share/texlive/texmf-dist/tex/latex/base/size12.clo)) (/usr/share/texlive/texmf-dist/tex/latex/base/inputenc.sty (/usr/share/texlive/texmf-dist/tex/latex/ucs/utf8x.def)) (/usr/share/texlive/texmf-dist/tex/latex/ucs/ucs.sty (/usr/share/texlive/texmf-dist/tex/latex/ucs/data/uni-global.def)) (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsmath.sty For additional information on amsmath, use the `?' option. (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amstext.sty (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsgen.sty)) (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsbsy.sty) (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsopn.sty)) (/usr/share/texlive/texmf-dist/tex/latex/amscls/amsthm.sty) (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/amssymb.sty (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/amsfonts.sty)) (/usr/share/texlive/texmf-dist/tex/latex/anyfontsize/anyfontsize.sty) (/usr/share/texlive/texmf-dist/tex/latex/tools/bm.sty) (/usr/share/texlive/texmf-dist/tex/latex/eepic/epic.sty Enhancements to Picture Environment. Version 1.2 - Released June 1, 1986 ) (/usr/share/texlive/texmf-dist/tex/generic/xypic/xy.sty (/usr/share/texlive/texmf-dist/tex/generic/xypic/xy.tex Bootstrap'ing: catcodes, docmode, (/usr/share/texlive/texmf-dist/tex/generic/xypic/xyrecat.tex ) (/usr/share/texlive/texmf-dist/tex/generic/xypic/xyidioms.tex) Xy-pic version 3.8.9 <2013/10/06> Copyright (c) 1991-2013 by Kristoffer H. Rose <krisrose@tug.org> and others Xy-pic is free software: see the User's Guide for details. Loading kernel: messages; fonts; allocations: state, direction, utility macros; pictures: \xy, positions, objects, decorations; kernel objects: directionals, circles, text; options; algorithms: directions, edges, connections; Xy-pic loaded) (/usr/share/texlive/texmf-dist/tex/generic/oberdiek/ifpdf.sty) (/usr/share/texlive/texmf-dist/tex/generic/xypic/xyall.tex Xy-pic option: All features v.3.8 (/usr/share/texlive/texmf-dist/tex/generic/xypic/xycurve.tex Xy-pic option: Curve and Spline extension v.3.12 curve, circles, loaded) (/usr/share/texlive/texmf-dist/tex/generic/xypic/xyframe.tex Xy-pic option: Frame and Bracket extension v.3.14 loaded) (/usr/share/texlive/texmf-dist/tex/generic/xypic/xycmtip.tex Xy-pic option: Computer Modern tip extension v.3.7 (/usr/share/texlive/texmf-dist/tex/generic/xypic/xytips.tex Xy-pic option: More Tips extension v.3.11 loaded) loaded) (/usr/share/texlive/texmf-dist/tex/generic/xypic/xyline.tex Xy-pic option: Line styles extension v.3.10 loaded) (/usr/share/texlive/texmf-dist/tex/generic/xypic/xyrotate.tex Xy-pic option: Rotate and Scale extension v.3.8 loaded) (/usr/share/texlive/texmf-dist/tex/generic/xypic/xycolor.tex Xy-pic option: Colour extension v.3.11 loaded) (/usr/share/texlive/texmf-dist/tex/generic/xypic/xymatrix.tex Xy-pic option: Matrix feature v.3.14 loaded) (/usr/share/texlive/texmf-dist/tex/generic/xypic/xyarrow.tex Xy-pic option: Arrow and Path feature v.3.9 path, \ar, loaded) (/usr/share/texlive/texmf-dist/tex/generic/xypic/xygraph.tex Xy-pic option: Graph feature v.3.11 loaded) loaded)) (/usr/share/texlive/texmf-dist/tex/latex/movie15/movie15.sty (/usr/share/texlive/texmf-dist/tex/latex/graphics/keyval.sty) (/usr/share/texlive/texmf-dist/tex/latex/base/ifthen.sty) (/usr/share/texlive/texmf-dist/tex/latex/oberdiek/ifdraft.sty) (/usr/share/texlive/texmf-dist/tex/latex/ms/everyshi.sty)) (./math.aux) (/usr/share/texlive/texmf-dist/tex/latex/ucs/ucsencs.def) ! Package movie15 Error: Package `hyperref' has not been loaded yet . See the movie15 package documentation for explanation. Type H <return> for immediate help. ... l.44 \begin{document} ! Undefined control sequence. \@begindocumenthook ...hyperref}}\ifHy@colorlinks \gdef \@MXV@pdfborder {/Bo... l.44 \begin{document} ! Extra \else. \@begindocumenthook ...der {/Border [0 0 0]}\else \gdef \@MXV@pdfborder {}\f... l.44 \begin{document} ! Extra \fi. \@begindocumenthook ...gdef \@MXV@pdfborder {}\fi \@EveryShipout@Init l.44 \begin{document} ABD: EveryShipout initializing macros (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/umsa.fd) (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/umsb.fd) Package movie15 Warning: @@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ (movie15) @@ Package `movie15' is obsolete and @@ (movie15) @@ superseded by `media9'. @@ (movie15) @@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@. [1] (./math.aux) ) (see the transcript file for additional information) Output written on math.dvi (1 page, 236 bytes). Transcript written on math.log.
. Trier 1000 éléments est 15 fois plus lent que trier 100 éléments.

Vocabulaire

Voir Glossaire.