From faa8419df5a8d17a44a87a25f9fe5aa9fefb9a1f Mon Sep 17 00:00:00 2001 From: team3 Date: Tue, 21 Jul 2026 09:15:32 +0200 Subject: [PATCH] update --- lernmaterial/cheatsheet.aux | 2 +- lernmaterial/cheatsheet.log | 14 +++---- lernmaterial/cheatsheet.pdf | Bin 201458 -> 219818 bytes lernmaterial/cheatsheet.tex | 71 ++++++++++++++++++++++++++++++++++++ 4 files changed, 79 insertions(+), 8 deletions(-) diff --git a/lernmaterial/cheatsheet.aux b/lernmaterial/cheatsheet.aux index 9752b69..4d2adf2 100644 --- a/lernmaterial/cheatsheet.aux +++ b/lernmaterial/cheatsheet.aux @@ -2,4 +2,4 @@ \providecommand\babel@aux[2]{} \@nameuse{bbl@beforestart} \babel@aux{ngerman}{} -\gdef \@abspage@last{5} +\gdef \@abspage@last{7} diff --git a/lernmaterial/cheatsheet.log b/lernmaterial/cheatsheet.log index 7c69b7e..da0e9d0 100644 --- a/lernmaterial/cheatsheet.log +++ b/lernmaterial/cheatsheet.log @@ -1,4 +1,4 @@ -This is pdfTeX, Version 3.141592653-2.6-1.40.25 (TeX Live 2023/Debian) (preloaded format=pdflatex 2026.5.23) 21 JUL 2026 08:48 +This is pdfTeX, Version 3.141592653-2.6-1.40.25 (TeX Live 2023/Debian) (preloaded format=pdflatex 2026.5.23) 21 JUL 2026 09:11 entering extended mode restricted \write18 enabled. %&-line parsing enabled. @@ -257,8 +257,8 @@ LaTeX Font Info: Trying to load font information for U+msb on input line 30. File: umsb.fd 2013/01/14 v3.01 AMS symbols B ) [1 -{/var/lib/texmf/fonts/map/pdftex/updmap/pdftex.map}] [2] [3] [4] [5] (./cheatsh -eet.aux) +{/var/lib/texmf/fonts/map/pdftex/updmap/pdftex.map}] [2] [3] [4] [5] [6] [7] (. +/cheatsheet.aux) *********** LaTeX2e <2023-11-01> patch level 1 L3 programming layer <2024-01-22> @@ -267,7 +267,7 @@ L3 programming layer <2024-01-22> Here is how much of TeX's memory you used: 4645 strings out of 476182 69369 string characters out of 5795595 - 1939975 words of memory out of 5000000 + 1940975 words of memory out of 5000000 26619 multiletter control sequences out of 15000+600000 572268 words of font info for 73 fonts, out of 8000000 for 9000 14 hyphenation exceptions out of 8191 @@ -292,10 +292,10 @@ st/fonts/type1/public/amsfonts/cm/cmr7.pfb> -Output written on cheatsheet.pdf (5 pages, 201458 bytes). +Output written on cheatsheet.pdf (7 pages, 219818 bytes). 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z(+o;F!Kb=gs_9Gbz-M~o^GonW60FUFFULaJ1X!nXuFr!F#jtT5Y^w3jC`QDJICC~D z-!0|v)iL;gx*p0e!`59;5f0xx4BJ$a?ONm8@vuW1-`N^=b%F1~;Co%}PJlfs_Fj$c zz2z+#FYXZ?&Dp2k-5&r4QsIZ0a4=omghNx|r!x3iDI8WZN9IFi0aPjTqiV-rba_li z`|Am))|nq)3nx^e-{Rn;GOAT{sub#0!08G&(+B>4F!*yBoIM5g1L52(IImj$T?H2$ zxY!FW6+y#txU93hqN%PfgvL#9P0et<5^fxZoBQFm4&`5U`+vtBGs>X!6U4nD(aaLP zu_Vz&Br{gc!|r16nFGE{z>meP3Egf@r+`1nSip}E$gjYfv0ZCUTGnC*1nIe8&C+r= zgs=mDYo*sh=Rz3CSZh9U3+J>fB3@jFNMG?gwA}|$_1@OHiSD9@Svz{eqE|utG>E0n z7RQlUJax8?fzataNVo-wxzL$2vLsH%x==bx)^@wm8`eDpdMxtR=p}lK`^JMf z(A=`SYdmD`fvsLN*xVzoV{R4y7xx&Q?N*P16XT$U{&Va26S`*(!C5-OJ;&GWUZ8>9 ohPz)>wz-Yd9rFol \frac{B-1}{B}$ $\to$ Greedy packt Item 1; Item 2 passt nicht mehr\\ +-- $GA = 1$, $\mathrm{OPT} = B - 1$ $\to$ Rate $B - 1 \to \infty$ + +\pblock{GÜTE: MODIFIEDGREEDY (Knapsack)} +\feld{Algorithmus} $\mathrm{MGA} = \max\{\text{Greedy-Lösung},\ \text{profitreichstes Einzelitem}\}$. +\feld{Güte} $2$, d.h.\ $\mathrm{OPT} \le 2\,\mathrm{MGA}$. +\feld{Beweis}\\ +-- $k{+}1$ = erstes Item, das Greedy nicht mehr packt\\ +-- fraktionale Relaxierung: $\mathrm{OPT} \le \mathrm{OPT}_f \le p_1 + \dots + p_k + p_{k+1}$\\ +-- $p_1 + \dots + p_k \le GA$ und $p_{k+1} \le p_{\max}$\\ +-- $\mathrm{OPT} \le GA + p_{\max} \le 2 \max\{GA, p_{\max}\} = 2\,\mathrm{MGA}$ + +\pblock{GÜTE: LISTSCHEDULING ($P\,\|\,C_{\max}$)} +\feld{Algorithmus} Jobs der Reihe nach auf die aktuell leerste Maschine. +\feld{Güte} $2 - \frac1m$. +\feld{Beweis}\\ +-- $J_k$ = letzter Job auf der vollsten Maschine (Last $L$); bei Platzierung war sie die leerste\\ +-- alle Maschinen hatten Last $\ge L - p_k$ $\to$ $\sum_i p_i \ge m(L - p_k) + p_k$\\ +-- $\mathrm{OPT} \ge \frac1m \sum_i p_i \ge L - (1 - \frac1m) p_k$\\ +-- $p_k \le \mathrm{OPT}$ einsetzen $\to$ $L \le (2 - \frac1m)\,\mathrm{OPT}$ +\feld{Scharf} (Skript!) $m(m{-}1)$ Einser + $1$ Job der Größe $m$: $LS = 2m - 1$, $\mathrm{OPT} = m$. + +\pblock{GÜTE: LPT} +\feld{Algorithmus} Jobs absteigend sortieren, dann ListScheduling. +\feld{Güte} $\frac43 - \frac1{3m}$. +\feld{Beweis}\\ +-- wie LS: $\mathrm{LPT} \le \mathrm{OPT} + (1 - \frac1m) p_n$ ($p_n$ = Job, der den Makespan setzt)\\ +-- Fall $p_n \le \frac13 \mathrm{OPT}$: einsetzen $\to$ $(\frac43 - \frac1{3m})\,\mathrm{OPT}$\\ +-- Fall $p_n > \frac13 \mathrm{OPT}$: alle Jobs $> \frac13 \mathrm{OPT}$ $\to$ optimal höchstens 2 Jobs pro Maschine $\to$ LPT ist optimal + +\pblock{GÜTE: 2ApproxVC} +\feld{Algorithmus} Kanten durchgehen; sind beide Endpunkte neu, nimm beide in $C$ auf. +\feld{Güte} $2$, d.h.\ $|C| \le 2\,|C^*|$. +\feld{Beweis}\\ +-- $A$ = aufgenommene Kanten $\to$ $|C| = 2|A|$\\ +-- $A$ ist Matching: gemeinsamer Endpunkt wäre schon in $C$ gewesen\\ +-- $C^*$ braucht pro Kante aus $A$ einen \emph{eigenen} Knoten $\to$ $|C^*| \ge |A|$\\ +-- $|C| = 2|A| \le 2\,|C^*|$ + +\pblock{GÜTE: MAX-3-SAT} +\feld{Algorithmus} Werte $\beta_0$ (alles falsch) und $\beta_1$ (alles wahr) aus; gib die bessere zurück. +\feld{Güte} $2$, d.h.\ $v(A(\varphi)) \ge \frac12 v(\mathrm{OPT})$. +\feld{Beweis}\\ +-- jede Klausel hat $\ge 1$ Literal; negatives Literal $\to$ $\beta_0$ erfüllt, positives $\to$ $\beta_1$\\ +-- also $v(\beta_0) + v(\beta_1) \ge m$\\ +-- $\max \ge$ Durchschnitt $\ge \frac m2 \ge \frac12 v(\mathrm{OPT})$ +\feld{Scharf} $(x_1 \vee x_2 \vee x_3) \wedge (\bar x_1 \vee \bar x_2 \vee \bar x_3)$: $A = 1$, $\mathrm{OPT} = 2$ (via $x_1 = 1, x_2 = 0$). + \end{document}