From 1f1529176108f6f57b61427397f9f85a9568c07f Mon Sep 17 00:00:00 2001 From: Team3 Date: Fri, 24 Jul 2026 11:25:13 +0200 Subject: [PATCH] update --- lernmaterial/cheatsheet.log | 57 ++++++++------- lernmaterial/cheatsheet.pdf | Bin 219818 -> 280532 bytes lernmaterial/cheatsheet.tex | 136 +++++++++++++++++++----------------- 3 files changed, 99 insertions(+), 94 deletions(-) diff --git a/lernmaterial/cheatsheet.log b/lernmaterial/cheatsheet.log index da0e9d0..8f5c6a6 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 09:11 +This is pdfTeX, Version 3.141592653-2.6-1.40.25 (TeX Live 2023/Debian) (preloaded format=pdflatex 2026.5.23) 21 JUL 2026 13:24 entering extended mode restricted \write18 enabled. %&-line parsing enabled. @@ -257,45 +257,44 @@ LaTeX Font Info: Trying to load font information for U+msb on input line 30. 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z4u=Ar+V&GVv|b_0T~ZH%VG!)B=JM}zi;y^dZV$b?*B>yNLM&p;HdXrdPiof}#{*LV z;*lw;$<~w>GiFWSlwrkx12HqC48k)z?>BuP{rfL6`XgA*Gcp{5v+kek^Rm^*%yihRS=xuB8^TDfCc&WvAH#@zL?WLh1lF$^t!zCV0zxC zqPo4k%Tw_`lqYp;?)P?Z`={(bteyX4FnV2xj0_kiDIzQ=1QrsK5fqe`mJ$ZPHohi1maH{{a*)19$}vUPD@yjo{6I$NP16>ZhU&DB3})Df>ZQF78`S# z3W9q?BYX_4zcJ)+vIM1>RFsZqCUSZ*e=lYfh-Fg?$40;;2^iv^ylD8Eq0`XPt;d$D zyT&EwQL(KrX{_kjTs+O5;$wW}A5+bYfGWVfA%Ggh6Hsn~lLKmJku=A($Jy^uWn2 zIke|>WbXsZ$9hQ$lUD0*nxEWvUw%#DLiBtofPwv+5M{7(pz+@F%7tBlgPOD^=_JgR z$awL%v+2q*r4EnDu5IlXj}uX(4nUA4)Rstk3^VlLd)|}Vp>Zo>hYQii8r}Cl$usI; z<#7ZlAfEWg@f4jfN19bc%=7~(Fa_&@uwVpQ_B_LG_x5`5HHuILSY;cJNz zLe^$Xb4oUxIsRE#5W`m-yZ1+iVRU;K!a=?F%;uFNp0axo*y>o9UAQny ztk7?S{r8#Z`cn0MP@ID!x!y!2w5BZ`gKOe>W=7po(;$@3k%hA455V~N@vzi3XwP|B zhCIW%wSeZKkodZkeO_JA?v0eYCcwfGK`ahS^(U&tP?jJRFw3#Td-D?2e75Sx-HQML zZ0%D$QVebOW0|AIm(iL(7ZQ+l;bBmfe-K1 zTPR6`$5SlAS&Rj%5>dkV!&sV8`>~lec;!BLg9;oK*1ngYyZa4Ja7qzXO?lZ?$?JwM z&9Mhu-s)l=2*p8kKeCx=bpO=Q)eBKh`#o56X1DD$5;P^r91bv7mTV<%=Q6uZqn@u% z)EfMgptF0vr$u>-01HZBUMx%W4Tt#4U;frID(j%>xG?hEFy7|8iXw^S&`HA6-A&8+ zrPQ)3vjg`!o6r9-d$VR2Z|wiuE5XGEhhNz_OOTABWg%)#$LLE9Rdr?C9E}}v5$#TX zKL0DmBZpI#;+eP L$ $\to$ Tour $\le L$ nutzt nur Kanten aus $E$ $\to$ HC. \feld{ETH} $|V'| = |V|$, $|E'| = O(|V|^2)$, $L = |V|$. \pblock{$k$-COLOR} -\feld{Def} Färbung $f: V \to [k]$ mit $f(u) \ne f(v)$ für Kanten? +\feld{Def} Graph $G$, Zahl $k$; existiert Färbung $f: V \to [k]$ mit $f(u) \ne f(v)$ für Kanten? \feld{Bsp} $C_5$: 3 Farben nötig; $C_4$: 2 reichen. \feld{Reduktion} von 3-\prob{Sat} (Skript, vollständig). \feld{Konstruktion}\\ --- $V' = \{x_i, \bar x_i, v_i \mid i \in [n]\} \cup \{F_j \mid j \in [m]\} \cup \{z\}$\\ --- $E' = \{\{v_i, v_j\}, \{v_i, x_j\}, \{v_i, \bar x_j\} \mid i \ne j\} \cup \{\{x_i, \bar x_i\}\} \cup \{\{\ell, F_j\} \mid \text{Literal } \ell \notin C_j\} \cup \{\{v_i, z\}, \{F_j, z\}\}$\\ --- $k' = n{+}1$\\ +-- $V' = \{x_i, \bar x_i, v_i \mid i \in [n]\} \cup \{F_j \mid j \in [m]\} \cup \{z\}$ \quad $O(n + m)$\\ +-- $E' = \{\{v_i, v_j\}, \{v_i, x_j\}, \{v_i, \bar x_j\} \mid i \ne j\} \cup \{\{x_i, \bar x_i\}\} \cup \{\{\ell, F_j\} \mid \text{Literal } \ell \notin C_j\} \cup \{\{v_i, z\}, \{F_j, z\}\}$ \quad $O(n^2 + nm)$\\ +-- $k' = n{+}1$ \quad $O(1)$\\ Vorüberlegung: obdA $f(v_i) = i$, $f(z) = n{+}1$; dann $f(x_j), f(\bar x_j) \in \{j, n{+}1\}$ -- kodiert die Belegung. \hin Wahres Literal bekommt Farbe $i$, falsches $n{+}1$; $F_j$ färbt man mit der Farbe eines erfüllenden Literals (keine Kante dorthin). \rueck Wäre $F_j$ unerfüllt, sind alle Literale in $F_j$ mit $n{+}1$ gefärbt; $F_j$ sieht dann alle Farben $1, \dots, n{+}1$ -- Widerspruch. \feld{ETH} $|V'| = O(m)$, $|E'| = O(m^2)$. \pblock{$k$-COLOR-PRECOLORING} -\feld{Def} Zusätzlich: vorgegebene Knoten $v_1, \dots, v_k$ mit $f(v_i) = i$. +\feld{Def} Graph $G$, Zahl $k$, Knoten $v_1, \dots, v_k$; existiert $k$-Färbung mit $f(v_i) = i$? \feld{Bsp} Dreieck $v_1, v_2, w$ mit $k = 3$, $f(v_1){=}1$, $f(v_2){=}2$: $w$ bekommt Farbe 3 -- ja. \feld{Reduktion} von $k$-\prob{Color}. \feld{Konstruktion}\\ --- $V' = V \cup \{v_1, \dots, v_k\}$\\ --- $E' = E \cup \{\{v_i, v_j\} \mid i \ne j\}$\\ --- $k' = k$\\ --- ausgezeichnete Knoten: $v_1, \dots, v_k$ (die frischen) +-- $V' = V \cup \{v_1, \dots, v_k\}$ \quad $O(|V| + k)$\\ +-- $E' = E \cup \{\{v_i, v_j\} \mid i \ne j\}$ \quad $O(|E| + k^2)$\\ +-- $k' = k$ \quad $O(1)$\\ +-- ausgezeichnete Knoten: $v_1, \dots, v_k$ (die frischen) \quad $O(k)$ \hin $k$-Färbung von $G$ bleibt gültig; neue Clique bekommt Farben $1..k$ per Definition. \rueck Einschränkung der Precoloring-Färbung auf $G$ ist $k$-Färbung. \feld{ETH} $|V'| = O(|V|)$, $|E'| = O(|V|^2)$, $k' = k$. \pblock{COVERCLIQUE} -\feld{Def} $k$ paarweise disjunkte Cliquen, die $V$ überdecken? +\feld{Def} Graph $G$, Zahl $k$; existieren $k$ paarweise disjunkte Cliquen, die $V$ überdecken? \feld{Bsp} $C_4$ (Viereck): 2 Cliquen (gegenüberliegende Kanten) -- ja für $k{=}2$. \feld{Reduktion} von $k$-\prob{Color} (SS25-Klausur). \feld{ETH} $|V'| = |V|$, $|E'| \le |V|^2$, $k' = k$. @@ -176,95 +182,95 @@ Vorüberlegung: obdA $f(v_i) = i$, $f(z) = n{+}1$; dann $f(x_j), f(\bar x_j) \in \feld{Bsp} $\{3, 5, 7, 11\}$, $T = 12$: ja ($5 + 7$). \feld{Reduktion} von X3C (Vorlesung). \feld{Konstruktion}\\ --- $c_j = \sum_{u_i \in S_j} (n{+}1)^{i-1}$ (Menge als Bitvektor zur Basis $n{+}1$)\\ --- $K = \sum_{j=0}^{3m-1} (n{+}1)^{j}$ (überall Ziffer 1) +-- $c_j = \sum_{u_i \in S_j} (n{+}1)^{i-1}$ (Menge als Bitvektor zur Basis $n{+}1$) \quad $O(|F| \cdot |U|)$\\ +-- $K = \sum_{j=0}^{3m-1} (n{+}1)^{j}$ (überall Ziffer 1) \quad $O(|U|)$ \hin Exact Cover $\to$ jede Position genau einmal überdeckt $\to$ Summe $= K$. \rueck $\le n$ Summanden, Basis $n{+}1$ $\to$ kein Übertrag $\to$ jede Position genau eine 1 $\to$ gewählte Mengen überdecken exakt. \feld{ETH} $n' = |F|$. \pblock{SUBSETSUMCARDINALITY} -\feld{Def} \prob{SubsetSum} mit Zusatz $|S| = n/2$ ($n$ gerade). +\feld{Def} Größen $c_1, \dots, c_n$ ($n$ gerade), Ziel $K$; existiert $S$ mit Summe $K$ und $|S| = n/2$? \feld{Bsp} $\{1, 2, 3, 8\}$, $K = 9$, $|S| = 2$: ja ($1 + 8$). \feld{Reduktion} von \prob{SubsetSum} (Präsenz 11.3). \feld{Konstruktion}\\ --- $c_i' = c_i + 1$ für $i \in [n]$\\ --- $n$ Einser-Items: $c_{n+1}' = \dots = c_{2n}' = 1$\\ --- $K' = K + n$ +-- $c_i' = c_i + 1$ für $i \in [n]$ \quad $O(n)$\\ +-- $n$ Einser-Items: $c_{n+1}' = \dots = c_{2n}' = 1$ \quad $O(n)$\\ +-- $K' = K + n$ \quad $O(1)$ \hin Lösung $S$ plus $n - |S|$ Einser: Kardinalität $n$, Summe $K + |S| + (n - |S|) = K + n$. \rueck $S = S' \cap [n]$: Einser tragen $n - |S|$ bei $\to$ $\sum_{i \in S} c_i = K$. \feld{ETH} $n' = O(n)$. \pblock{$(a_1 = 1)$-SUBSETSUM} -\feld{Def} \prob{SubsetSum}, wobei $a_1 = 1$ gilt. +\feld{Def} \prob{SubsetSum}-Instanz mit $a_1 = 1$; Teilmenge mit Summe genau $K$? \feld{Bsp} $\{1, 4, 6\}$, $K = 7$: ja ($1 + 6$). \feld{Reduktion} von \prob{SubsetSum}. \feld{Konstruktion}\\ --- $a_1' = 1$\\ --- $a_{i+1}' = 2 c_i$ für $i \in [n]$\\ --- $K' = 2K$ +-- $a_1' = 1$ \quad $O(1)$\\ +-- $a_{i+1}' = 2 c_i$ für $i \in [n]$ \quad $O(n)$\\ +-- $K' = 2K$ \quad $O(1)$ \hin Lösung $S$ $\to$ verdoppelte Items summieren zu $2K$, ohne die 1. \rueck $2K$ gerade, alle Items außer der 1 gerade $\to$ 1 nie nutzbar (Parität); halbieren ergibt Summe $K$. \feld{ETH} $n' = O(n)$. \pblock{SUBSETSUM MIT TEILBARKEIT (durch 3 oder 7)} -\feld{Def} \prob{SubsetSum}, jede Größe durch 3 oder durch 7 teilbar. +\feld{Def} \prob{SubsetSum}-Instanz, jede Größe durch 3 oder 7 teilbar; Teilmenge mit Summe genau $K$? \feld{Bsp} $\{3, 7, 21\}$, $K = 10$: ja ($3 + 7$). \feld{Reduktion} von \prob{SubsetSum}. \feld{Konstruktion}\\ --- $a_i' = 21\, c_i$\\ --- $K' = 21\, K$ +-- $a_i' = 21\, c_i$ \quad $O(n)$\\ +-- $K' = 21\, K$ \quad $O(1)$ \hin Lösung skaliert mit: Summe $21K$. \rueck Summe $21K$ durch 21 teilen $\to$ Summe $K$; $21c$ ist durch 3 und 7 teilbar. \feld{ETH} $n' = n$. \pblock{SUBSETSUM OHNE ZWEIERPOTENZEN} -\feld{Def} \prob{SubsetSum}, keine Größe ist eine Zweierpotenz $2^t$. +\feld{Def} \prob{SubsetSum}-Instanz, keine Größe eine Zweierpotenz $2^t$; Teilmenge mit Summe genau $K$? \feld{Bsp} $\{3, 5, 6\}$, $K = 9$: ja ($3 + 6$). \feld{Reduktion} von \prob{SubsetSum}. \feld{Konstruktion}\\ --- $a_i' = 3\, c_i$\\ --- $K' = 3K$ +-- $a_i' = 3\, c_i$ \quad $O(n)$\\ +-- $K' = 3K$ \quad $O(1)$ \hin Lösung skaliert mit: Summe $3K$. \rueck Summe $3K$ durch 3 teilen; $3c$ ist nie $2^t$, da $3 \nmid 2^t$. \feld{ETH} $n' = n$. \pblock{PARTITION} -\feld{Def} Zahlen in zwei Hälften gleicher Summe teilen? +\feld{Def} Zahlen $c_1, \dots, c_n$; existiert $S$ mit $\sum_{i \in S} c_i = \frac12 \sum_i c_i$? \feld{Bsp} $\{1, 2, 3, 4\}$: $\{1,4\} | \{2,3\}$ -- ja. \feld{Reduktion} von \prob{SubsetSum} (Skript). \feld{Konstruktion}\\ --- $N = \sum_{j=1}^n c_j + 1$\\ --- $c_{n+1} = N - K$, $c_{n+2} = K + 1$\\ +-- $N = \sum_{j=1}^n c_j + 1$ \quad $O(n)$\\ +-- $c_{n+1} = N - K$, $c_{n+2} = K + 1$ \quad $O(1)$\\ -- Gesamtsumme $= 2N$, Hälfte $= N$ \hin Lösung $S$: $S \cup \{c_{n+1}\}$ summiert zu $K + (N - K) = N$. \rueck $c_{n+1}, c_{n+2}$ nie zusammen (Summe $N + 1 > N$); obdA $c_{n+1} \in S$ $\to$ Rest von $S$ summiert zu $N - (N - K) = K$. \feld{ETH} $n' = O(n)$. \pblock{$P2\,\|\,C_{\max}$ (2 Maschinen)} -\feld{Def} Jobs auf 2 Maschinen, Makespan $\le T$? +\feld{Def} Jobs $p_1, \dots, p_n$, Schranke $T$; Schedule auf 2 Maschinen mit Makespan $\le T$? \feld{Bsp} Jobs $3,3,2,2,2$, $T = 6$: ja ($\{3,3\}$ und $\{2,2,2\}$). \feld{Reduktion} von \prob{Partition}. \feld{Konstruktion}\\ --- Jobs = Zahlen\\ --- $m = 2$\\ --- $T = \frac12 \sum p_j$ +-- Jobs = Zahlen \quad $O(n)$\\ +-- $m = 2$ \quad $O(1)$\\ +-- $T = \frac12 \sum p_j$ \quad $O(n)$ \hin Partition $=$ perfekt balancierter Schedule. \rueck Makespan $\frac12 \sum$ erzwingt exakte Balance $=$ Partition. \feld{ETH} $n' = n$, $T = O(\textstyle\sum a_i)$. \pblock{KNAPSACK (Entscheidung)} -\feld{Def} Gewichte $w_i$, Profite $p_i$; Auswahl mit $\sum w_i \le K$ und $\sum p_i \ge P$? +\feld{Def} Gewichte $w_i$, Profite $p_i$, Schranken $K, P$; Auswahl mit $\sum w_i \le K$ und $\sum p_i \ge P$? \feld{Bsp} Items $(w,p)$: $(2,3), (3,4)$; $K = 3$, $P = 4$: ja (zweites Item). \feld{Reduktion} von \prob{SubsetSum}. \feld{Konstruktion}\\ --- $w_i = p_i = a_i$\\ --- $K = P = T$ +-- $w_i = p_i = a_i$ \quad $O(n)$\\ +-- $K = P = T$ \quad $O(1)$ \hin Summe $= T$ $\to$ Gewicht $\le K$ und Profit $\ge P$. \rueck Gewicht $\le T$ und Profit $\ge T$ zugleich $\to$ Summe exakt $T$. \feld{ETH} $n' = n$, $K = P = T$. \pblock{HITTINGSET} -\feld{Def} Universum $U$, Mengen $F_1, \dots, F_r$, Zahl $k$; $H$, $|H| \le k$, trifft jede $F_i$? +\feld{Def} Universum $U$, Mengen $F_1, \dots, F_r$, Zahl $k$; existiert $H$, $|H| \le k$, das jede $F_i$ trifft? \feld{Bsp} $U = \{1..5\}$, $F = \{1,2\}, \{2,3\}, \{4,5\}$: $H = \{2, 4\}$, $k{=}2$ ja. \feld{Reduktion} von 3-\prob{Sat} (HA 12.2 / Probeklausur). \feld{ETH} $|U| = O(n)$, $r = O(m)$, $k = n$. @@ -276,12 +282,12 @@ Vorüberlegung: obdA $f(v_i) = i$, $f(z) = n{+}1$; dann $f(x_j), f(\bar x_j) \in \feld{ETH} $n' = O(n)$, $m' = m$. \pblock{HALT$_{\mathrm{TM}}$ (nur NP-schwer!)} -\feld{Def} Hält DTM $M$ auf Wort $w$? +\feld{Def} DTM $M$, Wort $w$; hält $M$ auf $w$? \feld{Bsp} $M$ = Endlosschleife auf jeder Eingabe: nein für jedes $w$. \feld{Reduktion} von 3-\prob{Sat} (HA 11.2). \feld{Konstruktion}\\ --- $M_\varphi$: probiere alle $2^n$ Belegungen; falls eine erfüllt, halte; sonst Endlosschleife\\ --- $w = \langle \varphi \rangle$ +-- $M_\varphi$: probiere alle $2^n$ Belegungen; falls eine erfüllt, halte; sonst Endlosschleife \quad $O(|\varphi|)$\\ +-- $w = \langle \varphi \rangle$ \quad $O(|\varphi|)$ \hin Erfüllbar $\to$ $M_\varphi$ findet Belegung und hält. \rueck $M_\varphi$ hält nur im Erfolgsfall $\to$ erfüllbar.