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zNl7T5dYkRw1RLyx@`kR%Kf?KGyV!~4dEbO%!1?3uDpc_>0ssLT01kPujSGN}`+psz{`LuTNlVD^@BlyXOG$I_b4o~pfHHi44P2Wu9KgUiw!nPi z@1}}Nkffl&1Il3232(xe{um-2`Ozo9{7n!%e36;;g3&dvHdGAt7SQJ(-I<)qw97vk zQK+8SzSZ3~zINR?)i|Bs?tyd-zDHCM+rHM#8Ix3Xi-G*pVgeNyB4>jz^H8 RbMo>6fv9wJ(yB73{{?O*B0&HE diff --git a/übung_3/aufgabe_1.tex b/übung_3/aufgabe_1.tex index 0bf9bb5..f36a6d8 100644 --- a/übung_3/aufgabe_1.tex +++ b/übung_3/aufgabe_1.tex @@ -1,6 +1,55 @@ \section{Triangulierungen und Dynamische Programmierung} \begin{tasks} \item + $$ + T(P)=\begin{cases} + 0 &\textnormal{falls } P \textnormal{ nur 2 Ecken hat}\\ + \min_{i\in \{2,...,n-1\}} [T(p_{1,i})+T(P_{i,n})+ \textnormal{Diagonalkosten}(i)] &\textnormal{sonst} + \end{cases} + $$ + $$ + \textnormal{Diagonalkosten}(i)= \begin{cases} + d(p_2, p_n) &\textnormal{falls } i=2\\ + d(p_1, p_{n-1}) &\textnormal{falls } i=n-1\\ + d(p_1, p_i)+d(p_i, p_{n}) &\textnormal{sonst} + \end{cases} + $$ + Die Kosten der minimalen Triangulierung $T(P)$ lassen sich aus der minimalsten Summe der konstenminimalsten Triangulierung der entstehenden Teilpolygone $T(p_{1,i})$ und $T(p_{i,n})$ und den entstehenden Diagonalenkosten berechnen.\\ + Die Diagonalenkosten unterscheiden sich, je nachdem ob eine oder zwei der drei Kanten Polygonkanten sind. Sobald ein (Teil-)Polygon nur noch zwei Ecken hat bricht die Rekursion ab. + \points{2} + \item + Man nutzt die Idee aus a.\\ + \textbf{Tabelle $A[i,j]$} speichert die kostenminimalen Triangulierung des Teilpolygons mit den Ecken $p_i,...,p_j$\\ + Für alle $i$ gilt $A[i,i+1]=0$, da diese Teilpolygone mit nur zwei Ecken darstellen, welche keine Triangulierung benötigen.\\ + Für $j>i+1$ gilt: + $$ + A[i,j]=\min_{k\in\{i+1,...,j-1\}}[A[i,k]+A[k,j]+\textnormal{Diagonalkosten}(i,k,j)] + $$ + $$ + \textnormal{Diagonalkosten}(i,k,j)=\begin{cases} + d(p_k,p_j) &\textnormal{falls } k=i+1\\ + d(p_i,p_k) &\textnormal{falls } k=j-1\\ + d(p_i, p_k)+d(p_k, p_j) &\textnormal{sonst} + \end{cases} + $$ + Man berechnet dabei die Einträge nach nach wachsendem Abstand $r=j-i$, also von $r=2$ bis $r=n-1$. Dadurch sind die Einträge $A[i,k]$ und $A[k,j]$ immer bereits berechnet wenn man den $A[i,j]$ benötigt, da $k-i