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Zusammenfassung Algorithmen und Berechnungskomplexität1-Präsenzzettel2-Algo1 $3.18
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Zusammenfassung Algorithmen und Berechnungskomplexität1-Präsenzzettel2-Algo1

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Dieses Dokument enthält die Lösungen zum 2. Präsenzblatt des Moduls Algorithmen und Berechnungskomplexität 1 sowie zusätzliche Mitschriften zur besseren Verständlichkeit.

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  • December 31, 2024
  • 10
  • 2024/2025
  • Summary
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Algorithmen und Berechnungshomplexität
Präsenzblatt 2
21
Aufgabe ,




begeben sei die folgende Rekursionsvorschrift TIN IN mit :




falls
[cn-1)
, n =1
T(m) =




+ 3ni
falls na


Ziel ist es nun , eine
geschlossene Form TCns für diese
Rekursionsvorschrift zu finden und per
vollständige Induktion zu beweisen .

Betrachtet man die ersten Werte :
TG) =
3

T(2) TH) + 3 2 3 + 6
= .
= =
9
T(3) T(2) + 3 3 g + 9 18
= . = =




T(4) +(3) + 3 4 18 + 12 38
= .
=
=

, Differenzen TCn) -TCn-1) In
Die nahe =


legen ,


dass Tens die Summe einer arithmetischen Folget

T(n) 3 + 3 =
.
2+ 3 3 +... + 3n
1




Dies entspricht der Summe :


T(n) = 3h =
3 .

hin
+)
_
3n(n)
2
k 1
=




Die Behauptung kann per vollständiger Induktion bewiesen werden :




Induktionsanfang
Fürn =
+ 1)
311(1 3
12 2 3
,


T(1) = = = =




Induktionsannahme
Es gelte die
geschlossene Rekussionsvorschrift
für ein beliebiges n21 :




3n(n + 1)
T(n) =

2

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