Algorithmen und Datenstrukturen by Karsten Weicker, Nicole Weicker

By Karsten Weicker, Nicole Weicker

Statt des üblichen theoretischen Zugangs vermittelt dieses Lehrbuch Algorithmen und Datenstrukturen durch die Geschichte einer jungen Informatikerin. Der Stoff einer traditionellen Einführungsveranstaltung Informatik wird so ausgehend von der praktischen Anwendung lebendig und humorvoll vermittelt. So schlägt das Buch eine Brücke von Alltagserfahrungen zu den Konzepten von Datenstrukturen und Algorithmen.

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Bei f (n) = n und g(n) = n · log n gilt O(n + n · log n) = O(n · log n) wegen f (n) ∈ O(g(n)). Analog können die stärker wachsenden Summanden bei einer Abschätzung nach unten entfallen. 3. 3 ermittelt. Da 3 + 27 2 · n ∈ O( 2 · n ) gilt, folgt T (n) ∈ O( 2 · 2 2 n ). Und damit folgt letztendlich: T (n) ∈ O(n ). 28: Berechnungsregel f (n) ∈ R+ ⇒ f (n) ∈ Θ(g(n)) g(n) f (n) = 0 ⇒ f (n) ∈ O(g(n)). 29: √ Betrachten wir f (n) = log n und g(n) = n. log n lim √ = lim n→∞ n n→∞ 1 n 1 2 1 · n− 2 √ (Regel von l’Hôptial) n = 2·n 1 √ = 0.

50 Das Symbol O wurde vom Mathematiker Bachmann (1894) eingeführt. Die Bezeichnung »Landau-Notation« geht auf das Buch von Landau (1909) zurück, welches der Verbreitung der Notation Vorschub geleistet hat. In der Informatik wurde die Notation durch Knuth (1976) um die anderen Symbole erweitert und durchgesetzt. Bei der asymptotischen Laufzeitanalyse möchte man einen einfachen Ausdruck für das Laufzeitverhalten haben, der eng an den exakten Werten ist. Es gelten die folgenden Regeln. 19: Komplementäre Symbole O und Ω sind komplementär zueinander: f (n) ∈ O(g(n)) ⇔ g(n) ∈ Ω( f (n)).

62 Übungsaufgaben . . . . . . . . . . . . . . . . . 64 Datenstrukturen für das Mengenproblem Bei dem Mengenproblem müssen verschiedene Datenelemente so gespeichert werden, dass die Menge leicht verwaltbar ist. 1 wurde dieses Problem so eingeführt, dass doppelte Schlüssel nicht erlaubt sind. Zur Vereinfachung wollen wir in diesem Kapitel diese Forderung nicht beachten. Andernfalls müsste bei jedem Einfügen geprüft werden, dass das Element noch nicht enthalten ist – was gerade bei den einfachen Verfahren den Blick auf die wesentliche Vorgehensweise der Algorithmen verstellt.

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