# Algorithmen - kurz gefasst (German Edition) by Uwe Schöning

By Uwe Schöning

In kompakter shape macht das Buch mit den wesentlichen Themen vertraut, die in einer Vorlesung über Algorithmen behandelt werden. Im Mittelpunkt stehen dabei die verschiedensten sequentiellen Algorithmen, deren Komplexitätsanalyse und allgemeine Algoithmen-Paradigma. Prof. Schöning gelingt es, kurz, konkret und verständlich die wichtigsten algorithmischen Aufgabenstellungen (Selektion, Sortieren, Hashing), Algorithmen auf Graphen, algebraische und zahlentheoretische Verfahren zu behandeln. Hinzu kommen heuristische Algorithmenprinzipien wie z.B. genetisches Programmieren.

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Sample text

6. The problem is that it is not clear how to compute P(dl,... ,dj) efficiently. Hence, we use a different weight function U with the following properties: 1. U 0 < 1. 2. U(dl,... ,dj) >1m i n { U ( d l , . . ,dj,0), U(dl,... ,dj, 1)}. 3. U is an upper bound on P. 4. U can be computed in polynomial time. 50 Chapter 3. Derandomization The first two properties ensure that we can use U as a weight function. By the same inequality as for P we obtain U ( d l , . . , dr) < 1. Since U is an upper bound on P, this implies P ( d l , .

We only prove the first statement. The second statement can be proved in a similar way. We apply Markov's inequality to the moment generating function e tx and obtain for each t > 0: Prob[X > (1 + \$)m] = Prob[e tx > e t(l+6)m] < e - t ( l + 8 ) m E[etX]. Now we exploit the independence of the Xj. 48 Chapter 3. Derandomization e-tO+~) m E[e tx] = = e-tO+~)m E[e~,lxl . . . E[~o~xq = e-t(l+~)'~H~jeta' +(1-pj)l]. 3) j-----1 Now we may choose t = In(1 + 5) and obtain r r (1 + 5)-(1+~)~ 1 ] [PJ (1 + 5)~ + 1 - pj] <.

2. 2. Las Vegas and Monte Carlo Algorithms 31 Note that a Las Vegas algorithm is a Monte Carlo algorithm with error probability 0. 3. The efficiency of such Las Vegas algorithm depends heavily on the time needed to verify the correctness of a solution to a problem. Thus, for decision problems, a Las Vegas algorithm derived from a Monte Carlo algorithm is in general useless. We may apply these definitions to the complexity classes introduced in the previous chapter: - The problems in the class ZT~P have efficient Las Vegas decision algorithms.