Read e-book online Algorithms and Complexity (Second edition) PDF

By Herbert S. Wilf

ISBN-10: 1568811780

ISBN-13: 9781568811789

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Extra resources for Algorithms and Complexity (Second edition)

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37) and cancelling common factors, we find that the quadratic equation for√α is, in this case, α2 =√α+1. 36). It remains to determine the constants c1 , c2 from the initial conditions F0 = 0, F1 = 1. From the form of the general solution, we have F0 = 0 = c1 + c2 and F1 = 1 = c1 α+ + c2 α− . 4. Recurrence Relations 31 of the general solution, we obtain an explicit formula for the nth Fibonacci number: √ ¶n ¾ √ ¶n µ ½µ 1 1− 5 1+ 5 − Fn = √ (n = 0, 1, . ). 38) 2 2 5 The Fibonacci numbers are in fact 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, .

Mathematicians have known for years that induction is a marvelous method for proving theorems, making constructions, etc. Now computer scientists and programmers can profitably think recursively too, because recursive compilers allow them to express such thoughts in a natural way, and as a result, many methods of great power are being formulated recursively, methods which, in many cases, might not have been developed if recursion were not readily available as a practical programming tool. Observe next that the ‘trivial case,’ where n = 1, is handled separately, in the recursive form of the n!

An independent set S is maximal if it is not a proper subset of another independent set of vertices of G. Dually, if a vertex subset S induces a complete graph, then we speak of a complete subgraph of G. A maximal complete subgraph of G is called a clique. A graph might be labeled or unlabeled. The vertices of a labeled graph are numbered 1, 2, . . , n. One difference that this makes is that there are a lot more labeled graphs than there are unlabeled graphs. 10). Most counting problems on graphs are much easier for labeled than for unlabeled graphs.

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Algorithms and Complexity (Second edition) by Herbert S. Wilf

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