By Herbert S. Wilf

ISBN-10: 1568811780

ISBN-13: 9781568811789

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**Additional resources for Algorithms and Complexity (Second edition)**

**Sample text**

By induction on n. 46) by (1 + x) to obtain X µn¶ X µn¶ xk + xk+1 (1 + x)n+1 = k k k k X µn¶ Xµ n ¶ = xk + xk k k−1 k k X ½µn¶ µ n ¶¾ = + xk k k−1 k X µn + 1¶ = xk k k which completes the proof. - Now let’s ask how big the binomial coeﬃcients are, as an exercise in asymptotics. , 0 1 n as the coeﬃcients of order n. 46) with x = 1), the sum of all of the coeﬃcients of order n is 2n . 1, that the largest one(s) of the coeﬃcients of order n is (are) the one(s) in the middle. ¢ precisely, ¡ n if ¢n is odd, then the largest coeﬃcients of order n¡ are ¢ ¡ More n n and , whereas if n is even, the largest one is uniquely n/2 .

Find a graph G of n vertices, other than the complete graph, whose chromatic number is equal to 1 plus the maximum degree of any vertex of G. 13. Let n be a multiple of 3. Consider a labeled graph G that consists of n/3 connected components, each of them a K3 . How many maximal independent sets does G have? 14. Describe the complement of the graph G in Exercise 13 above. How many cliques does it have? 15. In how many labeled graphs of n vertices is the subgraph that is induced by vertices {1, 2, 3} a triangle?

The usual decimal system represents numbers by using the 20 1. Mathematical Preliminaries digits 0, 1, . . , 9. For the purpose of representing whole numbers, we can imagine that the powers of 10 are displayed before us like this: . . , 100000, 10000, 1000, 100, 10, 1. Then, to represent an integer, we can specify how many copies of each power of 10 we would like to have. If we write 237, for example, then that means that we want 2 100s, 3 10s, and 7 1s. In general, if we write out the string of digits that represents a number in the decimal system, as dm dm−1 · · · d1 d0 , then the number that is being represented by that string of digits is: n= m X di 10i .

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

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