By Donald E. Knuth, Daniel H. Greene

**Publish 12 months note:** First released January 1st 1980

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This monograph collects a few basic mathematical thoughts which are required for the research of algorithms. It builds at the basics of combinatorial research and intricate variable thought to offer a number of the significant paradigms utilized in the correct research of algorithms, emphasizing the more challenging notions.

The authors hide recurrence relatives, operator equipment, and asymptotic research in a structure that's concise sufficient for simple reference but specified adequate for people with little heritage with the fabric.

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**Extra resources for Mathematics for the Analysis of Algorithms (3rd Edition)**

**Example text**

22) is the probability of obtaining a factorization into irreducible parts of distinct sizes kl, k 2 , . . , kin. 22), and since all of the events are disjoint these probabilities are summed. T h u s the generating function properly determines h,,,, the limiting probability t h a t a polynomial of degree n factors into irreducible parts of distinct sizes modulo a large prime. 21) does not give us a closed form for hn, and there does not seem to be one, so instead we seek an asymptotic formula as n ---} co.

Treating each polynomial this way gives a total of pkl pk2 kl k2 pkm km pn -- kl k2 . . 23) polynomials whose irreducible factors have the appropriate sizes. Since there are a total of pn monic polynomial of size n, this means that the coefficient 1 kl k2 . . 22) is the probability of obtaining a factorization into irreducible parts of distinct sizes kl, k 2 , . . , kin. 22), and since all of the events are disjoint these probabilities are summed. T h u s the generating function properly determines h,,,, the limiting probability t h a t a polynomial of degree n factors into irreducible parts of distinct sizes modulo a large prime.

36) A further iteration of bootstrapping yields Pn = O log n n . 38) " In the last step we computed p(1) by summing the infinite series 1. 30) We estimated the sum ~ a > n O( ~ k~ 2 J by considering its integral counterpart oo log x (log n)2 x dx = 0 . 42) 52 ASYMPTOTIC ANALYSIS This expression can be bootstrapped through another iteration to obtain the slightly better approximation Pn = -e 2n2 /x2 +0 ( logn~ ~ ) . 311. 44) where s(z)=exp z z4 3 z5 4 b-g-... 45) r(z)=exp 3k 3 4k 4 t-... The expression for s(z) can be reworked, ( s(z) = exp l n ( l + z ) - z + - ~ - .