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**Read Online or Download Introduction to Algorithms (MIT Electrical Engineering and Computer Science) PDF**

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**Additional resources for Introduction to Algorithms (MIT Electrical Engineering and Computer Science)**

**Sample text**

19k n n€ b. nk en c. [ii d. 2n n sin n 2n/2 e. n lg m m lg n f. ) 19(nn) 2-3 Ordering by asymptotic growth rates a. Rank the following functions by order of growth; that is, find an arrangement gl> g2, ... , g30 of the functions satisfying gl = Q(g2), g2 = Q(g3), ... , g29 = Q(g30). Partition your list into equivalence classes such that j(n) and g(n) are in the same class if and only if j(n) = 8(g(n)). 39 Problems for Chapter 2 19(1g* n) 21g' n (V2)lgn n2 n! ) nlglgll (lg n)! nJ/lgn Inn 1 (lgn)lgII en 41g n lg"(Ig n) 2~ n 2n (n + I)!

Nlglgll (lg n)! nJ/lgn Inn 1 (lgn)lgII en 41g n lg"(Ig n) 2~ n 2n (n + I)! JIgn nlgn 2 2" + 1 b. Give an example of a single nonnegative function j( n) such that for all functions gj(n) in part (a), j(n) is neither O(gi(n» nor n(gi(n». 2-4 Asymptotic notation properties Let j(n) and g(n) be asymptotically positive functions. Prove or disprove each of the following conjectures. a. j(n) = O(g(n» implies g(n) b. j(n) + g(n) = O(f(n». = 8(min(j(n),g(n»). = O(g(n» implies 19(f(n» O(lg(g(n»), where 19(9(n» > 0 and j( n) 2:: 1 for all sufficiently large n.

Nlglgll (lg n)! nJ/lgn Inn 1 (lgn)lgII en 41g n lg"(Ig n) 2~ n 2n (n + I)! JIgn nlgn 2 2" + 1 b. Give an example of a single nonnegative function j( n) such that for all functions gj(n) in part (a), j(n) is neither O(gi(n» nor n(gi(n». 2-4 Asymptotic notation properties Let j(n) and g(n) be asymptotically positive functions. Prove or disprove each of the following conjectures. a. j(n) = O(g(n» implies g(n) b. j(n) + g(n) = O(f(n». = 8(min(j(n),g(n»). = O(g(n» implies 19(f(n» O(lg(g(n»), where 19(9(n» > 0 and j( n) 2:: 1 for all sufficiently large n.