# ps6-comments - University of Virginia - cs3102: Theory of...

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University of Virginia - cs3102: Theory of Computation Spring 2010 Problem Set 6 - Comments Problem 1: Asymptotic Notation. The way we advocated using asymptotic notation to de- scribe algorithm complexity in Classes 22 and 23 is different from how Sipser uses it in Sec- tion 7.1. Identify at least two ways in which Sispers use of the asymptotic operators differs from the way we advocated using them in class, and illustrate each with a speciﬁc example from Sipsers text. Answer. Sipsers book sometimes uses the asymptotic notations as sets, but sometimes as if they produce actual numbers. For example, “Each scan uses O ( n ) steps.” (p. 251). This is certainly easier than saying, “A function from the input size n to the number of steps used in each scan would be a member of O(n).”, but it is not strictly correct if we view the asymptotic operators are producing sets of functions. Sipser uses equality (e.g., f ( n ) = O ( n 3 ) ) to show a function is a member of a O ( g ( n )) class. In class, we advocated using set membership (e.g., f ( n ) O ( n 3 ) ) to clearly distinguish between the case where a function is a member of a set and the case where two sets are equal (e.g., O ( n ) O ( logn ) = O ( n ) ). Sipser uses O to analyze algorithms, when a tight Θ bound could be easily stated. For ex- ample, in the ﬁrst example above, the number of steps is also known to be in Θ ( n ) . It is not incorrect to say it is in O ( n ) , but provides much less information than providing the tight bound. Problem 2: Asymptotic Propositions. For each sub-part below, state whether the statement is true , false , or unknown (where unknown means no one on Earth knows the answer, not just that you don’t know it!). Include a brief but convincing argument supporting your an- swer. a. Θ ( n log n ) O ( n 2 ) Answer. True . Intuitively, O ( n 2 ) is the set of all functions that grow no faster than n 2 , and Θ ( n log n ) is the set of functions that grow as fast as n log n . Since n log n grows slower than n 2 , Θ ( n log n ) is a proper subset of O ( n 2 ) . More formally, we will ﬁrst show that Θ ( n log n ) O ( n 2 ) and then that the inclusion is proper. Recall from the deﬁnition of Θ that Θ ( n log n ) = Ω ( n log n ) O ( n log n ) . So, f ( n ) Θ ( n log n ) implies f ( n ) O ( n log n ) . By the deﬁnition of O this means that there exist positive numbers c , n 0 such that n n 0 , f ( n ) cn log n . For all n > 1 , n log n < n 2 . So, f ( n ) cn log n > cn 2 implies f ( n ) O ( n 2 ) . PS6C-1

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This proves Θ ( n log n ) O ( n 2 ) . To prove the inclusion is proper (that is, ), we need to identify some function that is in O ( n 2 ) but not in Θ ( n log n ) . One such function is f ( n ) = n 2 . This is obviously in O ( n 2 ) (choose c = 1, n 0 = 1 ), but is not in Θ ( n log n ) . b.
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ps6-comments - University of Virginia - cs3102: Theory of...

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