# 21 - 14:57:33 CS 61B Lecture 21 Wednesday ASYMPTOTIC...

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03/03/09 14:57:33 1 21 CS 61B: Lecture 21 Wednesday, March 11, 2009 ASYMPTOTIC ANALYSIS (continued): More Formalism ================================================ |-----------------------------------------------------------------------------| | Omega(f(n)) is the set of all functions T(n) that satisfy: | | | | There exist positive constants d and N such that, for all n >= N, | | T(n) >= d f(n) | |-----------------------------------------------------------------------------| ^^^^^^^^^^ Compare with the definition of Big-Oh: T(n) <= c f(n). ^^^^^^^^^^^ Omega is the reverse of Big-Oh. If T(n) is in O(f(n)), f(n) is in Omega(T(n)). 2n is in Omega(n) BECAUSE n is in O(2n). n^2 is in Omega(n) BECAUSE n is in O(n^2). n^2 is in Omega(3 n^2 + n log n) BECAUSE 3 n^2 + n log n is in O(n^2). Omega gives us a LOWER BOUND on a function, just as Big-Oh gives us an UPPER BOUND. Big-Oh says, "Your algorithm is at least this good." Omega says, "Your algorithm is at least this bad." If we have both--say, T(n) is in O(f(n)) and is also in Omega(g(n))--then T(n) is effectively sandwiched between the two, below c f(n) and above d g(n). If f(n) = g(n), we say that T(n) is in Theta(f(n)): |-----------------------------------------------------------------------------| | Theta(f(n)) is the set of all functions T(n) that are in both of | | | | O(f(n)) and Omega(f(n)). | |-----------------------------------------------------------------------------| But how can a function be sandwiched between f(n) and f(n)? Easy: we choose different constants (c and d) for the upper bound and lower bound. For instance, here is a function T(n) in Theta(n): c f(n) = 10 n ^ / | / T(n) | / ** | / * * | / *** * ** | / * * * | *** / * * * | ** ** / * * * |* ** * * * * / * * ** * | / ** *** ˜˜˜ | / ˜˜˜˜˜ | / ˜˜˜˜˜ | / ˜˜˜˜˜ | / ˜˜˜˜˜ d f(n) = 2 n | / ˜˜˜˜˜ |/ ˜˜˜˜˜ O˜˜------------------------------> n If we extend this graph infinitely far to the right, and find that T(n) remains always sandwiched between 2n and 10n, then T(n) is in Theta(n). If T(n) is an

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## This note was uploaded on 02/21/2010 for the course CS 61B taught by Professor Canny during the Spring '01 term at Berkeley.

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21 - 14:57:33 CS 61B Lecture 21 Wednesday ASYMPTOTIC...

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