Abstract Data Types and Algorithms by Manoochehr Azmoodeh (auth.)

By Manoochehr Azmoodeh (auth.)

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The asymptotic time complexity of this statement can now be determined using the above rules. 1) = O(I) n Time for the outer loop = ~ 0(1) 1=1 Using the second rule =oC~1 1) =0 = O(l/2n 2 (n(;+l)) + 1/2n) = O(l/2n 2 ) =O(n 2 ) Therefore, the whole statement has asymptotic time complexity O(n 2 ). The Big 0 notation enables us to define an upper bound for the growth rate of time complexity of an algorithm. To find a lower bound for time complexity, we define Big n as: T(n) = n (f(n)) if there exists a constant C such that T(n) for infinitely many n ~ Cf(n) The reason that Big n is defined for infinitely many n and not for all n greater than a given number is that many algorithms are very fast on some sizes of input but not all.

Thus statement 1 will be executed n-1 times. We cannot find the number of times the for loop (inside the while loop) will be executed as a function of i. However, if s; is the value of s inside the while loop for a given value of i, then the total number of times statemenh will be executed is ~ s1• The sum of the s1 t values will be ~ I s; = m, since size is initially set tom and in each iteration of the while loop s; is subtracted from size. Thus the total time due to statement2 will be m x O(n).

62)n for some k. 62n) This indicates that algorithm A requires at least an exponential amount of time. To calculate T(n): T(n) = 1 + T(n) n ~ t=O F(i) > 1 +x +x 2 +x 3 + ... 62n+l ). Again this suggests that the procedure seql is an exponential algorithm. However, the procedure seq2 can be 20 Abstract Data Types and Algorithms easily shown to be O(n). 1 can be shown to be O(n). 2. As n gets larger, O(n) gets larger much faster than does the O(log n) algorithm. 62n+l) behaves similarly when compared with the O(n) algorithm of the seq2 program.

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