By M. H. Alsuwaiyel
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Extra resources for Algorithms: Design Techniques and Analysis (Lecture Notes Series on Computing)
End while 8. end for 9. return count 6. , an integer whose square root is integer. Algorithm PSUM computes for each perfect square j between 1 and n the sum i. (Obviously, this sum can be computed more efficiently). can be computed in 0(1)time. We compute the We will assume that running time of the algorithm as follows. The outer and inner for loops are executed k = f i and j2 times, respectively. Hence, the number of iterations performed by the inner for loop is c:=l It follows that the running time of the algorithm is Q ( ~ L ' .
2 Counting the frequency of basic operations In some algorithms, it is cumbersome, or even impossible, to make use of the previous method in order to come up with a tight estimate of its running time. Unfortunately, at this point we have not covered good examples of such algorithms. Good examples that will be covered in subsequent chapters include the single-source shortest path problem, Prim’s algorithm for finding minimum spanning trees, depth-first search, computing convex hulls and others. However, Algorithm MERGE will serve as a reasonable candidate.
For j t l t o n count t cmnt 4. 5. end for 6. nc-n/2 7. end while 8. return count +1 Basic Concepts in Algorithmic Analysis 36 + The while loop is executed k 1 times, where k = logn. The for loop is executed n times, and then n/2,n/4,. . , l . 11 (page 78) on this geometric series, the number of times Step 4 is executed is k k j=0 j=O Since the running time is proportional to count, we conclude that it is Q(n). 23 Consider Algorithm COUNT2, which consists of two nested loops and a variable count which counts the number of iterations performed by the algorithm on input n, which is a positive integer.