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Additional info for Algorithms: Design Techniques and Analysis (Lecture Notes Series on Computing)
Piil Fig. 4 Example of bottom-up merge sorting when n is not a power of 2. Algorithm BOTTOMUPSORT implements this idea. T h e algorithm maintains the variable s which is the size of sequences t o be merged. Initially, s is set to 1, and is doubled in each iteration of the outer while loop. i 1, i s and a t define the boundaries of the two sequences to be merged. Step 8 is needed in t h e case when n is not a multiple oft. In this case, if the number of remaining elements, which is n - i, is greater than s, then one more merge is applied on a sequence of size s and the remaining elements.
End while 8. A[j l ] t z 9. end for +- + Unlike Algorithm SELECTIONSORT, the number of element comparisons done by Algorithm INSERTIONSORT depends on the order of the input elements. It is easy to see that the number of element comparisons is minimum when the array is already sorted in nondecreasing order. In this case, the number of element comparisons is exactly n - 1 , as each element A [ i ] , 2 5 i 5 n , is compared with A[i - 11 only. On the other hand, the maximum number of element comparisons occurs if the array is already sorted in decreasing order and all elements are distinct.
G. how and on what machine the algorithm is implemented and in what language or even what compiler or programmer's skills, to mention a few. Therefore, we should be content with only an approximation of the exact time, But, first of all, when assessing an algorithm's efficiency, do we have to deal with exact or even approximate times? It turns out that we redly do not need even approximate times. This is supported by many factors, some of which are the following. First, when analyzing the running time of an algorithm, we usually compare its behavior with another algo- 22 Basic Concepts in Algorithmic Analysis rithm that solves the same problem, or even a different problem.