A Practical Introduction to Data Structures and Algorithm Analysis

Sec. 7.2 Three Θ(n 2 ) Sorting Algorithms 243Insertion Bubble SelectionComparisons:Best Case Θ(n) Θ(n 2 ) Θ(n 2 )Average Case Θ(n 2 ) Θ(n 2 ) Θ(n 2 )Worst Case Θ(n 2 ) Θ(n 2 ) Θ(n 2 )Swaps:Best Case 0 0 Θ(n)Average Case Θ(n 2 ) Θ(n 2 ) Θ(n)Worst Case Θ(n 2 ) Θ(n 2 ) Θ(n)Figure 7.5 A comparison of the asymptotic complexities for three simple sortingalgorithms.7.2.4 The Cost of Exchange SortingFigure 7.5 summarizes the cost of Insertion, Bubble, and Selection Sort in terms oftheir required number of comparisons and swaps 1 in the best, average, and worstcases. The running time for each of these sorts is Θ(n 2 ) in the average and worstcases.The remaining sorting algorithms presented in this chapter are significantly betterthan these three under typical conditions. But before continuing on, it is instructiveto investigate what makes these three sorts so slow. The crucial bottleneckis that only adjacent records are compared. Thus, comparisons and moves (in allbut Selection Sort) are by single steps. Swapping adjacent records is called an exchange.Thus, these sorts are sometimes referred to as exchange sorts. The costof any exchange sort can be at best the total number of steps that the records in thearray must move to reach their “correct” location (i.e., the number of inversions foreach record).What is the average number of inversions? Consider a list L containing n values.Define L R to be L in reverse. L has n(n−1)/2 distinct pairs of values, each ofwhich could potentially be an inversion. Each such pair must either be an inversionin L or in L R . Thus, the total number of inversions in L and L R together is exactlyn(n − 1)/2 for an average of n(n − 1)/4 per list. We therefore know with certaintythat any sorting algorithm which limits comparisons to adjacent items will cost atleast n(n − 1)/4 = Ω(n 2 ) in the average case.1 There is a slight anomaly with Selection Sort. The supposed advantage for Selection Sort is itslow number of swaps required, yet Selection Sort’s best-case number of swaps is worse than that forInsertion Sort or Bubble Sort. This is because the implementation given for Selection Sort does notavoid a swap in the case where record i is already in position i. The reason is that it usually takesmore time to repeatedly check for this situation than would be saved by avoiding such swaps.