A general framework for the realistic analysis of sorting and searching algorithms. Application to some popular algorithms

We describe a general framework for realistic analysis of sorting and searching algorithms, and we apply it to the average-case analysis of five basic algorithms: three sorting algorithms (QuickSort, InsertionSort, BubbleSort) and two selection algorithms (QuickMin and SelectionMin). Usually, the analysis deals with the mean number of key comparisons, but, here, we view keys as words produced by the same source, which are compared via their symbols in the lexicographic order. The “realistic” cost of the algorithm is now the total number of symbol comparisons performed by the algorithm, and, in this context, the average–case analysis aims to provide estimates for the mean number of symbol comparisons used by the algorithm. For sorting algorithms, and with respect to key comparisons, the average-case complexity of QuickSort is asymptotic to 2n logn, InsertionSort to n2/4 and BubbleSort to n2/2. With respect to symbol comparisons, we prove that their average-case complexity becomes Θ(n log2 n),Θ(n2),Θ(n2 logn). For selection algorithms, and with respect to key comparisons, the average-case complexity of QuickMin is asymptotic to 2n, of SelectionMin is n − 1. With respect to symbol comparisons, we prove that their average-case complexity remains Θ(n). In these five cases, we describe the dominant constants which exhibit the probabilistic behaviour of the source (namely, entropy, and various notions of coincidence) with respect to the algorithm. 1998 ACM Subject Classification F2.2: Pattern matching, sorting and searching – G2.1: Generating functions, permutations – G4: Algorithm design and analysis – H1.1: Information theory – I1.2: Analysis of algorithms