Basic MOQA Operations

The present chapter introduces the basicMOQAoperations, including the Random Product, the Random Deletion and Percolation, the Random Projection, the Random Split and the Top and Bot operations. These are sufficient to implement many wellknown algorithms as illustrated in Chapter 8.We recall that the operationsMOQAproduct operation and the MOQAdeletion operation have been described in an informal way in Section 1.8.1, while theMOQAsplit operation has been described in Section 1.7.1.

Each of theMOQAoperations is shown to be random bag preserving. Deletion operations typically are not included in the context of automated average-case analysis, since the analysis of deletions with respect to average-case time is well-known to be problematic, even in the context of traditional average-case analysis. Hence the Random Deletion opens up the way for the inclusion of novel algorithms, such as Percolating Heapsort and Treapsort, which are analyzed in Chapter 9. The Extension Theorem of Chapter 4 is applied to extend these operations from local applications on isolated subsets to applications over the entire random structure. Uniformly random bag preserving operations are singled out as of particular interest, since this type of operations enables simplifications of probability computations in later chapters. TheMOQAoperations are shown to preserve series-parallel data structures which yields a characterization of the so-called MOQAatomic-constructible data structures as series-parallel orders. Finally, some simplifications for the series-parallel case are obtained in the context of the computation of cardinalities of random structures. Such simplifications for series-parallel orders will also be useful in the context of Chapter 6, which regards the average-case analysis of the basicMOQAoperations. Finally, separative functions as a sufficient condition for random bag preservation are discussed in relation to the basic MOQAoperations.