Average Case Analysis of Heap Building by Repeated Insertion

An array A[1], . . . , A[n] is a (min) heap if A[⌊i/2⌋] ≤ A[i], for i = 2, . . . , n. The heap is a much used and much studied data structure (for example, see Knuth [K]). Williams’ method for constructing a heap involves repeatedly inserting a key at the bottom of the heap and ’bubbling’ it up. This method requires Θ(n logn) time in the worst case. Let Wn denote the number of ’swaps’ (or ’promotions’) performed in the building of a heap with n keys (see Appendix 1 for a precise definition). We make the usual assumption that each initial permutation of the keys is equally likely. It is shown by Bollobás and Simon [BS] and Frieze [F] that for perfect heaps (that is, for n equal to 2k−1) the expected number E[Wn]