Heaps on heaps

Abstract

As part of a study of the general issue of complexity of comparison based problems, as well as interest in the specific problem, we consider the task of performing the basic priority queue operations on a heap. We show that in the worst case:

• (i) log log n comparisons are necessary and sufficient to insert an element into a heap. (This improves the previous upper and lower bounds of log n and 0(1).)

• (ii) log n+g(n)−ε(n) comparisons are necessary and sufficient to replace the maximum in a heap. (ε(n) denotes a function in the range [0,1]. This improves the previous upper and lower bounds of 2 log n and log n.)

• (iii) 1.625n+0(log n ^* g(n)) comparisons are sufficient to create a heap. 1.37 ... n comparisons are necessary not only in the worst case but also on the average.