The amortized cost of finding the minimum

We obtain an essentially optimal tradeoff between the amortized cost of the three basic priority queue operations insert, delete and find-min in the comparison model. More specifically, we show that

for any fixed ε > 0, where n is the number of items in the priority queue and A(insert), A(delete) and A(find-min) are the amortized costs of the insert, delete and find-min operations, respectively. In particular, if A(insert) + A(delete) = O(1), then A(find-min) = Ω(n), and A(find-min) = O(n^α), for some α < 1, only if A(insert) + A(delete) = Ω(log n). (We can, of course, have A(insert) = O(1), A(delete) = O(log n), or vice versa, and A(find-min) = O(1).) Our lower bound holds even if randomization is allowed. Surprisingly, such fundamental bounds on the amortized cost of the operations were not known before. Brodal, Chaudhuri and Rad-hakrishnan, obtained similar bounds for the worst-case complexity of find-min.