Abstract

A Fibonacci heap is a deterministic data structure implementing a priority queue with op-timal amortized asymptotic operation costs. An unaesthetic aspect of Fibonacci heaps is that they must maintain a “mark bit ” which serves only to ensure efficiency of heap operations, not their correctness. Karger proposed a simple randomized variant of Fibonacci heaps in which mark bits are replaced by coin flips. This modified data structure still has expected amortized cost O(1) for insert, decrease-key, and merge. Karger conjectured that this data structure has expected amortized cost O(log s) for delete-min, where s is the number of heap operations. In this paper, we give a tight analysis of randomized Fibonacci heaps, resolving Karger’s conjecture. Specifically, we obtain matching upper and lower bounds of Θ(log2 s / log log s) for the runtime of delete-min. We also prove a tight lower bound of Ω( n) on delete-min in terms of the number of heap elements n. Finally, we give a simple additional modification to these heaps which yields a tight runtime O(log2 n / log log n) for delete-min.