Replacing Mark Bits with Randomness in Fibonacci Heaps

Abstract

A Fibonacci heap is a deterministic data structure implementing a priority queue with optimal amortized operation costs. An unfortunate aspect of Fibonacci heaps is that they must maintain a “mark bit” which serves only to ensure efficiency of heap operations, not correctness. Karger proposed a simple randomized variant of Fibonacci heaps in which mark bits are replaced by coin flips. This variant 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.

We give a tight analysis of Karger’s randomized Fibonacci heaps, resolving Karger’s conjecture. Specifically, we obtain matching upper and lower bounds of $\varTheta (\log ^2 s / \log \log s)$ for the runtime of delete-min. We also prove a tight lower bound of $\varOmega (\sqrt{n})$ on delete-min in terms of the number of heap elements n. The request sequence used to prove this bound also solves an open problem of Fredman on whether cascading cuts are necessary. Finally, we give a simple additional modification to these heaps which yields a tight runtime $O(\log ^2 n / \log \log n)$ for delete-min.