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A Fibonacci heap is a deterministic data structure implementing a priorityqueue with optimal amortized operation costs. An unfortunate aspect ofFibonacci heaps is that they must maintain a "mark bit" which serves only toensure efficiency of heap operations, not correctness. Karger proposed a simplerandomized variant of Fibonacci heaps in which mark bits are replaced by coinflips. This variant still has expected amortized cost $O(1)$ for insert,decrease-key, and merge. Karger conjectured that this data structure hasexpected amortized cost $O(\log s)$ for delete-min, where $s$ is the number ofheap operations.  We give a tight analysis of Karger's randomized Fibonacci heaps, resolvingKarger's conjecture. Specifically, we obtain matching upper and lower bounds of$\Theta(\log^2 s / \log \log s)$ for the runtime of delete-min. We also prove atight lower bound of $\Omega(\sqrt{n})$ on delete-min in terms of the number ofheap elements $n$. The request sequence used to prove this bound also solves anopen problem of Fredman on whether cascading cuts are necessary. Finally, wegive a simple additional modification to these heaps which yields a tightruntime $O(\log^2 n / \log \log n)$ for delete-min.

Replacing Mark Bits with Randomness in Fibonacci Heaps