Replacing Mark Bits with Randomness in Fibonacci Heaps

TL;DR

This paper provides a tight analysis of a randomized variant of Fibonacci heaps that replaces mark bits with coin flips, resolving a conjecture and establishing new bounds on delete-min operation costs.

Contribution

It proves tight bounds for the randomized Fibonacci heap's delete-min operation, confirming Karger's conjecture and addressing open problems about cascading cuts.

Findings

Expected amortized delete-min cost is Θ(log^2 s / log log s)

Lower bound of Ω(√n) for delete-min in terms of heap size

A simple modification achieves O(log^2 n / log log n) delete-min runtime

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 $\Theta(\log^2 s / \log \log s)$ for the runtime of delete-min. We also prove a tight lower bound of $\Omega(\sqrt{n})$ 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.