A Simpler Proof that Pairing Heaps Take O(1) Amortized Time per Insertion

TL;DR

This paper provides a simpler and more efficient proof that pairing heaps support insertions in amortized constant time, improving understanding and analysis of their performance.

Contribution

The paper offers a simplified proof of Iacono's result that pairing heaps have O(1) amortized insertion time, using a natural heap representation.

Findings

Insertion operation takes O(1) amortized time.

Simplified proof reduces constant factors.

Analysis uses natural heap representation.

Abstract

The pairing heap is a simple "self-adjusting" implementation of a heap (priority queue). Inserting an item into a pairing heap or decreasing the key of an item takes O(1) time worst-case, as does melding two heaps. But deleting an item of minimum key can take time linear in the heap size in the worst case. The paper that introduced the pairing heap proved an O(log n) amortized time bound for each heap operation, where n is the number of items in the heap or heaps involved in the operation, by charging all but O(log n) of the time for each deletion to non-deletion operations, O(log n) to each. Later Iacono found a way to reduce the amortized time per insertion to O(1) and that of meld to zero while preserving the O(log n) amortized time bound for the other update operations. We give a simpler proof of Iacono's result with significantly smaller constant factors. Our analysis uses the natural representation of pairing heaps instead of the conversion to a binary tree used in the original analysis and in Iacono's.