Efficiency of Self-Adjusting Heaps

TL;DR

This paper proves that recent self-adjusting heaps like slim and smooth heaps support decrease-key operations in O(log log n) time, significantly improving previous bounds and matching known lower bounds.

Contribution

It establishes tight amortized bounds for slim and smooth heaps, demonstrating their efficiency for decrease-key and other operations, and extends analysis to the multipass pairing heap.

Findings

Slim and smooth heaps support decrease-key in O(log log n) time.

Bounds for delete-min and delete are O(log n); insert and meld are O(1).

Results are tight and improve upon previous bounds.

Abstract

Since the invention of the pairing heap by Fredman, Sedgewick, Sleator, and Tarjan, it has been an open question whether this or any other simple "self-adjusting" heap supports decrease-key operations in $O(\log\log n)$ time, where $n$ is the number of heap items. Using powerful new techniques, we answer this question in the affirmative. We prove that both slim and smooth heaps, recently introduced self-adjusting heaps, support heap operations in the following amortized time bounds: $O(\log n)$ for delete-min and delete, $O(\log\log n)$ for decrease-key, and $O(1)$ for all other heap operations, including insert and meld, where $n$ is the number of heap items that are eventually deleted: Items inserted but never deleted do not count in the bounds. We also analyze the multipass pairing heap, a variant of pairing heaps. For this heap implementation, we obtain the same bounds except for decrease-key, for which our bound is $O(\log\log n \log\log\log n)$, where again items that are never deleted do not count in $n$. Our bounds significantly improve the best previously known bounds for all three data structures. For slim and smooth heaps our bounds are tight, since they match lower bounds of Iacono and \"Ozkan.