Analysis of Smooth Heaps and Slim Heaps

TL;DR

This paper introduces a simplified variant of the smooth heap, called the slim heap, and provides a comprehensive analysis showing it has optimal amortized bounds, with experimental results indicating competitive performance.

Contribution

The paper presents the slim heap, a simpler variant of the smooth heap, along with a new analysis establishing optimal amortized bounds for these self-adjusting heaps.

Findings

Slim heaps match the best known bounds for self-adjusting heaps.

Smooth heaps and slim heaps are competitive with pairing heaps in experiments.

Slim heaps outperform pairing heaps in some scenarios.

Abstract

The smooth heap is a recently introduced self-adjusting heap [Kozma, Saranurak, 2018] similar to the pairing heap [Fredman, Sedgewick, Sleator, Tarjan, 1986]. The smooth heap was obtained as a heap-counterpart of Greedy BST, a binary search tree updating strategy conjectured to be \emph{instance-optimal} [Lucas, 1988], [Munro, 2000]. Several adaptive properties of smooth heaps follow from this connection; moreover, the smooth heap itself has been conjectured to be instance-optimal within a certain class of heaps. Nevertheless, no general analysis of smooth heaps has existed until now, the only previous analysis showing that, when used in \emph{sorting mode} ($n$ insertions followed by $n$ delete-min operations), smooth heaps sort $n$ numbers in $O(n\lg n)$ time. In this paper we describe a simpler variant of the smooth heap we call the \emph{slim heap}. We give a new, self-contained analysis of smooth heaps and slim heaps in unrestricted operation, obtaining amortized bounds that match the best bounds known for self-adjusting heaps. Previous experimental work has found the pairing heap to dominate other data structures in this class in various settings. Our tests show that smooth heaps and slim heaps are competitive with pairing heaps, outperforming them in some cases, while being comparably easy to implement.