A Tight Analysis of Slim Heaps and Smooth Heaps

TL;DR

This paper provides a precise analysis of smooth and slim heaps, showing they are optimal self-adjusting heap implementations within a broad theoretical model, with tight bounds on their operation times.

Contribution

It introduces tight amortized bounds for smooth and slim heaps, demonstrating they match the lower bounds for all self-adjusting heaps in a comprehensive theoretical model.

Findings

Smooth and slim heaps have optimal amortized bounds.

They match the lower bounds for self-adjusting heaps.

Analysis extends to all heaps in the pure heap model.

Abstract

The smooth heap and the closely related slim heap are recently invented self-adjusting implementations of the heap (priority queue) data structure. We analyze the efficiency of these data structures. We obtain the following amortized bounds on the time per operation: $O(1)$ for make-heap, insert, find-min, and meld; $O(\log\log n)$ for decrease-key; and $O(\log n)$ for delete-min and delete, where $n$ is the current number of items in the heap. These bounds are tight not only for smooth and slim heaps but for any heap implementation in Iacono and \"{O}zkan's pure heap model, intended to capture all possible "self-adjusting" heap implementations. Slim and smooth heaps are the first known data structures to match Iacono and \"{O}zkan's lower bounds and to satisfy the constraints of their model. Our analysis builds on Pettie's insights into the efficiency of pairing heaps, a classical self-adjusting heap implementation.