Heaps Simplified

TL;DR

This paper explores the design space of comparison-based, amortized-efficient heap implementations, introducing new variants and a simplified approach that combines performance with ease of understanding.

Contribution

It presents a simple derivation of binomial queues from tournaments, introduces two new heap variants, and develops the rank-pairing heap with improved performance guarantees.

Findings

Introduces the binomial queue in a natural way from tournaments.

Develops two new heap variants: one-tree and one-pass versions.

Creates the rank-pairing heap, combining Fibonacci heap performance with simplicity.

Abstract

The heap is a basic data structure used in a wide variety of applications, including shortest path and minimum spanning tree algorithms. In this paper we explore the design space of comparison-based, amortized-efficient heap implementations. From a consideration of dynamic single-elimination tournaments, we obtain the binomial queue, a classical heap implementation, in a simple and natural way. We give four equivalent ways of representing heaps arising from tournaments, and we obtain two new variants of binomial queues, a one-tree version and a one-pass version. We extend the one-pass version to support key decrease operations, obtaining the {\em rank-pairing heap}, or {\em rp-heap}. Rank-pairing heaps combine the performance guarantees of Fibonacci heaps with simplicity approaching that of pairing heaps. Like pairing heaps, rank-pairing heaps consist of trees of arbitrary structure, but these trees are combined by rank, not by list position, and rank changes, but not structural changes, cascade during key decrease operations.