Amortized Analysis of Leftist Heaps

TL;DR

This paper analyzes the amortized complexity of leftist heaps, comparing them with skew heaps, and shows how weight-biased variants match skew heaps' bounds, offering a trade-off between worst-case and amortized efficiency.

Contribution

It provides the first detailed amortized analysis of leftist heaps, establishing bounds comparable to skew heaps and exploring the impact of weight-biased variants.

Findings

Weight-biased leftist heaps satisfy the same amortized bounds as skew heaps.

Storage of weights limits worst-case complexity without affecting amortized bounds.

Open problem remains whether rank-biased leftist heaps' bounds are tight.

Abstract

Leftist heaps and skew heaps are two well-known data structures for mergeable priority queues. Leftist heaps are constructed for efficiency in the worst-case sense whereas skew heaps are self-adjusting, designed for efficiency in the amortized sense. In this paper, we analyze the amortized complexity of leftist heaps to initiate a full performance comparison with skew heaps. We consider both the leftist heaps originally developed by Crane and Knuth, which are also referred to as rank-biased (or, height-biased) leftist heaps, and the weight-biased leftist heaps introduced by Cho and Sahni. We show how weight-biased leftist heaps satisfy the same exact amortized bounds as skew heaps. With these matching bounds we establish a nice trade-off in which storage of weights is used to limit the worst-case complexity of leftist heaps, without affecting the amortized complexity compared to skew heaps. For rank-biased leftist heaps, we obtain the same amortized lower bounds as for skew heaps, but whether these bounds are tight is left as an open problem.