Pairing heaps: the forward variant

TL;DR

This paper analyzes the forward variant of pairing heaps, showing that insertions and deletions operate in amortized time significantly better than the previous bound, using a new potential function for analysis.

Contribution

It provides the first improved amortized time bounds for the forward variant of pairing heaps, previously only bounded by a quadratic bound.

Findings

Insertion and deletion take amortized time O(log n * 4^{√log n})

First improvement over the previous O(√n) bound by Fredman et al.

Introduces a new potential function based on parent-child rank differences.

Abstract

The pairing heap is a classical heap data structure introduced in 1986 by Fredman, Sedgewick, Sleator, and Tarjan. It is remarkable both for its simplicity and for its excellent performance in practice. The "magic" of pairing heaps lies in the restructuring that happens after the deletion of the smallest item. The resulting collection of trees is consolidated in two rounds: a left-to-right pairing round, followed by a right-to-left accumulation round. Fredman et al. showed, via an elegant correspondence to splay trees, that in a pairing heap of size $n$ all operations take $O(\log{n})$ amortized time. They also proposed an arguably more natural variant, where both pairing and accumulation are performed in a combined left-to-right round (called the forward variant of pairing heaps). The analogy to splaying breaks down in this case, and the analysis of the forward variant was left open. In this paper we show that inserting an item and deleting the minimum in a forward-variant pairing heap both take amortized time $O(\log{n} \cdot 4^{\sqrt{\log{n}}} )$. This is the first improvement over the $O(\sqrt{n})$ bound showed by Fredman et al. three decades ago. Our analysis relies on a new potential function that tracks parent-child rank-differences in the heap.