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Pairing heaps: the forward variant

Authors Dani Dorfman, Haim Kaplan, László Kozma, Uri Zwick

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Dani Dorfman
  • Blavatnik School of Computer Science, Tel Aviv University, Israel
Haim Kaplan
  • Blavatnik School of Computer Science, Tel Aviv University, Israel
László Kozma
  • Eindhoven University of Technology, The Netherlands
Uri Zwick
  • Blavatnik School of Computer Science, Tel Aviv University, Israel

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Dani Dorfman, Haim Kaplan, László Kozma, and Uri Zwick. Pairing heaps: the forward variant. In 43rd International Symposium on Mathematical Foundations of Computer Science (MFCS 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 117, pp. 13:1-13:14, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2018)


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 heap 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) * 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.

Subject Classification

ACM Subject Classification
  • Theory of computation → Data structures design and analysis
  • data structure
  • priority queue
  • pairing heap


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