## Abstract

We present a new connection between self-adjusting binary search trees (BSTs) and heaps, two fundamental, extensively studied, and practically relevant families of data structures (Allen, Munro, 1978; Sleator, Tarjan, 1983; Fredman, Sedgewick, Sleator, Tarjan, 1986; Wilber, 1989; Fredman, 1999; Iacono, Özkan, 2014). Roughly speaking, we map an arbitrary heap algorithm within a broad and natural model, to a corresponding BST algorithm with the same cost on a dual sequence of operations (i.e. the same sequence with the roles of time and key-space switched). This is the first general transformation between the two families of data structures. There is a rich theory of dynamic optimality for BSTs (i.e. the theory of competitiveness between BST algorithms). The lack of an analogous theory for heaps has been noted in the literature (e.g. Pettie; 2005, 2008). Through our connection, we transfer all instance-specific lower bounds known for BSTs to a general model of heaps, initiating a theory of dynamic optimality for heaps. On the algorithmic side, we obtain a new, simple and efficient heap algorithm, which we call the smooth heap. We show the smooth heap to be the heap-counterpart of Greedy, the BST algorithm with the strongest proven and conjectured properties from the literature, widely believed to be instance-optimal (Lucas, 1988; Munro, 2000; Demaine et al., 2009). Assuming the optimality of Greedy, the smooth heap is also optimal within our model of heap algorithms. Intriguingly, the smooth heap, although derived from a non-practical BST algorithm, is simple and easy to implement (e.g. it stores no auxiliary data besides the keys and tree pointers). It can be seen as a variation on the popular pairing heap data structure, extending it with a “power-of-two-choices” type of heuristic. For the smooth heap we obtain instance-specific upper bounds, with applications in adaptive sorting, and we see it as a promising candidate for the long-standing question of a simpler alternative to Fibonacci heaps. The paper is dedicated to Raimund Seidel on occasion of his sixtieth birthday.

Original language | English |
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Title of host publication | STOC 2018 - Proceedings of the 50th Annual ACM SIGACT Symposium on Theory of Computing |

Place of Publication | New York |

Publisher | Association for Computing Machinery, Inc |

Pages | 801-814 |

Number of pages | 13 |

ISBN (Electronic) | 978-1-4503-5559-9 |

DOIs | |

Publication status | Published - 20 Jun 2018 |

Event | 50th Annual ACM Symposium on Theory of Computing, STOC 2018 - Los Angeles, United States Duration: 25 Jun 2018 → 29 Jun 2018 |

### Conference

Conference | 50th Annual ACM Symposium on Theory of Computing, STOC 2018 |
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Country/Territory | United States |

City | Los Angeles |

Period | 25/06/18 → 29/06/18 |

## Keywords

- Binary search trees
- Heaps
- Self-adjusting data structures
- Sorting
- sorting
- self-adjusting data structures
- binary search trees
- heaps