## Pairing Heaps with Costless Meld (2009)

### BibTeX

@MISC{Elmasry09pairingheaps,

author = {Amr Elmasry},

title = {Pairing Heaps with Costless Meld},

year = {2009}

}

### OpenURL

### Abstract

Improving the structure and analysis in [1], we give a variation of the pairing heaps that has amortized zero cost per meld (compared to an O(log log n) in [1]) and the same amortized bounds for all other operations. More precisely, the new pairing heap requires: no cost per meld, O(1) per find-min and insert, O(log n) per delete-min, and O(log log n) per decrease-key. These bounds are the best known for any self-adjusting heap, and match the lower bound proven by Fredman for a family of such heaps. Moreover, our structure is even simpler than that in [1].

### Citations

591 |
Fibonacci heaps and their uses in improved network optimization algorithms
- Fredman, Tarjan
(Show Context)
Citation Context ...priority queues [8, 9] or with some of its variants [2, 3, 12]. Such experiments illustrate that the pairing heaps are practically efficient and superior to other heaps, including the Fibonacci heaps =-=[6]-=-. In this paper, we give a variation of the pairing heaps that achieves the best known bounds for any self-adjusting heap for all operations. Namely, our amortized bounds are: zero cost per meld, O(1)... |

143 |
Amortized computational complexity
- TARJAN
- 1983
(Show Context)
Citation Context ...white. 1. Inactive nodes: Every node x with w(x) = 0. 2. Active nodes: Other nodes. To bound the cost of the heap operations, we use a combination of the potential function and the accounting methods =-=[13]-=-. 3.1 The potential function Consider the link between a node x and its parent p(x). Let w ′ (x) be the number of white descendants of p(x) restricted to the subtrees of the right siblings of x, inclu... |

87 |
An empirical comparison of priority-queues and event-set implementations
- Jones
- 1986
(Show Context)
Citation Context ... O(log n) per delete-min, and O(log log n) per decrease-key and meld. See Table 1. Several experiments were conducted on the pairing heaps, either comparing its performance with other priority queues =-=[8, 9]-=- or with some of its variants [2, 3, 12]. Such experiments illustrate that the pairing heaps are practically efficient and superior to other heaps, including the Fibonacci heaps [6]. In this paper, we... |

47 |
Self-Adjusting Heaps
- Sleator, Tarjan
- 1986
(Show Context)
Citation Context ... O(log log n) zero The original analysis of the pairing heaps [5] showed an O(log n) amortized cost for all operations. Another self-adjusting heap that requires O(log n) amortized cost per operation =-=[11]-=- is the skew heap. Theoretical results concerning the pairing heaps were later obtained through the years. Stasko and Vitter [12] suggested a variant that achieves O(1) amortized cost per insert. The ... |

41 | An empirical assessment of algorithms for constructing a minimal spanning tree
- Moret, Shapiro
- 1994
(Show Context)
Citation Context ... O(log n) per delete-min, and O(log log n) per decrease-key and meld. See Table 1. Several experiments were conducted on the pairing heaps, either comparing its performance with other priority queues =-=[8, 9]-=- or with some of its variants [2, 3, 12]. Such experiments illustrate that the pairing heaps are practically efficient and superior to other heaps, including the Fibonacci heaps [6]. In this paper, we... |

31 |
Pairing heaps: Experiments and analysis
- Stasko
- 1987
(Show Context)
Citation Context ...oldt Fellowship. 1Table 1: Previous results for upper bounds on pairing-heap’s operations insert delete-min decrease-key meld Fredman et al. [5] O(log n) O(log n) O(log n) O(log n) Stasko and Vitter =-=[12]-=- O(1) O(log n) O(log n) O(log n) Iacono [7] O(1) O(log n) O(log n) zero Pettie [10] O(2 2 √ √ √ log log n) 2 log log O(log n) O(2 n) 2 log log O(2 n) Elmasry [1] O(1) O(log n) O(log log n) O(log log n... |

22 |
The pairing heap: a new form of self adjusting heap. Algorithmica 1(1
- Fredman, Sedgewick, et al.
- 1986
(Show Context)
Citation Context ...st known for any self-adjusting heap, and match the lower bound proven by Fredman for a family of such heaps. Moreover, our structure is even simpler than that in [1]. 1 Introduction The pairing heap =-=[5]-=- is a self-adjusting heap that is implemented as a single heap-ordered multi-way tree. The basic operation on a pairing heap is the linking operation in which two trees are combined by linking the roo... |

18 |
Improved upper bounds for pairing heaps
- Iacono
- 2000
(Show Context)
Citation Context ... for upper bounds on pairing-heap’s operations insert delete-min decrease-key meld Fredman et al. [5] O(log n) O(log n) O(log n) O(log n) Stasko and Vitter [12] O(1) O(log n) O(log n) O(log n) Iacono =-=[7]-=- O(1) O(log n) O(log n) zero Pettie [10] O(2 2 √ √ √ log log n) 2 log log O(log n) O(2 n) 2 log log O(2 n) Elmasry [1] O(1) O(log n) O(log log n) O(log log n) This paper O(1) O(log n) O(log log n) zer... |

12 |
On the efficiency of pairing heaps and related data structures
- Fredman
- 1999
(Show Context)
Citation Context ...[12] suggested a variant that achieves O(1) amortized cost per insert. The bounds for the standard implementation were later improved by Iacono [7] to: O(1) per inset, and zero cost per meld. Fredman =-=[4]-=- showed that Ω(log log n) amortized comparisons, in the decision-tree model, would be necessary per decrease-key operation for a family of heaps that generalizes the pairing heaps. Pettie [10] proved ... |

10 | Towards a final analysis of pairing heaps
- Pettie
- 2005
(Show Context)
Citation Context ...rations insert delete-min decrease-key meld Fredman et al. [5] O(log n) O(log n) O(log n) O(log n) Stasko and Vitter [12] O(1) O(log n) O(log n) O(log n) Iacono [7] O(1) O(log n) O(log n) zero Pettie =-=[10]-=- O(2 2 √ √ √ log log n) 2 log log O(log n) O(2 n) 2 log log O(2 n) Elmasry [1] O(1) O(log n) O(log log n) O(log log n) This paper O(1) O(log n) O(log log n) zero The original analysis of the pairing h... |

5 | Pairing heaps with O(log log n) decrease cost
- Elmasry
- 2009
(Show Context)
Citation Context ...s with Costless Meld arXiv:0903.4130v2 [cs.DS] 9 Apr 2009 Amr Elmasry ∗ Max-Planck Institut für Informatik Saarbrücken, Germany elmasry@mpi-inf.mpg.de Abstract Improving the structure and analysis in =-=[1]-=-, we give a variation of the pairing heaps that has amortized zero cost per meld (compared to an O(log log n) in [1]) and the same amortized bounds for all other operations. More precisely, the new pa... |

5 |
Parameterized self-adjusting heaps
- Elmasry
(Show Context)
Citation Context ...inked in order from right to left, where each tree is linked with the tree resulting from the linkings of the trees to its right. Other variants with different delete-min implementation were given in =-=[2, 4, 6]-=-. ∗ A. Elmasry was supported by an Alexander von Humboldt Fellowship. 1• meld. Link the two trees representing the two heaps. The original analysis of the pairing heaps [6] showed an O(log n) amortiz... |

3 |
A priority queue transform
- Fredman
- 1999
(Show Context)
Citation Context ...inked in order from right to left, where each tree is linked with the tree resulting from the linkings of the trees to its right. Other variants with different delete-min implementation were given in =-=[2, 3, 5]-=-. ∗ Supported by an Alexander von Humboldt Fellowship. 1Table 1: Previous results for upper bounds on pairing-heap’s operations insert delete-min decrease-key meld Fredman et al. [5] O(log n) O(log n... |

3 |
Adaptive properties of pairing heaps
- Elmasry
- 2001
(Show Context)
Citation Context ...operation for a family of heaps that generalizes the pairing heaps. Pettie [11] proved amortized costs of: O(log n) per delete-min, and O(2 2√ log log n ) for other operations. Iacono [8] and Elmasry =-=[3]-=- studied some distribution-sensitive properties of the pairing heaps. Recently, Elmasry [1] introduced a variant that achieves the following amortized bounds: O(1) per insert, O(log n) per deletemin, ... |